Following is a list of some mathematically

well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...

shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...

Algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
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Cubic plane curve In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an ...

Quartic plane curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one of ...

Rational curves

Degree 2

Conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...

s *

Unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

Unit hyperbola In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative ra ...

Degree 3

Folium of Descartes In geometry, the folium of Descartes (; named for René Decartes) is an algebraic curve defined by the implicit equation :x^3 + y^3 - 3 a x y = 0. History The curve was first proposed and studied by René Descartes in 1638. Its claim to fam ...

Cissoid of Diocles In geometry, the cissoid of Diocles (; named for Diocles) is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be de ...

Conchoid of de Sluze In algebraic geometry, the conchoids of de Sluze are a family of plane curves studied in 1662 by Walloon mathematician René François Walter, baron de Sluze.. The curves are defined by the polar equation :r=\sec\theta+a\cos\theta \,. In cartes ...

Right strophoid In geometry, a strophoid is a curve generated from a given curve and points (the fixed point) and (the pole) as follows: Let be a variable line passing through and intersecting at . Now let and be the two points on whose distance from ...

Semicubical parabola In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form : y^2 - a^2 x^3 = 0 (with ) in some Cartesian coordinate system. Solving for leads to the ''explicit form'' : y = ...

Serpentine curve A serpentine curve is a curve whose equation is of the form :x^2y+a^2y-abx=0, \quad ab > 0. Equivalently, it has a parametric representation :x=a\cot(t), y=b\sin (t)\cos(t), or functional representation :y=\frac. The curve has an inflection po ...

Trident curve In mathematics, a trident curve (also trident of Newton or parabola of Descartes) is any member of the family of curves that have the formula: :xy+ax^3+bx^2+cx=d Trident curves are cubic plane curves with an ordinary double point in the real pro ...

Trisectrix of Maclaurin In algebraic geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a ...

Tschirnhausen cubic In algebraic geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, by the polar equation :r = a\sec^3 \left(\frac\right) where is the secant function. History The curve was studied by von ...

Witch of Agnesi In mathematics, the witch of Agnesi () is a cubic plane curve defined from two diametrically opposite points of a circle. It gets its name from Italian mathematician Maria Gaetana Agnesi, and from a mistranslation of an Italian word for a sail ...

Degree 4

Ampersand curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one o ...

Bean curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one o ...

Bicorn In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation y^2 \left(a^2 - x^2\right) = \left(x^2 + 2ay - a^2\right)^2. It has two cusps and is symmetric about ...

Bow curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one ...

Bullet-nose curve In mathematics, a bullet-nose curve is a unicursal quartic curve with three inflection points, given by the equation :a^2y^2-b^2x^2=x^2y^2 \, The bullet curve has three double points in the real projective plane, at and , and , and and , and ...

Cruciform curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one o ...

* Deltoid curve *

Devil's curve In geometry, a Devil's curve, also known as the Devil on Two Sticks, is a curve defined in the Cartesian plane by an equation of the form : y^2(y^2 - b^2) = x^2(x^2 - a^2). The polar equation of this curve is of the form :r = \sqrt = \sqrt. D ...

Hippopede In geometry, a hippopede () is a plane curve determined by an equation of the form :(x^2+y^2)^2=cx^2+dy^2, where it is assumed that and since the remaining cases either reduce to a single point or can be put into the given form with a rotatio ...

Kampyle of Eudoxus The Kampyle of Eudoxus (Greek: καμπύλη �ραμμή meaning simply "curved ine curve") is a curve with a Cartesian equation of :x^4 = a^2(x^2+y^2), from which the solution ''x'' = ''y'' = 0 is excluded. Alternative parameterizations In ...

Kappa curve In geometry, the kappa curve or Gutschoven's curve is a two-dimensional algebraic curve resembling the Greek letter . The kappa curve was first studied by Gérard van Gutschoven around 1662. In the history of mathematics, it is remembered as one o ...

Lemniscate of Booth In geometry, a hippopede () is a plane curve determined by an equation of the form :(x^2+y^2)^2=cx^2+dy^2, where it is assumed that and since the remaining cases either reduce to a single point or can be put into the given form with a rotation. ...

Lemniscate of Gerono In algebraic geometry, the lemniscate of Gerono, or lemniscate of Huygens, or figure-eight curve, is a plane algebraic curve of degree four and genus zero and is a lemniscate In algebraic geometry, a lemniscate is any of several figure-eight o ...

Lemniscate of Bernoulli In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. ...

Limaçon In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. I ...

Cardioid In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal ...

Limaçon trisectrix In geometry, a limaçon trisectrix is the name for the quartic plane curve that is a trisectrix that is specified as a limaçon. The shape of the limaçon trisectrix can be specified by other curves particularly as a rose, conchoid or ep ...

* Trifolium curve

Degree 5

* Quintic of l'Hospital

Degree 6

Astroid In mathematics, an astroid is a particular type of roulette curve: a hypocycloid with four cusps. Specifically, it is the locus of a point on a circle as it rolls inside a fixed circle with four times the radius. By double generation, it ...

Atriphtaloid An atriphtaloid, also called an atriphtothlassic curve, is type of sextic plane curve. It is given by the equation x^4 \left(x^2 + y^2\right) - \left(ax^2 - b\right)^2 = 0, where ''a'' and ''b'' are positive numbers. References

Sextic c ...

* Nephroid *

Quadrifolium The quadrifolium (also known as four-leaved clover) is a type of rose curve with an angular frequency of 2. It has the polar equation: :r = a\cos(2\theta), \, with corresponding algebraic equation :(x^2+y^2)^3 = a^2(x^2-y^2)^2. \, Rotated c ...

Families of variable degree

* Epicycloid *

Epispiral The epispiral is a plane curve with polar equation :\ r=a \sec. There are ''n'' sections if ''n'' is odd and 2''n'' if ''n'' is even. It is the polar or circle inversive geometry, inversion of the rose (mathematics), rose curve. In astronomy the ...

Epitrochoid In geometry, an epitrochoid ( or ) is a roulette traced by a point attached to a circle of radius rolling around the outside of a fixed circle of radius , where the point is at a distance from the center of the exterior circle. The parametric ...

Hypocycloid In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid cre ...

Lissajous curve A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations : x=A\sin(at+\delta),\quad y=B\sin(bt), which describe the superposition of two perpendicular oscillations in x and y dire ...

Poinsot's spirals In mathematics, Poinsot's spirals are two spirals represented by the polar equations : r = a\ \operatorname (n\theta) : r = a\ \operatorname (n\theta) where csch is the hyperbolic cosecant, and sech is the hyperbolic secant. They are named after ...

Rational normal curve In mathematics, the rational normal curve is a smooth, rational curve of degree in projective n-space . It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For it is the ...

Rose curve A rose is either a woody perennial flowering plant of the genus ''Rosa'' (), in the family Rosaceae (), or the flower it bears. There are over three hundred Rose species, species and Garden roses, tens of thousands of cultivars. They form a ...

Curves of genus one

Bicuspid curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one o ...

Cassini oval In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points ( foci) is constant. This may be contrasted with an ellipse, for which the ''sum'' of t ...

Cassinoide In geometry, a Cassini oval is a quartic plane curve defined as the locus (mathematics), locus of points in the plane (geometry), plane such that the Product_(mathematics), product of the distances to two fixed points (Focus (geometry), foci) is ...

Cubic curve In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an eq ...

Elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...

Watt's curve In mathematics, Watt's curve is a tricircular plane algebraic curve of degree six. It is generated by two circles of radius ''b'' with centers distance 2''a'' apart (taken to be at (±''a'', 0)). A line segment of length 2''c'' attaches to a ...

Curves with genus greater than one

* Butterfly curve * Elkies trinomial curves *

Hyperelliptic curve In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...

Klein quartic In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms ...

* Classical modular curve *

Bolza surface In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus (mathematics), genus 2 with the highest possible order of the conformal map, conformal automorphism group in thi ...

* Macbeath surface

Curve families with variable genus

Polynomial lemniscate In mathematics, a polynomial lemniscate or ''polynomial level curve'' is a plane algebraic curve of degree 2n, constructed from a polynomial ''p'' with complex coefficients of degree ''n''. For any such polynomial ''p'' and positive real number ' ...

* Fermat curve *

Sinusoidal spiral In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates :r^n = a^n \cos(n \theta)\, where is a nonzero constant and is a rational number other than 0. With a rotation about the origin, ...

* Superellipse *

Hurwitz surface In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(''g'' − 1) automorphisms, where ''g'' is the genus of the surface. This number is maximal by virt ...

Transcendental curves

* Bowditch curve * Brachistochrone * Butterfly curve *

Catenary In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, superfici ...

* Clélies *

Cochleoid In geometry, a cochleoid is a snail-shaped curve similar to a strophoid which can be represented by the polar equation :r=\frac, the Cartesian equation :(x^2+y^2)\arctan\frac=ay, or the parametric equations :x=\frac, \quad y=\frac. The cochleo ...

Cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another cu ...

Horopter The horopter was originally defined in geometric terms as the locus of points in space that make the same angle at each eye with the fixation point, although more recently in studies of binocular vision it is taken to be the locus of points in spa ...

* Isochrone ** Isochrone of Huygens (Tautochrone) ** Isochrone of Leibniz ** Isochrone of Varignon *

Lamé curve A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape. In the C ...

Pursuit curve In geometry, a curve of pursuit is a curve constructed by analogy to having a point or points representing pursuers and pursuees; the curve of pursuit is the curve traced by the pursuers. With the paths of the pursuer and pursuee parameterize ...

Rhumb line In navigation, a rhumb line, rhumb (), or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north. Introduction The effect of following a rhumb l ...

Spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a ...

Cornu spiral An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros, clothoids, or Cornu spirals. Eu ...

Cotes' spiral Introduction In physics and in the mathematics of plane curves, a Cotes's spiral (also written Cotes' spiral and Cotes spiral) is one of a family of spirals classified by Roger Cotes. Cotes introduces his analysis of these curves as follows: “ ...

Fermat's spiral A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance ...

Galileo's spiral Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...

Hyperbolic spiral A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation :r=\frac of a hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in ...

Lituus The word ''lituus'' originally meant a curved augural staff, or a curved war-trumpet in the ancient Latin language. This Latin word continued in use through the 18th century as an alternative to the vernacular names of various musical instruments ...

Logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...

Nielsen's spiral In mathematics, trigonometric integrals are a indexed family, family of integrals involving trigonometric functions. Sine integral The different sine integral definitions are \operatorname(x) = \int_0^x\frac\,dt \operatorname(x) = -\int ...

Syntractrix A syntractrix is a curve of the form :x+\sqrt= a \ln \frac. It is the locus of a point on the tangent of a tractrix at a constant distance from the point of tangency, as the point of tangency is moved along the curve.Dionysius Lardner Prof ...

Tractrix In geometry, a tractrix (; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a pulling point (the ''tractor'') that moves at a right ...

Trochoid In geometry, a trochoid () is a roulette curve formed by a circle rolling along a line. It is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line. If the ...

Piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
constructions

Bézier curve A Bézier curve ( ) is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape ...

* Splines **

B-spline In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expresse ...

Nonuniform rational B-spline Non-uniform rational basis spline (NURBS) is a mathematical model using basis splines (B-splines) that is commonly used in computer graphics for representing curves and surfaces. It offers great flexibility and precision for handling both analyt ...

Ogee An ogee ( ) is the name given to objects, elements, and curves—often seen in architecture and building trades—that have been variously described as serpentine-, extended S-, or sigmoid-shaped. Ogees consist of a "double curve", the combinat ...

* Loess curve *

Lowess Local regression or local polynomial regression, also known as moving regression, is a generalization of the moving average and polynomial regression. Its most common methods, initially developed for scatterplot smoothing, are LOESS (locally es ...

* Polygonal curve **

Maurer rose In geometry, the concept of a Maurer rose was introduced by Peter M. Maurer in his article titled ''A Rose is a Rose... A Maurer rose consists of some lines that connect some points on a rose curve. Definition Let ''r'' = sin(''nθ'') ...

Reuleaux triangle A Reuleaux triangle is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circular disks, each having its center on the boundary of the ...

Bézier triangle A Bézier triangle is a special type of Bézier surface that is created by (linear, quadratic, cubic or higher degree) interpolation of control points. ''n''th-order Bézier triangle A general ''n''th-order Bézier triangle has (''n'' +1)(''n ...

Curves generated by other curves

Caustic Caustic most commonly refers to: * Causticity, a property of various corrosive substances ** Sodium hydroxide, sometimes called ''caustic soda'' ** Potassium hydroxide, sometimes called ''caustic potash'' ** Calcium oxide, sometimes called ''caust ...

including Catacaustic and Diacaustic *

Cissoid In geometry, a cissoid (() is a plane curve generated from two given curves , and a point (the pole). Let be a variable line passing through and intersecting at and at . Let be the point on so that \overline = \overline. (There are actua ...

* Conchoid *

Evolute In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that cur ...

Glissette In geometry, a glissette is a curve determined by either the locus of any point, or the envelope of any line or curve, that is attached to a curve that slides against or along two other fixed curves. Examples Ellipse A basic example is that of a ...

Inverse curve In inversive geometry, an inverse curve of a given curve is the result of applying an inverse operation to . Specifically, with respect to a fixed circle with center and radius the inverse of a point is the point for which lies on the ray ...

Involute In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from o ...

Isoptic In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle. Examples: # The orthoptic of a parabola is its directrix (proof: see below), # The orthoptic of an ellipse \tfrac + \ ...

including Orthoptic * Orthotomic *

Negative pedal curve In geometry, a negative pedal curve is a plane curve that can be constructed from another plane curve ''C'' and a fixed point ''P'' on that curve. For each point ''X'' ≠ ''P'' on the curve ''C'', the negative pedal curve has a tange ...

Pedal curve A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to: Computers and other equipment * Footmouse, a foot-operated computer mouse * In medical transcription, a pedal is used to control ...

Parallel curve A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of '' parallel (straight) lines''. It can also be defined as a curve whose points are at a constant ''normal distance'' f ...

* Radial curve *

Roulette Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...

Strophoid In geometry, a strophoid is a curve generated from a given curve and points (the fixed point) and (the pole) as follows: Let be a variable line passing through and intersecting at . Now let and be the two points on whose distance from ...

Space curves

* Conchospiral *

Helix A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helic ...

Tendril perversion Tendril perversion is a geometric phenomenon sometimes observed in helical structures in which the direction of the helix transitions between left-handed and right-handed. Such a reversal of chirality is commonly seen in helical plant tendril ...

(a transition between back-to-back helices) ** Hemihelix, a quasi-helical shape characterized by multiple tendril perversions *

Seiffert's spiral Seiffert's spherical spiral is a curve on a sphere made by moving on the sphere with constant speed and angular velocity with respect to a fixed diameter. If the selected diameter is the line from the north pole to the south pole, then the requir ...

* Slinky spiral *

Twisted cubic In mathematics, a twisted cubic is a smooth, rational curve ''C'' of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (''the'' twisted cubic, therefore). ...

Viviani's curve In mathematics, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and ...

Surfaces in 3-space

* Plane * Quadric surfaces **

Cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...

Cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an ...

Ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as th ...

***

Spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ...

****

Sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...

Hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...

Paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plan ...

Bicylinder In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. The intersection of two cylinders ...

* Tricylinder *

Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and A ...

Torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does n ...

Minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
s

* Catalan's minimal surface * Costa's minimal surface *

Catenoid In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally descri ...

Enneper surface In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by: \begin x &= \tfrac u \left(1 - \tfracu^2 + v^2\right), \\ y &= \tfrac v \left(1 - \tfracv^2 + u^2\righ ...

* Gyroid *

Helicoid The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity ...

* Lidinoid * Riemann's minimal surface *

Saddle tower In differential geometry, a saddle tower is a minimal surface family generalizing the singly periodic Scherk's second surface so that it has ''N''-fold (''N'' > 2) symmetry around one axis. These surfaces are the only properly embedded ...

Scherk surface In mathematics, a Scherk surface (named after Heinrich Scherk) is an example of a minimal surface. Scherk described two complete embedded minimal surfaces in 1834; his first surface is a doubly periodic surface, his second surface is singly peri ...

* Schwarz minimal surface * Triply periodic minimal surface

Non-orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
surfaces

Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a ...

Real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...

** Cross-cap **

Roman surface In mathematics, the Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plan ...

Boy's surface In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. He discovered it on assignment from David Hilbert to prove that the projective plane ''could not'' be immersed in 3-space ...

Quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
s

Oblate spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ci ...

Hyperboloid of one sheet In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...

* Hyperboloid of two sheets *

Hyperbolic paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plan ...

(a ruled surface) *

Sphericon In solid geometry, the sphericon is a solid that has a continuous developable surface with two congruent, semi-circular edges, and four vertices that define a square. It is a member of a special family of rollers that, while being rolle ...

* Oloid

Pseudospherical surfaces

Dini's surface In geometry, Dini's surface is a surface with constant negative curvature that can be created by twisting a pseudosphere. It is named after Ulisse Dini and described by the following parametric equations: : \begin x&=a \cos u \sin v \\ y&=a \ ...

* Pseudosphere

Algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
s

''See the list of algebraic surfaces.'' * Cayley cubic * Barth sextic *

Clebsch cubic In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 exceptional lines can be defined over the real numbers. The term Klein's icosahedral sur ...

* Monkey saddle (saddle-like surface for 3 legs.) *

Dupin cyclide In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered by (and named after) Charl ...

(inversion of a torus) *

Whitney umbrella frame, Section of the surface In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface placed in three d ...

Miscellaneous surfaces

Right conoid In geometry, a right conoid is a ruled surface generated by a family of straight lines that all intersect perpendicularly to a fixed straight line, called the ''axis'' of the right conoid. Using a Cartesian coordinate system in three-dimensiona ...

ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, t ...

)

Fractals

Apollonian gasket In mathematics, an Apollonian gasket or Apollonian net is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek ...

Apollonian sphere packing Apollonian sphere packing is the three-dimensional equivalent of the Apollonian gasket. The principle of construction is very similar: with any four spheres that are Tangent, cotangent to each other, it is then possible to construct two more sphe ...

Blancmange curve In mathematics, the blancmange curve is a self-affine curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who described it in 1901, or as the Takagi–Landsberg curve, a generalization of the cur ...

Cantor dust In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Th ...

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...

* Cantor tesseract *

Circle inversion A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

fractal *

De Rham curve In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham. The Cantor function, Cesàro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve, and Koch curve are all spe ...

* Douady rabbit *

Dragon curve A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The dragon curve is probably most commonly thought of as the shape that is generated from rep ...

Fibonacci word A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet). The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition. It is a para ...

fractal * Flame fractal *

Fractal curve A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectif ...

Gosper curve The Gosper curve, named after Bill Gosper, also known as the Peano-Gosper Curve and the flowsnake (a spoonerism of snowflake), is a space-filling curve whose limit set is rep-7. It is a fractal curve similar in its construction to the dragon c ...

Gosper island The Gosper curve, named after Bill Gosper, also known as the Peano-Gosper Curve and the flowsnake (a spoonerism of snowflake), is a space-filling curve whose limit set is rep-7. It is a fractal curve similar in its construction to the dragon cu ...

* H-fractal * Hénon map *

Hexaflake An ''n''-flake, polyflake, or Sierpinski ''n''-gon, is a fractal constructed starting from an ''n''-gon. This ''n''-gon is replaced by a flake of smaller ''n''-gons, such that the scaled polygons are placed at the vertices, and sometimes in the c ...

* Hilbert curve *

Ikeda map In physics and mathematics, the Ikeda map is a discrete-time dynamical system given by the complex map : z_ = A + B z_n e^ The original map was proposed first by Kensuke Ikeda as a model of light going around across a nonlinear optical res ...

attractor *

Iterated function system In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981. IFS fractals, ...

* Jerusalem cube *

Julia set In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wi ...

Koch curve The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...

Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...

L-system An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into som ...

* Lévy C curve * Feigenbaum attractor *

Lorenz attractor The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the L ...

* Lyapunov fractal *

Mandelbrot set The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This ...

* Mandelbrot tree *

Mandelbulb The Mandelbulb is a three-dimensional fractal, constructed for the first time in 1997 by Jules Ruis and in 2009 further developed by Daniel White and Paul Nylander using spherical coordinates. A canonical 3-dimensional Mandelbrot set does not e ...

Menger sponge In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Si ...

* Monkeys tree *

Moore curve A Moore curve (after E. H. Moore) is a continuous fractal space-filling curve which is a variant of the Hilbert curve. Precisely, it is the loop version of the Hilbert curve, and it may be thought as the union of four copies of the Hilbert curves ...

N-flake An ''n''-flake, polyflake, or Sierpinski ''n''-gon, is a fractal constructed starting from an ''n''-gon. This ''n''-gon is replaced by a flake of smaller ''n''-gons, such that the scaled polygons are placed at the vertices, and sometimes in the c ...

* Pascal triangle *

Peano curve In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not in ...

Penrose tiling A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of the plane by non-overlapping polygons or other shapes, and ''aperiodic'' means that shifting any tiling with these shapes by any finite distance, without ...

Pinwheel tiling In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many or ...

* Pythagoras tree *

Rauzy fractal In mathematics, the Rauzy fractal is a fractal set associated with the Tribonacci substitution : s(1)=12,\ s(2)=13,\ s(3)=1 \,. It was studied in 1981 by Gérard Rauzy, with the idea of generalizing the dynamic properties of the Fibonacci mor ...

* Rössler attractor * Sierpiński arrowhead curve * Sierpinski carpet * Sierpiński curve * Sierpinski triangle *

Smith–Volterra–Cantor set In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Vol ...

T-square A T-square is a technical drawing instrument used by draftsmen primarily as a guide for drawing horizontal lines on a drafting table. The instrument is named after its resemblance to the letter T, with a long shaft called the "blade" and a sh ...

* Takagi or Blancmange curve * Triflake *

Vicsek fractal In mathematics the Vicsek fractal, also known as Vicsek snowflake or box fractal, is a fractal arising from a construction similar to that of the Sierpinski carpet, proposed by Tamás Vicsek. It has applications including as compact antennas, par ...

* von Koch curve *

Weierstrass function In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. The Weierstr ...

Z-order curve In mathematical analysis and computer science, functions which are Z-order, Lebesgue curve, Morton space-filling curve, Morton order or Morton code map multidimensional data to one dimension while preserving locality of the data points. It ...

Random fractals

* von Koch curve with random interval * von Koch curve with random orientation *

polymer A polymer (; Greek '' poly-'', "many" + '' -mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...

shapes *

diffusion-limited aggregation Diffusion-limited aggregation (DLA) is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by T.A. Witten Jr. and L.M. Sander in 1981, is ap ...

* Self-avoiding random walk *

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

Lichtenberg figure A Lichtenberg figure (German ''Lichtenberg-Figuren''), or Lichtenberg dust figure, is a branching electric discharge that sometimes appears on the surface or in the interior of insulating materials. Lichtenberg figures are often associated w ...

Percolation theory In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnecte ...

Multiplicative cascade In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process. Definition The plots above are examples of multiplicative cascade multifractals. To create thes ...

Regular polytopes

This table shows a summary of regular

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

counts by dimension. There are no nonconvex Euclidean regular tessellations in any number of dimensions.

Polytope elements

The elements of a polytope can be considered according to either their own dimensionality or how many dimensions "down" they are from the body. *

Vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

, a 0-dimensional element *

Edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed ...

, a 1-dimensional element *

Face The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...

, a 2-dimensional element *

Cell Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery ...

, a 3-dimensional element * Hypercell or Teron, a 4-dimensional element *

Facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...

, an (''n''-1)-dimensional element *

Ridge A ridge or a mountain ridge is a geographical feature consisting of a chain of mountains or hills that form a continuous elevated crest for an extended distance. The sides of the ridge slope away from the narrow top on either side. The line ...

, an (''n''-2)-dimensional element * Peak, an (''n''-3)-dimensional element For example, in a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...

(3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak. *

Vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...

: not itself an element of a polytope, but a diagram showing how the elements meet.

Tessellations

The classical convex polytopes may be considered

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...

s, or tilings, of spherical space. Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.

Zero dimension

* Point

One-dimensional regular polytope

There is only one polytope in 1 dimension, whose boundaries are the two endpoints of a

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...

, represented by the empty

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...

Two-dimensional regular polytopes

Polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...

Equilateral In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

* Cyclice polygon *

Convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...

Star polygon In geometry, a star polygon is a type of non- convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operatio ...

Pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...

Convex

Regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

Equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

Simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

Square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

Cross-polytope In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahe ...

Hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...

Pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be sim ...

Hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

Heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than '' hepta-'', a Greek-derived nu ...

Octagon In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, w ...

* Enneagon *

Decagon In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. A self-intersecting ''regular decagon'' i ...

Hendecagon In geometry, a hendecagon (also undecagon or endecagon) or 11-gon is an eleven-sided polygon. (The name ''hendecagon'', from Greek ''hendeka'' "eleven" and ''–gon'' "corner", is often preferred to the hybrid ''undecagon'', whose first part is f ...

Dodecagon In geometry, a dodecagon or 12-gon is any twelve-sided polygon. Regular dodecagon A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational s ...

Tridecagon In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon. Regular tridecagon A '' regular tridecagon'' is represented by Schläfli symbol . The measure of each internal angle of a regular tridecagon is approximately ...

* Tetradecagon * Pentadecagon *

Hexadecagon In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon. Regular hexadecagon A '' regular hexadecagon'' is a hexadecagon in which all angles are equal and all sides are congruent. Its Schläfli symb ...

* Heptadecagon * Octadecagon *

Enneadecagon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

Icosagon In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees. Regular icosagon The regular icosagon has Schläfli symbol , and can also be constructed as a truncated decagon, , or a ...

* Hectogon *

Chiliagon In geometry, a chiliagon () or 1000-gon is a polygon with 1,000 sides. Philosophers commonly refer to chiliagons to illustrate ideas about the nature and workings of thought, meaning, and mental representation. Regular chiliagon A '' regular c ...

= Degenerate (spherical)

= * Monogon *

Digon In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...

Non-convex

star polygon In geometry, a star polygon is a type of non- convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operatio ...

* Heptagram *

Octagram In geometry, an octagram is an eight-angled star polygon. The name ''octagram'' combine a Greek numeral prefix, '' octa-'', with the Greek suffix '' -gram''. The ''-gram'' suffix derives from γραμμή (''grammḗ'') meaning "line". Deta ...

Enneagram Enneagram is a compound word derived from the Greek neoclassical stems for "nine" (''ennea'') and something "written" or "drawn" (''gramma''). Enneagram may refer to: * Enneagram (geometry), a nine-sided star polygon with various configurations ...

* Decagram

Tessellation

Apeirogon In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to t ...

Three-dimensional regular polytopes

Convex

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...

Tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...

, the 3-space

Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

, the 3-space

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...

Octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

, the 3-space

Dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

Icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

Degenerate (spherical)

hosohedron In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune ha ...

* dihedron * Henagon#In spherical geometry

Non-convex

* Kepler–Poinsot polyhedra **

Small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeti ...

Great dodecahedron In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagon ...

Great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each ve ...

Great icosahedron In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeti ...

Tessellations

= Euclidean tilings

= *

Square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of th ...

Triangular tiling In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...

Hexagonal tiling In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling). English mathemati ...

* Dihedron

= Hyperbolic tilings

= *

Lobachevski plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

* Hyperbolic tiling

= Hyperbolic star-tilings

= *

Order-7 heptagrammic tiling In geometry, the order-7 heptagrammic tiling is a tiling of the hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulat ...

* Heptagrammic-order heptagonal tiling * Order-9 enneagrammic tiling * Enneagrammic-order enneagonal tiling

Four-dimensional regular polytopes

*convex

regular 4-polytope In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star reg ...

5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It ...

, the 4-space

8-cell In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...

, the 4-space

16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the ...

, the 4-space

24-cell In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, o ...

120-cell In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, he ...

600-cell In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from ...

Degenerate (spherical)

Ditope A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with ...

* Hosotope *

3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...

Non-convex

*Star or (Schläfli–Hess)

** Icosahedral 120-cell **

Small stellated 120-cell In geometry, the small stellated 120-cell or stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. Related polytopes It has the same edge arrangement as the great gran ...

** Great 120-cell ** Grand 120-cell **

Great stellated 120-cell In geometry, the great stellated 120-cell or great stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. It is one of four ''regular star 4-polytopes'' discovered by L ...

Grand stellated 120-cell In geometry, the grand stellated 120-cell or grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. It is also one of two such polytopes that is self-dual. Relat ...

Great grand 120-cell In geometry, the great grand 120-cell or great grand polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. Related polytopes It has the same edge arrangement as the small ste ...

Great icosahedral 120-cell In geometry, the great icosahedral 120-cell, great polyicosahedron or great faceted 600-cell is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. Related polytopes It has the same edge arran ...

Grand 600-cell In geometry, the grand 600-cell or grand polytetrahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. It is the only one with 600 cells. It is one of four ''regular star 4-polytopes'' disco ...

Great grand stellated 120-cell In geometry, the great grand stellated 120-cell or great grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol , one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the 10 for having 600 vertices, and ha ...

Tessellations of Euclidean 3-space

Honeycomb A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of honey ...

Cubic honeycomb The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a r ...

Degenerate tessellations of Euclidean 3-space

Hosohedron In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune ha ...

* Dihedron * Order-2 apeirogonal tiling *

Apeirogonal hosohedron In geometry, an apeirogonal hosohedron or infinite hosohedronConway (2008), p. 263 is a tiling of the plane consisting of two vertices at infinity. It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol ...

* Order-4 square hosohedral honeycomb * Order-6 triangular hosohedral honeycomb * Hexagonal hosohedral honeycomb * Order-2 square tiling honeycomb * Order-2 triangular tiling honeycomb * Order-2 hexagonal tiling honeycomb

Tessellations of hyperbolic 3-space

* Order-4 dodecahedral honeycomb * Order-5 dodecahedral honeycomb * Order-5 cubic honeycomb * Icosahedral honeycomb * Order-3 icosahedral honeycomb * Order-4 octahedral honeycomb *

Triangular tiling honeycomb The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. It is called ''paracompact'' because it has infinite cells and vertex figures, with all vertices as ideal points ...

Square tiling honeycomb In the geometry of hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called ''paracompact'' because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point a ...

Order-4 square tiling honeycomb In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb is one of 11 paracompact regular honeycombs. It is ''paracompact'' because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. Given b ...

Order-6 tetrahedral honeycomb In hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation (or honeycomb). It is ''paracompact'' because it has vertex figures composed of an infinite number of faces, and has all vertices as idea ...

* Order-6 cubic honeycomb * Order-6 dodecahedral honeycomb *

Hexagonal tiling honeycomb In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is ''paracompact'' because it has cells composed of an infinite number of faces. Each cell is a ...

Order-4 hexagonal tiling honeycomb In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is ''paracompact'' because it has cells composed of an infinite number of faces. ...

* Order-5 hexagonal tiling honeycomb *

Order-6 hexagonal tiling honeycomb In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is ''paracompact'' because it has cells with an infinite number of faces. Each cell is a ...

Five-dimensional regular polytopes and higher

5-polytope In geometry, a five-dimensional polytope (or 5-polytope) is a polytope in five-dimensional space, bounded by ( 4-polytope) facets, pairs of which share a polyhedral cell. Definition A 5-polytope is a closed five-dimensional figure with vertic ...

Tetracomb In geometry, a honeycomb is a ''space filling'' or ''close packing'' of polyhedral or higher-dimensional ''cells'', so that there are no gaps. It is an example of the more general mathematical ''tiling'' or '' tessellation'' in any number of di ...

Tessellations of Euclidean 4-space

* honeycombs * Tesseractic honeycomb *

16-cell honeycomb In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol , and constructed by a 4-dimensional packing of 16-cell facets, three arou ...

24-cell honeycomb In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean space by regular 24-cells. It can be represented by Schläfli symbol . ...

Tessellations of Euclidean 5-space and higher

Hypercubic honeycomb In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in -dimensional spaces with the Schläfli symbols and containing the symmetry of Coxeter group (or ) for . The tessellation is constructed from 4 - hypercu ...

* Tesseractic honeycomb *

5-cube honeycomb In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an ''order-4 penteractic hon ...

6-cube honeycomb The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 6-space. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space. Constructions There are m ...

7-cube honeycomb The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-space. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space. There are many different W ...

* 8-cube honeycomb *

Tessellations of hyperbolic 4-space

* honeycombs * Order-5 5-cell honeycomb *

120-cell honeycomb In the geometry of hyperbolic 4-space, the 120-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol , it has three 120-cells around each face. Its dual is the order-5 5-cell honeycomb ...

Order-5 tesseractic honeycomb In the geometry of Hyperbolic space, hyperbolic 4-space, the order-5 tesseractic honeycomb is one of five compact regular polytope, regular space-filling tessellations (or honeycomb (geometry), honeycombs). With Schläfli symbol , it has five 8-cel ...

* Order-4 120-cell honeycomb * Order-5 120-cell honeycomb * Order-4 24-cell honeycomb * Cubic honeycomb honeycomb *

Small stellated 120-cell honeycomb In the geometry of hyperbolic 4-space, the small stellated 120-cell honeycomb is one of four regular star- honeycombs. With Schläfli symbol , it has three small stellated 120-cells around each face. It is dual to the pentagrammic-order 600-ce ...

Pentagrammic-order 600-cell honeycomb In the geometry of hyperbolic 4-space, the pentagrammic-order 600-cell honeycomb is one of four regular star- honeycombs. With Schläfli symbol , it has five 600-cells around each face in a pentagrammic arrangement. It is dual to the small stell ...

* Order-5 icosahedral 120-cell honeycomb * Great 120-cell honeycomb

Tessellations of hyperbolic 5-space

* 5-orthoplex honeycomb * 24-cell honeycomb honeycomb *

16-cell honeycomb honeycomb In the geometry of hyperbolic 5-space, the 16-cell honeycomb honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs). It is called ''paracompact'' because it has infinite vertex figures, with all vertices as idea ...

* Order-4 24-cell honeycomb honeycomb * Tesseractic honeycomb honeycomb

Apeirotopes

* Apeirotope **

** Apeirohedron **

Regular skew polyhedron In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhe ...

Abstract polytopes

Abstract polytope In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be ...

11-cell In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli symbol , with 3 hemi-icosahedra ...

57-cell In mathematics, the 57-cell (pentacontakaiheptachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has 57 vertices, 171 edges and 171 two-dimensional faces. The symmetry or ...

Non-regular polytopes

2D with 1D surface

Concave polygon A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive. Polyg ...

Constructible polygon In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinite ...

Cyclic polygon In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every poly ...

Equiangular polygon In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal (that is, if it is also equilateral) then it is a regular polygon. Isogonal polygons are equiangular polygons w ...

Equilateral polygon In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon ...

* Penrose tile *

Polyform In recreational mathematics, a polyform is a plane figure or solid compound constructed by joining together identical basic polygons. The basic polygon is often (but not necessarily) a convex plane-filling polygon, such as a square or a triangle ...

* Balbis *

Gnomon A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields. History A painted stick dating from 2300 BC that was excavated at the astronomical site of Taosi is the ...

Golygon A golygon, or more generally a serial isogon of 90°, is any polygon with all right angles (a rectilinear polygon) whose sides are consecutive integer lengths. Golygons were invented and named by Lee Sallows, and popularized by A.K. Dewdney in a ...

Star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

without crossing lines *

Hexagram , can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram ( Greek language, Greek) or sexagram ( Latin) is a six-pointe ...

***

Star of David The Star of David (). is a generally recognized symbol of both Jewish identity and Judaism. Its shape is that of a hexagram: the compound of two equilateral triangles. A derivation of the ''seal of Solomon'', which was used for decorative ...

** Heptagram **

*** Star of Lakshmi ** Decagram **

Polygons named for their number of sides * Monogon — 1 sided *

— 2 sided *

Triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

Acute triangle An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's an ...

Isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

Obtuse triangle An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...

Rational triangle An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational length; any such rational triangle can be integrally rescaled (ca ...

Right triangle A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...

***

30-60-90 triangle A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45� ...

***

Isosceles right triangle A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45° ...

***

Kepler triangle A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is \sqrt\varphi where \varphi=(1+\sqrt)/2 is the golden ratio, and the progression can be written: or approximately . Squa ...

Scalene triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collin ...

Quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

Cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...

***

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

kite A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the fac ...

Parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...

***

Rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...

(equilateral parallelogram) ****

Lozenge Lozenge or losange may refer to: *Lozenge (shape), a type of rhombus *Throat lozenge, a tablet intended to be dissolved slowly in the mouth to suppress throat ailments *Lozenge (heraldry), a diamond-shaped object that can be placed on the field of ...

***

Rhomboid Traditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are non-right angled. A parallelogram with sides of equal length (equilateral) is a rhombus but not a rhomboi ...

***

Rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram contain ...

****

(regular quadrilateral) **

Tangential quadrilateral In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called ...

Trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eu ...

or trapezium ***

Isosceles trapezoid In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defin ...

Regular pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

** Lemoine hexagon *

** Regular octagon *

Nonagon In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ''nonagon'' is a prefix hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in French ''nonogo ...

** Regular decagon *

* Triskaidecagon * Tetradecagon * Pentadecagon *

* Heptadecagon * Octadecagon *

Triacontagon In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees. Regular triacontagon The '' regular triacontagon'' is a constructible polygon, by an edge-bisection of a regular ...

Tetracontagon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...

* Pentacontagon * Hexacontagon * Heptacontagon * Octacontagon * Enneacontagon * Hectogon * 257-gon *

* Myriagon *

65537-gon In geometry, a 65537-gon is a polygon with 65,537 (216 + 1) sides. The sum of the interior angles of any non– self-intersecting is 11796300°. Regular 65537-gon The area of a ''regular '' is (with ) :A = \frac t^2 \cot \frac A whole regu ...

* Megagon *

Tilings

List of uniform tilings This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their dual ...

* Uniform tilings in hyperbolic plane * Archimedean tiling **

Truncated square tiling In geometry, the truncated square tiling is a semiregular tiling, semiregular tiling by regular polygons of the Euclidean plane with one square (geometry), square and two octagons on each vertex (geometry), vertex. This is the only edge-to-edge ti ...

Snub square tiling In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. Its Schläfli symbol is ''s''. Conway calls it a snub quadrille, constructed by a snub operation applie ...

Trihexagonal tiling In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2 ...

Truncated hexagonal tiling In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex. As the name implies this tiling is constructed by a truncation operation applies to a he ...

Rhombitrihexagonal tiling In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr. John Conway calls it a rhombihexadeltille.Conway, ...

Truncated trihexagonal tiling In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of ''tr''. Names Uniform colorings ...

Snub hexagonal tiling In geometry, the snub hexagonal tiling (or ''snub trihexagonal tiling'') is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol ''sr''. The snub tetrahexagonal tiling is a ...

** Elongated triangular tiling

Uniform polyhedra

Regular polyhedron A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...

***

Kepler–Poinsot polyhedron In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figures. ...

(regular star polyhedra) ***

***

** Abstract regular polyhedra (

Projective polyhedron In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids. Projec ...

) *** Hemicube *** Hemi-octahedron ***

Hemi-dodecahedron A hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 6 pentagons), which can be visualized by const ...

***

Hemi-icosahedron A hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 10 triangles), which can be visualized by const ...

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are compose ...

** Truncated tetrahedron **

Cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...

Truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edg ...

Truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 ...

Rhombicuboctahedron In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at ea ...

Truncated cuboctahedron In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its fa ...

Snub cube In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron; that is, it has two distinct forms, which are mirr ...

Icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 i ...

** Truncated dodecahedron **

Truncated icosahedron In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares ...

Rhombicosidodecahedron In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has 20 regular triangular faces, 30 square faces, 12 regular ...

Truncated icosidodecahedron In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron,Wenninger Model Number 16 great rhombicosidodecahedron,Williams (Section 3-9, p. 94)Cromwell (p. 82) omnitruncated dodecahedron or omnitruncated icosahedronNorman Wooda ...

Snub dodecahedron In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces. The snub dodecahedron has 92 faces (the mo ...

Prismatic uniform polyhedron In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in two infinite families, the uniform prisms and the uniform antiprisms. All have their vertices in parallel planes and are therefore prismatoid ...

** Prism **

Antiprism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...

Uniform star polyhedron In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...

** Cubitruncated cuboctahedron **

Cubohemioctahedron In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. It has 10 faces (6 squares and 4 regular hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral. It is given Wythoff symbol 4 , ...

** Ditrigonal dodecadodecahedron ** Dodecadodecahedron **

Great cubicuboctahedron In geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14. It has 20 faces (8 triangles, 6 squares and 6 octagrams), 48 edges, and 24 vertices. Its square faces and its octagrammic faces are parallel to those of a ...

Great dirhombicosidodecahedron In geometry, the great dirhombicosidodecahedron (or great snub disicosidisdodecahedron) is a nonconvex uniform polyhedron, indexed last as . It has 124 faces (40 triangles, 60 squares, and 24 pentagrams), 240 edges, and 60 vertices. This is ...

Great disnub dirhombidodecahedron In geometry, the great disnub dirhombidodecahedron, also called ''Skilling's figure'', is a degenerate uniform star polyhedron. It was proven in 1970 that there are only 75 uniform polyhedra other than the infinite families of prisms and antipri ...

Great ditrigonal dodecicosidodecahedron In geometry, the great ditrigonal dodecicosidodecahedron (or great dodekified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U42. It has 44 faces (20 triangles, 12 pentagons, and 12 decagrams), 120 edges, and 60 vertices. I ...

** Great ditrigonal icosidodecahedron ** Great dodecahemicosahedron **

Great dodecahemidodecahedron In geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U70. It has 18 faces (12 pentagrams and 6 decagrams), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilateral. Aside from the regular s ...

Great dodecicosahedron In geometry, the great dodecicosahedron (or great dodekicosahedron) is a nonconvex uniform polyhedron, indexed as U63. It has 32 faces (20 hexagons and 12 decagrams), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. It ...

** Great dodecicosidodecahedron **

Great icosicosidodecahedron In geometry, the great icosicosidodecahedron (or great icosified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U48. It has 52 faces (20 triangles, 12 pentagrams, and 20 hexagons), 120 edges, and 60 vertices. Its vertex fi ...

Great icosidodecahedron In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r. It is the rectification of the great stel ...

** Great icosihemidodecahedron **

Great inverted snub icosidodecahedron In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U69. It is given a Schläfli symbol sr, and Coxeter-Dynkin diagram . In the book '' Polyhedron Models'' by Ma ...

Great retrosnub icosidodecahedron In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as . It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläf ...

** Great rhombidodecahedron **

Great rhombihexahedron In geometry, the great rhombihexahedron (or great rhombicube) is a nonconvex uniform polyhedron, indexed as U21. It has 18 faces (12 Square, squares and 6 Octagram, octagrams), 48 edges, and 24 vertices. Its Duality (mathematics), dual is the gre ...

Great snub dodecicosidodecahedron In geometry, the great snub dodecicosidodecahedron (or great snub dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U64. It has 104 faces (80 triangles and 24 pentagrams), 180 edges, and 60 vertices. It has Coxeter diagram, . ...

Great snub icosidodecahedron In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It can be represented by a Schläfli symbol sr, and Coxeter-Dynkin di ...

** Great stellated truncated dodecahedron ** Great truncated cuboctahedron ** Great truncated icosidodecahedron **

Icosidodecadodecahedron In geometry, the icosidodecadodecahedron (or icosified dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U44. It has 44 faces (12 pentagons, 12 pentagrams and 20 hexagons), 120 edges and 60 vertices. Its vertex figure is a cros ...

** Icositruncated dodecadodecahedron **

Inverted snub dodecadodecahedron In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60. It is given a Schläfli symbol sr. Cartesian coordinates Cartesian coordinates for the vertices of an invert ...

Nonconvex great rhombicosidodecahedron In geometry, the nonconvex great rhombicosidodecahedron is a nonconvex uniform polyhedron, indexed as U67. It has 62 faces (20 triangles, 30 squares and 12 pentagrams), 120 edges, and 60 vertices. It is also called the quasirhombicosidodecahedron ...

** Nonconvex great rhombicuboctahedron **

Octahemioctahedron In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as . It has 12 faces (8 triangles and 4 hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral. It is one ...

** Rhombicosahedron ** Rhombidodecadodecahedron **

Small cubicuboctahedron In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces (8 triangles, 6 squares, and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral. The small cubicuboctahe ...

** Small ditrigonal dodecicosidodecahedron ** Small ditrigonal icosidodecahedron **

Small dodecahemicosahedron In geometry, the small dodecahemicosahedron (or great dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U62. It has 22 faces (12 pentagrams and 10 hexagons), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilate ...

** Small dodecahemidodecahedron ** Small dodecicosahedron ** Small dodecicosidodecahedron ** Small icosicosidodecahedron ** Small icosihemidodecahedron ** Small retrosnub icosicosidodecahedron ** Small rhombidodecahedron **

Small rhombihexahedron In geometry, the small rhombihexahedron (or small rhombicube) is a nonconvex uniform polyhedron, indexed as U18. It has 18 faces (12 squares and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is an antiparallelogram. Related polyhedr ...

Small snub icosicosidodecahedron In geometry, the small snub icosicosidodecahedron or snub disicosidodecahedron is a uniform star polyhedron, indexed as U32. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. Its stellation core is a truncated penta ...

** Small stellated truncated dodecahedron **

Snub dodecadodecahedron In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as . It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol as a snub great dodecahedron. ...

Snub icosidodecadodecahedron In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices. As the name indicates, it belongs to the family of sn ...

Stellated truncated hexahedron In geometry, the stellated truncated hexahedron (or quasitruncated hexahedron, and stellatruncated cube) is a uniform star polyhedron, indexed as U19. It has 14 faces (8 triangles and 6 octagrams), 36 edges, and 24 vertices. It is represented ...

Tetrahemihexahedron In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin dia ...

Truncated dodecadodecahedron In geometry, the truncated dodecadodecahedron (or stellatruncated dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U59. It is given a Schläfli symbol t0,1,2. It has 54 faces (30 squares, 12 decagons, and 12 decagrams), 180 edges ...

** Truncated great dodecahedron **

Truncated great icosahedron In geometry, the truncated great icosahedron (or great truncated icosahedron) is a nonconvex uniform polyhedron, indexed as U55. It has 32 faces (12 pentagrams and 20 hexagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t or t0,1 ...

Duals of uniform polyhedra

Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan so ...

Triakis tetrahedron In geometry, a triakis tetrahedron (or kistetrahedron) is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the truncated tetrahedron. The triakis tetrahedron can be se ...

Rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahed ...

** Triakis octahedron ** Tetrakis hexahedron **

Deltoidal icositetrahedron In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icosit ...

Disdyakis dodecahedron In geometry, a disdyakis dodecahedron, (also hexoctahedron, hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedron), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is fa ...

** Pentagonal icositetrahedron **

Rhombic triacontahedron In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Ca ...

Triakis icosahedron In geometry, the triakis icosahedron (or kisicosahedronConway, Symmetries of things, p.284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron. Cartesian coordinates Let \phi be the golden ratio. The 12 ...

Pentakis dodecahedron In geometry, a pentakis dodecahedron or kisdodecahedron is the polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. It is a Catalan solid, meaning that it ...

Deltoidal hexecontahedron In geometry, a deltoidal hexecontahedron (also sometimes called a ''trapezoidal hexecontahedron'', a ''strombic hexecontahedron'', or a ''tetragonal hexacontahedron'') is a Catalan solid which is the dual polyhedron of the rhombicosidodecahedron, ...

** Disdyakis triacontahedron ** Pentagonal hexecontahedron * non-convex **

Great complex icosidodecahedron In geometry, the great complex icosidodecahedron is a degenerate uniform star polyhedron. It has 12 vertices, and 60 (doubled) edges, and 32 faces, 12 pentagrams and 20 triangles. All edges are doubled (making it degenerate), sharing 4 faces, bu ...

** Great deltoidal hexecontahedron **

Great deltoidal icositetrahedron In geometry, the great deltoidal icositetrahedron (or great sagittal disdodecahedron) is the dual of the nonconvex great rhombicuboctahedron. Its faces are darts. Part of each dart lies inside the solid, hence is invisible in solid models. One o ...

** Great dirhombicosidodecacron **

** Great disdyakis dodecahedron ** Great disdyakis triacontahedron ** Great disnub dirhombidodecacron ** Great ditrigonal dodecacronic hexecontahedron ** Great dodecacronic hexecontahedron ** Great dodecahemicosacron **

Great dodecicosacron In geometry, the great dodecicosacron (or great dipteral trisicosahedron) is the dual of the great dodecicosahedron (U63). It has 60 intersecting bow-tie-shaped faces. Proportions Each face has two angles of \arccos(\frac+\frac\sqrt)\approx 30 ...

Great hexacronic icositetrahedron In geometry, the great hexacronic icositetrahedron is the dual of the great cubicuboctahedron. Its faces are kites. Part of each kite lies inside the solid, hence is invisible in solid models. Proportions The kites have two angles of \arccos( ...

** Great hexagonal hexecontahedron ** Great icosacronic hexecontahedron ** Great icosihemidodecacron ** Great inverted pentagonal hexecontahedron ** Great pentagonal hexecontahedron ** Great pentagrammic hexecontahedron ** Great pentakis dodecahedron **

Great rhombic triacontahedron In geometry, the great rhombic triacontahedron is a nonconvex Isohedral figure, isohedral, isotoxal polyhedron. It is the Dual polyhedron, dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombus, rhomb ...

** Great rhombidodecacron **

Great rhombihexacron In geometry, the great rhombihexacron (or great dipteral disdodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron (U21). It has 24 identical bow-tie-shaped faces, 18 vertices, and 48 edges.

Great stellapentakis dodecahedron In geometry, the great stellapentakis dodecahedron (or great astropentakis dodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces. Proportions The triangle ...

** Great triakis icosahedron ** Great triakis octahedron ** Great triambic icosahedron ** Medial deltoidal hexecontahedron **

Medial disdyakis triacontahedron In geometry, the medial disdyakis triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform truncated dodecadodecahedron. It has 120 triangular faces. Proportions The triangles have one angle of \arccos(-\frac)\approx ...

** Medial hexagonal hexecontahedron **

Medial icosacronic hexecontahedron In geometry, the medial icosacronic hexecontahedron (or midly sagittal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform A uniform is a variety of clothing worn by members of an organization while partici ...

Medial inverted pentagonal hexecontahedron In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60. It is given a Schläfli symbol sr. Cartesian coordinates Cartesian coordinates for the vertices of an invert ...

Medial pentagonal hexecontahedron In geometry, the medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces. Proportions Denote the golden ratio by \phi, and let \xi\a ...

Medial rhombic triacontahedron __NOTOC__ In geometry, the medial rhombic triacontahedron (or midly rhombic triacontahedron) is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual ...

Hexahemioctacron In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. It has 10 faces (6 squares and 4 regular hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral. It is given Wythoff symbol 4 , ...

Hemipolyhedron In geometry, a hemipolyhedron is a uniform star polyhedron some of whose faces pass through its center. These "hemi" faces lie parallel to the faces of some other symmetrical polyhedron, and their count is half the number of faces of that other p ...

Octahemioctacron In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as . It has 12 faces (8 triangles and 4 hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral. It is one of ...

** Rhombicosacron **

Small complex icosidodecahedron In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. ...

** Small ditrigonal dodecacronic hexecontahedron ** Small dodecacronic hexecontahedron ** Small dodecahemicosacron ** Small dodecahemidodecacron ** Small dodecicosacron ** Small hexacronic icositetrahedron ** Small hexagonal hexecontahedron ** Small hexagrammic hexecontahedron ** Small icosacronic hexecontahedron ** Small icosihemidodecacron ** Small rhombidodecacron ** Small rhombihexacron ** Small stellapentakis dodecahedron ** Small triambic icosahedron **

Tetrahemihexacron In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin dia ...

Johnson solids

* Augmented dodecahedron *

Augmented hexagonal prism In geometry, the augmented hexagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by augmenting a hexagonal prism by attaching a square pyramid () to one of its equatorial faces. When two or three such pyram ...

Augmented pentagonal prism In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field o ...

* Augmented sphenocorona *

Augmented triangular prism In geometry, the augmented triangular prism is one of the Johnson solids (). As the name suggests, it can be constructed by augmenting a triangular prism by attaching a square pyramid () to one of its equatorial faces. The resulting solid bears ...

Augmented tridiminished icosahedron In geometry, the augmented tridiminished icosahedron is one of the Johnson solids (). It can be obtained by joining a tetrahedron to another Johnson solid, the tridiminished icosahedron In geometry, the tridiminished icosahedron is one of th ...

Augmented truncated cube In geometry, the augmented truncated cube is one of the Johnson solids (). As its name suggests, it is created by attaching a square cupola () onto one octagonal face of a truncated cube. References *Norman W. Johnson Norman Woodason J ...

* Augmented truncated dodecahedron * Augmented truncated tetrahedron * Biaugmented pentagonal prism *

Biaugmented triangular prism In geometry, the biaugmented triangular prism is one of the Johnson solids (). As the name suggests, it can be constructed by augmenting a triangular prism by attaching square pyramids () to two of its equatorial faces. It is related to the a ...

* Biaugmented truncated cube * Bigyrate diminished rhombicosidodecahedron * Bilunabirotunda * Diminished rhombicosidodecahedron *

Disphenocingulum In geometry, the disphenocingulum or pentakis elongated gyrobifastigium is one of the Johnson solids (). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. ...

Elongated pentagonal bipyramid In geometry, the elongated pentagonal bipyramid or pentakis pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal bipyramid () by inserting a pentagonal prism between its cong ...

* Elongated pentagonal cupola * Elongated pentagonal gyrobicupola * Elongated pentagonal gyrobirotunda * Elongated pentagonal gyrocupolarotunda * Elongated pentagonal orthobicupola * Elongated pentagonal orthobirotunda * Elongated pentagonal orthocupolarotunda *

Elongated pentagonal pyramid In geometry, the elongated pentagonal pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal pyramid () by attaching a pentagonal prism to its base. Formulae The following formulae ...

* Elongated pentagonal rotunda *

Elongated square bipyramid In geometry, the elongated square bipyramid (or elongated octahedron) is one of the Johnson solids (). As the name suggests, it can be constructed by elongating an octahedron by inserting a cube between its congruent halves. It has been named ...

* Elongated square cupola *

Elongated square gyrobicupola In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (). It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same p ...

Elongated square pyramid In geometry, the elongated square pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a square pyramid () by attaching a cube to its square base. Like any elongated pyramid, it is topologically ( ...

* Elongated triangular bipyramid * Elongated triangular cupola * Elongated triangular gyrobicupola *

Elongated triangular orthobicupola In geometry, the elongated triangular orthobicupola or cantellated triangular prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a triangular orthobicupola () by inserting a hexagonal prism between i ...

Elongated triangular pyramid In geometry, the elongated triangular pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated Pyramid (geometry), pyramid, ...

* Gyrate bidiminished rhombicosidodecahedron *

Gyrate rhombicosidodecahedron In geometry, the gyrate rhombicosidodecahedron is one of the Johnson solids (). It is also a canonical polyhedron In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is ...

* Gyrobifastigium *

Gyroelongated pentagonal bicupola In geometry, the gyroelongated pentagonal bicupola is one of the Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same ...

* Gyroelongated pentagonal birotunda * Gyroelongated pentagonal cupola * Gyroelongated pentagonal cupolarotunda *

Gyroelongated pentagonal pyramid In geometry, the gyroelongated pentagonal pyramid is one of the Johnson solids (). As its name suggests, it is formed by taking a pentagonal pyramid and "gyroelongating" it, which in this case involves joining a pentagonal antiprism to its base. ...

* Gyroelongated pentagonal rotunda *

Gyroelongated square bicupola In geometry, the gyroelongated square bicupola is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating a square bicupola ( or ) by inserting an octagon In geometry, an octagon (from the Greek ὀκτάγ ...

* Gyroelongated square bipyramid * Gyroelongated square cupola *

Gyroelongated square pyramid In geometry, the gyroelongated square pyramid is one of the Johnson solids (). As its name suggests, it can be constructed by taking a square pyramid and "gyroelongating" it, which in this case involves joining a square antiprism to its base. ...

Gyroelongated triangular bicupola In geometry, the gyroelongated triangular bicupola is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating a triangular bicupola (either triangular orthobicupola, , or the cuboctahedron) by inserting a h ...

* Gyroelongated triangular cupola * Hebesphenomegacorona * Metabiaugmented dodecahedron * Metabiaugmented hexagonal prism * Metabiaugmented truncated dodecahedron *

Metabidiminished icosahedron In geometry, the metabidiminished icosahedron is one of the Johnson solids (). The name refers to one way of constructing it, by removing two pentagonal pyramids () from a regular icosahedron In geometry, a regular icosahedron ( or ) is a con ...

* Metabidiminished rhombicosidodecahedron * Metabigyrate rhombicosidodecahedron * Metagyrate diminished rhombicosidodecahedron * Parabiaugmented dodecahedron * Parabiaugmented hexagonal prism * Parabiaugmented truncated dodecahedron * Parabidiminished rhombicosidodecahedron * Parabigyrate rhombicosidodecahedron * Paragyrate diminished rhombicosidodecahedron *

Pentagonal bipyramid In geometry, the pentagonal bipyramid (or dipyramid) is third of the infinite set of face-transitive bipyramids, and the 13th Johnson solid (). Each bipyramid is the dual of a uniform prism. Although it is face-transitive, it is not a Plato ...

* Pentagonal cupola * Pentagonal gyrobicupola * Pentagonal gyrocupolarotunda * Pentagonal orthobicupola *

Pentagonal orthobirotunda In geometry, the pentagonal orthobirotunda is one of the Johnson solids (). It can be constructed by joining two pentagonal rotundae () along their decagonal faces, matching like faces. Related polyhedra The pentagonal orthobirotunda is al ...

* Pentagonal orthocupolarotunda *

Pentagonal pyramid In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the apex). Like any pyramid, it is self- dual. The ''regular'' pentagonal pyramid has a base that is a r ...

Pentagonal rotunda In geometry, the pentagonal rotunda is one of the Johnson solids (). It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces. Formulae The following formulae for volume, surface ar ...

Snub disphenoid In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some ve ...

* Snub square antiprism * Sphenocorona * Sphenomegacorona * Square cupola *

Square gyrobicupola In geometry, the square gyrobicupola is one of the Johnson solids (). Like the square orthobicupola (), it can be obtained by joining two square cupolae () along their bases. The difference is that in this solid, the two halves are rotated 45 de ...

Square orthobicupola In geometry, the square orthobicupola is one of the Johnson solids (). As the name suggests, it can be constructed by joining two square cupolae () along their octagonal bases, matching like faces. A 45-degree rotation of one cupola before the jo ...

Square pyramid In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyrami ...

Triangular bipyramid In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. As the name suggests, ...

Triangular cupola In geometry, the triangular cupola is one of the Johnson solids (). It can be seen as half a cuboctahedron. Formulae The following formulae for the volume (V), the surface area (A) and the height (H) can be used if all faces are regular, ...

Triangular hebesphenorotunda In geometry, the triangular hebesphenorotunda is one of the Johnson solids (). . It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it does have ...

* Triangular orthobicupola *

Triaugmented dodecahedron In geometry, the triaugmented dodecahedron is one of the Johnson solids (). It can be seen as a dodecahedron with three pentagonal pyramids () attached to nonadjacent faces. When pyramids are attached to a dodecahedron in other ways, they may r ...

Triaugmented hexagonal prism In geometry, the triaugmented hexagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by triply augmenting a hexagonal prism by attaching square pyramids () to three of its nonadjacent equatorial faces. S ...

* Triaugmented triangular prism * Triaugmented truncated dodecahedron * Tridiminished icosahedron * Tridiminished rhombicosidodecahedron * Trigyrate rhombicosidodecahedron

Other nonuniform polyhedra

Pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...

Bipyramid A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices. The "-gonal" in the name of a bipyramid does ...

Disphenoid In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same ...

Parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclid ...

Cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a c ...

Rhombohedron In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be us ...

Trapezohedron In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...

Frustum In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are ...

Trapezo-rhombic dodecahedron In geometry, the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has symmetry. A concave form can be constructed with an identical net, seen as excavating tri ...

Rhombo-hexagonal dodecahedron In geometry, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular de ...

* Truncated trapezohedron *

Deltahedron In geometry, a deltahedron (plural ''deltahedra'') is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle. There are infinitely many d ...

Zonohedron In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments i ...

Prismatoid In geometry, a prismatoid is a polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles. If both planes have the same number of vertices, and the lateral faces are either parallelograms or ...

Cupola In architecture, a cupola () is a relatively small, most often dome-like, tall structure on top of a building. Often used to provide a lookout or to admit light and air, it usually crowns a larger roof or dome. The word derives, via Italian, f ...

Bicupola In geometry, a bicupola is a solid formed by connecting two cupolae on their bases. There are two classes of bicupola because each cupola (bicupola half) is bordered by alternating triangles and squares. If similar faces are attached together ...

Spherical polyhedra

* Dihedron *

Honeycombs

;

Convex uniform honeycomb In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. Twenty-eight such honeycombs are known: * the familiar cubic honeycomb and 7 tr ...

* Truncated cubic honeycomb *

Bitruncated cubic honeycomb The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently, bitruncated cubes). It has 4 truncated octahedra around each vertex. Being composed entirely of ...

* Cantellated cubic honeycomb * Cantitruncated cubic honeycomb *

Rectified cubic honeycomb The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a r ...

* Runcitruncated cubic honeycomb *

Omnitruncated cubic honeycomb The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 3-space made up of cube, cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex ...

Tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names i ...

* Truncated alternated cubic honeycomb * Cantitruncated alternated cubic honeycomb * Runcinated alternated cubic honeycomb *

Quarter cubic honeycomb The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is ...

Gyrated tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names in ...

Gyrated triangular prismatic honeycomb The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed entirely of triangular prisms. It is constructed from a triangular tiling extruded into pri ...

* Gyroelongated alternated cubic honeycomb * Gyroelongated triangular prismatic honeycomb * Elongated triangular prismatic honeycomb * Elongated alternated cubic honeycomb * Hexagonal prismatic honeycomb *

Triangular prismatic honeycomb The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed entirely of triangular prisms. It is constructed from a triangular tiling extruded into p ...

Triangular-hexagonal prismatic honeycomb The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed entirely of triangular prisms. It is constructed from a triangular tiling extruded into p ...

Truncated hexagonal prismatic honeycomb The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed entirely of triangular prisms. It is constructed from a triangular tiling extruded into pri ...

* Truncated square prismatic honeycomb * Rhombitriangular-hexagonal prismatic honeycomb *

Omnitruncated triangular-hexagonal prismatic honeycomb The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed entirely of triangular prisms. It is constructed from a triangular tiling extruded into pri ...

* Snub triangular-hexagonal prismatic honeycomb * Snub square prismatic honeycomb ;Dual uniform honeycomb *

Disphenoid tetrahedral honeycomb The tetragonal disphenoid tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of identical tetragonal disphenoidal cells. Cells are face-transitive with 4 identical isosceles triangle faces. John Horton ...

*Rhombic dodecahedral honeycomb ;Others *Trapezo-rhombic dodecahedral honeycomb *Weaire–Phelan structure ;Convex uniform honeycombs in hyperbolic space * Order-4 dodecahedral honeycomb * Order-5 cubic honeycomb * Order-5 dodecahedral honeycomb * Icosahedral honeycomb

Other

*Apeirogonal prism * Apeirohedron *Bicupola (geometry), Bicupola *

*Bifrustum *Boerdijk–Coxeter helix *Császár polyhedron *Flexible polyhedron *Gyroelongated square dipyramid *Heronian tetrahedron *Hexagonal bifrustum *Hexagonal truncated trapezohedron *Hill tetrahedron *Holyhedron *Infinite skew polyhedron *Jessen's icosahedron *Near-miss Johnson solid *

*Pentagonal bifrustum *Polytetrahedron *Pyritohedron *Rhombic enneacontahedron *Rhombic icosahedron *

*Scalenohedron *Schönhardt polyhedron *Square bifrustum *Square truncated trapezohedron *Szilassi polyhedron *Tetradecahedron *Tetradyakis hexahedron *Tetrated dodecahedron *Triangular bifrustum * Triaugmented triangular prism *Truncated rhombic dodecahedron * Truncated trapezohedron *Truncated triakis tetrahedron *Tridyakis icosahedron *Trigonal trapezohedron *

*Waterman polyhedron *Wedge (geometry), Wedge

Regular and uniform compound polyhedra

;Polyhedral compound and Uniform polyhedron compound *Compound of cube and octahedron *Compound of dodecahedron and icosahedron *Compound of eight octahedra with rotational freedom *Compound of eight triangular prisms *Compound of five cubes *Compound of five cuboctahedra *Compound of five cubohemioctahedra *Compound of five great cubicuboctahedra *Compound of five great dodecahedra *Compound of five great icosahedra *Compound of five great rhombihexahedra *Compound of five icosahedra *Compound of five octahedra *Compound of five octahemioctahedra *Compound of five small cubicuboctahedra *Compound of five small rhombicuboctahedra *Compound of five small rhombihexahedra *Compound of five small stellated dodecahedra *Compound of five stellated truncated cubes *Compound of five tetrahedra *Compound of five tetrahemihexahedra *Compound of five truncated cubes *Compound of five truncated tetrahedra *Compound of five uniform great rhombicuboctahedra *Compound of four hexagonal prisms *Compound of four octahedra *Compound of four octahedra with rotational freedom *Compound of four tetrahedra *Compound of four triangular prisms *Compound of great icosahedron and great stellated dodecahedron *Compound of six cubes with rotational freedom *Compound of six decagonal prisms *Compound of six decagrammic prisms *Compound of six pentagonal prisms *Compound of six pentagrammic crossed antiprisms *Compound of six pentagrammic prisms *Compound of six tetrahedra *Compound of six tetrahedra with rotational freedom *Compound of small stellated dodecahedron and great dodecahedron *Compound of ten hexagonal prisms *Compound of ten octahedra *Compound of ten tetrahedra *Compound of ten triangular prisms *Compound of ten truncated tetrahedra *Compound of three cubes *Compound of three tetrahedra *Compound of twelve pentagonal antiprisms with rotational freedom *Compound of twelve pentagonal prisms *Compound of twelve pentagrammic prisms *Compound of twelve tetrahedra with rotational freedom *Compound of twenty octahedra *Compound of twenty octahedra with rotational freedom *Compound of twenty tetrahemihexahedra *Compound of twenty triangular prisms *Compound of two great dodecahedra *Compound of two great icosahedra *Compound of two great inverted snub icosidodecahedra *Compound of two great retrosnub icosidodecahedra *Compound of two great snub icosidodecahedra *Compound of two icosahedra *Compound of two inverted snub dodecadodecahedra *Compound of two small stellated dodecahedra *Compound of two snub cubes *Compound of two snub dodecadodecahedra *Compound of two snub dodecahedra *Compound of two snub icosidodecadodecahedra *Compound of two truncated tetrahedra *Prismatic compound of antiprisms *Prismatic compound of antiprisms with rotational freedom *Prismatic compound of prisms *Prismatic compound of prisms with rotational freedom *4-polytope **Hecatonicosachoron **Hexacosichoron **Hexadecachoron **Icositetrachoron **Pentachoron **Tesseract *Spherical cone ;Convex

, Tesseract,

;Abstract regular polytope *

;regular 4-polytope, Schläfli–Hess 4-polytope (Regular star 4-polytope) * Icosahedral 120-cell,

, Great 120-cell, Grand 120-cell,

;Uniform 4-polytope *Rectified 5-cell, Truncated 5-cell, Cantellated 5-cell, Runcinated 5-cell *Rectified tesseract, Truncated tesseract, Cantellated tesseract, Runcinated tesseract *Rectified 16-cell, Truncated 16-cell *Rectified 24-cell, Truncated 24-cell, Cantellated 24-cell, Runcinated 24-cell, Snub 24-cell *Rectified 120-cell, Truncated 120-cell, Cantellated 120-cell, Runcinated 120-cell *Rectified 600-cell, Truncated 600-cell, Cantellated 600-cell ;Prismatic uniform polychoron *Grand antiprism *Duoprism *Tetrahedral prism, Truncated tetrahedral prism *Truncated cubic prism, Truncated octahedral prism, Cuboctahedral prism, Rhombicuboctahedral prism, Truncated cuboctahedral prism, Snub cubic prism *Truncated dodecahedral prism, Truncated icosahedral prism, Icosidodecahedral prism, Rhombicosidodecahedral prism, Truncated icosidodecahedral prism, Snub dodecahedral prism *Uniform antiprismatic prism

Honeycombs

* Tesseractic honeycomb *

*Snub 24-cell honeycomb *Rectified 24-cell honeycomb *Truncated 24-cell honeycomb *

*5-cell honeycomb *Omnitruncated 5-cell honeycomb *Truncated 5-cell honeycomb *Omnitruncated 5-simplex honeycomb

5D with 4D surfaces

*regular 5-polytope **5-dimensional cross-polytope **5-dimensional

**5-dimensional simplex ;Five-dimensional space,

and uniform 5-polytope *5-simplex, Rectified 5-simplex, Truncated 5-simplex, Cantellated 5-simplex, Runcinated 5-simplex, Stericated 5-simplex *5-demicube, Truncated 5-demicube, Cantellated 5-demicube, Runcinated 5-demicube *5-cube, Rectified 5-cube, 5-cube, Truncated 5-cube, Cantellated 5-cube, Runcinated 5-cube, Stericated 5-cube *5-orthoplex, Rectified 5-orthoplex, Truncated 5-orthoplex, Cantellated 5-orthoplex, Runcinated 5-orthoplex ;Prismatic uniform 5-polytope: For each polytope of dimension ''n'', there is a prism of dimension ''n''+1.

Honeycombs

*5-cubic honeycomb *5-simplex honeycomb *Truncated 5-simplex honeycomb *5-demicubic honeycomb

Six dimensions

;Six-dimensional space, 6-polytope and uniform 6-polytope *6-simplex, Rectified 6-simplex, Truncated 6-simplex, Cantellated 6-simplex, Runcinated 6-simplex, Stericated 6-simplex, Pentellated 6-simplex *6-demicube, Truncated 6-demicube, Cantellated 6-demicube, Runcinated 6-demicube, Stericated 6-demicube *6-cube, Rectified 6-cube, 6-cube, Truncated 6-cube, Cantellated 6-cube, Runcinated 6-cube, Stericated 6-cube, Pentellated 6-cube *6-orthoplex, Rectified 6-orthoplex, Truncated 6-orthoplex, Cantellated 6-orthoplex, Runcinated 6-orthoplex, Stericated 6-orthoplex *1 22 polytope, 1₂₂ polytope, 2 21 polytope, 2₂₁ polytope

Honeycombs

*6-cubic honeycomb *6-simplex honeycomb *6-demicubic honeycomb *2 22 honeycomb, 2₂₂ honeycomb

Seven dimensions

;Seven-dimensional space, uniform 7-polytope *7-simplex, Rectified 7-simplex, Truncated 7-simplex, Cantellated 7-simplex, Runcinated 7-simplex, Stericated 7-simplex, Pentellated 7-simplex, Hexicated 7-simplex *7-demicube, Truncated 7-demicube, Cantellated 7-demicube, Runcinated 7-demicube, Stericated 7-demicube, Pentellated 7-demicube *7-cube, Rectified 7-cube, 7-cube, Truncated 7-cube, Cantellated 7-cube, Runcinated 7-cube, Stericated 7-cube, Pentellated 7-cube, Hexicated 7-cube *7-orthoplex, Rectified 7-orthoplex, Truncated 7-orthoplex, Cantellated 7-orthoplex, Runcinated 7-orthoplex, Stericated 7-orthoplex, Pentellated 7-orthoplex *1 32 polytope, 1₃₂ polytope, 2 31 polytope, 2₃₁ polytope, 3 21 polytope, 3₂₁ polytope

Honeycombs

*7-cubic honeycomb *7-demicubic honeycomb *3 31 honeycomb, 3₃₁ honeycomb, 1 33 honeycomb, 1₃₃ honeycomb

Eight dimension

;Eight-dimensional space, uniform 8-polytope *8-simplex, Rectified 8-simplex, Truncated 8-simplex, Cantellated 8-simplex, Runcinated 8-simplex, Stericated 8-simplex, Pentellated 8-simplex, Hexicated 8-simplex, Heptellated 8-simplex *8-orthoplex, Rectified 8-orthoplex, Truncated 8-orthoplex, Cantellated 8-orthoplex, Runcinated 8-orthoplex, Stericated 8-orthoplex, Pentellated 8-orthoplex, Hexicated 8-orthoplex *8-cube, Rectified 8-cube, Truncated 8-cube, Cantellated 8-cube, Runcinated 8-cube, Stericated 8-cube, Pentellated 8-cube, Hexicated 8-cube, Heptellated 8-cube *8-demicube, Truncated 8-demicube, Cantellated 8-demicube, Runcinated 8-demicube, Stericated 8-demicube, Pentellated 8-demicube, Hexicated 8-demicube *1 42 polytope, 1₄₂ polytope, 2 41 polytope, 2₄₁ polytope, 4 21 polytope, 4₂₁ polytope, Truncated 4 21 polytope, Truncated 4₂₁ polytope, Truncated 2 41 polytope, Truncated 2₄₁ polytope, Truncated 1 42 polytope, Truncated 1₄₂ polytope, Cantellated 4 21 polytope, Cantellated 4₂₁ polytope, Cantellated 2 41 polytope, Cantellated 2₄₁ polytope, Runcinated 4 21 polytope, Runcinated 4₂₁ polytope

Honeycombs

*8-cubic honeycomb *8-demicubic honeycomb *5 21 honeycomb, 5₂₁ honeycomb, 2 51 honeycomb, 2₅₁ honeycomb, 1 52 honeycomb, 1₅₂ honeycomb

Nine dimensions

;9-polytope *9-cube *9-demicube *9-orthoplex *9-simplex

Hyperbolic honeycombs

*E9 honeycomb, E₉ honeycomb

Ten dimensions

;10-polytope *10-cube *10-demicube *10-orthoplex *10-simplex

Dimensional families

;Regular polytope and List of regular polytopes *

;Uniform polytope *Demihypercube *Uniform 1 k2 polytope, Uniform 1_''k''2 polytope *Uniform 2 k1 polytope, Uniform 2_''k''1 polytope *Uniform k 21 polytope, Uniform ''k''₂₁ polytope ;Honeycombs *

*Alternated hypercubic honeycomb

Geometry

*Automedian triangle *Delaunay triangulation *

*Golden triangle (mathematics), Golden triangle *Hyperbolic triangle (non-Euclidean geometry) *

* Sierpinski triangle (fractal geometry) *Special right triangles *Spiral of Theodorus *Thomson cubic *

*Triangular prism *Triangular pyramid *

Geometry and other areas of mathematics

* Annulus (mathematics), Annulus * Apollonian circles *

* Arbelos * Borromean rings * Circle * Circular sector * Circular segment *

** Epicycloid ***

*** Nephroid *** Deferent and epicycle * Ex-tangential quadrilateral * Horocycle * Hypotrochoid **

***

*** Deltoid curve * Lune (mathematics), Lune * Pappus chain * Peaucellier–Lipkin linkage * Robbins pentagon * Salinon * Semicircle * Squircle * Steiner chain *

** Bicentric quadrilateral

Glyphs and symbols

* Borromean rings * Crescent * Vesica piscis * Arc (geometry), Arc *

* Conchoid * Cubic Hermite curve * Curve of constant width * Hedgehog (curve), hedgehog * Parametric curve **

** Spline (mathematics), Spline *** Hermite spline **** Beta spline *****

*** Higher-order spline *** NURBS * Line (mathematics)#Ray, Ray *

* Ribaucour curve

References

{{DEFAULTSORT:Mathematical shapes Lists of shapes, Mathematical

Algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...s

Rational curves

Degree 2

Degree 3

Degree 4

Degree 5

Degree 6

Families of variable degree

Curves of genus one

Curves with genus greater than one

Curve families with variable genus

Transcendental curves

Piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ... constructions

Curves generated by other curves

Space curves

Surfaces in 3-space

Minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...s

Non-orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ... surfaces

Quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...s

Pseudospherical surfaces

Algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...s

Miscellaneous surfaces

Fractals

Random fractals

Regular polytopes

Polytope elements

Tessellations

Zero dimension

One-dimensional regular polytope

Two-dimensional regular polytopes

Convex

= Degenerate (spherical)

Non-convex

Tessellation

Three-dimensional regular polytopes

Convex

Degenerate (spherical)

Non-convex

Tessellations

= Euclidean tilings

= Hyperbolic tilings

= Hyperbolic star-tilings

Four-dimensional regular polytopes

Degenerate (spherical)

Non-convex

Tessellations of Euclidean 3-space

Degenerate tessellations of Euclidean 3-space

Tessellations of hyperbolic 3-space

Five-dimensional regular polytopes and higher

Tessellations of Euclidean 4-space

Tessellations of Euclidean 5-space and higher

Tessellations of hyperbolic 4-space

Tessellations of hyperbolic 5-space

Apeirotopes

Abstract polytopes

Non-regular polytopes

2D with 1D surface

Tilings

Uniform polyhedra

Duals of uniform polyhedra

Johnson solids

Other nonuniform polyhedra

Spherical polyhedra

Honeycombs

Other

Regular and uniform compound polyhedra

Honeycombs

5D with 4D surfaces

Honeycombs

Six dimensions

Honeycombs

Seven dimensions

Honeycombs

Eight dimension

Honeycombs

Nine dimensions

Hyperbolic honeycombs

Ten dimensions

Dimensional families

Geometry

Algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
s

Piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
constructions

Minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
s

Non-orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
surfaces

Quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
s

Algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
s