A polytope is a geometric object with flat sides, which exists in any general number of dimensions. The following list of polygons, polyhedra and polytopes gives the names of various classes of polytopes and lists some specific examples.

Polytope elements

Polygon (2-polytope)

* Vertex *

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 ...

the ''facet'' or ''(n−1)-face'' of the polygon

Polyhedron (3-polytope)

* Vertex the ''peak'' or ''(n−3)-face'' of the polyhedron *

the ''ridge'' or ''(n−2)-face'' of the polyhedron *

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 ...

the ''facet'' or ''(n−1)-face'' of the polyhedron

4-polytope

* Vertex *

the ''peak'' or ''(n−3)-face'' of the 4-polytope *

the ''ridge'' or ''(n−2)-face'' of the 4-polytope *

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 ...

the ''facet'' or ''(n−1)-face'' of the 4-polytope

5-polytope

* Vertex the ''(n−5)-face'' of the 5-polytope *

the ''(n−4)-face'' of the 5-polytope *

the ''peak'' or ''(n−3)-face'' of the 5-polytope *

the ''ridge'' or ''(n−2)-face'' of the 5-polytope * Hypercell or Teron the ''facet'' or ''(n−1)-face'' of the 5-polytope

Other

* Point * Line segment *

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 ...

Peak Peak or The Peak may refer to: Basic meanings Geology * Mountain peak ** Pyramidal peak, a mountaintop that has been sculpted by erosion to form a point Mathematics * Peak hour or rush hour, in traffic congestion * Peak (geometry), an (''n''-3)-di ...

– ''(n−3)-face'' * Ridge – ''(n−2)-face'' *

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 ...

– ''(n−1)-face''

Two dimensional (
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 to ...
s)

;

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 ...

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 ...

* Isosceles triangle *

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-collinear ...

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 ...

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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, ...

* Rectangle **

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 ...

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 ...

* Parallelogram *

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 ...

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 ...

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 ...

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 ...

; Pentagon ;

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 ...

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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 ;

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 ...

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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 ...

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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 ...

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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 sym ...

; Triskaidecagon ; Tetradecagon ;

Pentadecagon In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon. Regular pentadecagon A ''regular polygon, regular pentadecagon'' is represented by Schläfli symbol . A Regular polygon, regular pentadecagon has interior angl ...

; Hexadecagon ;

Heptadecagon In geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon. Regular heptadecagon A '' regular heptadecagon'' is represented by the Schläfli symbol . Construction As 17 is a Fermat prime, the regular heptadecagon is a ...

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Octadecagon In geometry, an octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon. Regular octadecagon A '' regular octadecagon'' has a Schläfli symbol and can be constructed as a quasiregular truncated enneagon, t, which alternates tw ...

;

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 ;

Icosihenagon In geometry, a polygon is traditionally a plane (mathematics), plane Shape, figure that is bounded by a finite chain of straight line segments closing in a loop to form a Polygonal chain, closed chain. These segments are called its ''edges'' or ...

; Icosidigon ; Icositrigon ;

Icositetragon In geometry, an icositetragon (or icosikaitetragon) or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees. Regular icositetragon The '' regular icositetragon'' is represented by Schläfli symbol ...

; Icosipentagon ; Icosihexagon ; Icosiheptagon ; Icosioctagon ; Icosienneagon ;

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 ; 257-gon ; 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 ...

;

Myriagon In geometry, a myriagon or 10000-gon is a polygon with 10,000 sides. Several philosophers have used the regular myriagon to illustrate issues regarding thought. Meditation VI by Descartes (English translation). Regular myriagon A regular myriag ...

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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 regula ...

;

Megagon A megagon or 1,000,000-gon is a polygon with one million sides (mega-, from the Greek μέγας, meaning "great", being a unit prefix denoting a factor of one million). Regular megagon A Regular polygon, regular megagon is represented by the ...

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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 th ...

Star polygons

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 ...

* Hexagram * 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 * Decagram * Hendecagram * Dodecagram

Families

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 ...

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 ...

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 ...

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 ...

* Gnomon * Golygon

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 In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive ( transitive on its v ...

; Archimedean tiling *

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 equilate ...

* Hexagonal 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 * Rhombitrihexagonal tiling * Truncated trihexagonal tiling *

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

Three dimensional (
polyhedra 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 on ...
)

;

Three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...

Regular

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 ...

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 th ...

, Cube,

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 ...

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 *Regular

spherical polyhedron In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most ...

** Dihedron, Hosohedron *

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) **

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 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 ...

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 ...

*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 (geometry) In abstract geometry, a hemicube is an abstract, regular polyhedron, containing half the faces of a cube. Realization It can be realized as a projective polyhedron (a tessellation of the real projective plane by three quadrilaterals), which ca ...

, 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 ...

;

* Disphenoid ; Pentahedron * Square pyramid,

Triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A ...

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Hexahedron A hexahedron (plural: hexahedra or hexahedrons) or sexahedron (plural: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex. Ther ...

* Parallelepiped, Cuboid,

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 ...

Trigonal trapezohedron In geometry, a trigonal trapezohedron is a rhombohedron (a polyhedron with six rhombus-shaped faces) in which, additionally, all six faces are congruent. Alternative names for the same shape are the ''trigonal deltohedron'' or ''isohedral rhomboh ...

, Cube,

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 regu ...

Triangular dipyramid 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, ...

, quadrilateral

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 ...

; Heptahedron * hexagonal pyramid,

pentagonal prism In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. As a semiregular (or uniform) polyhedron If faces are all regular, the pentagonal prism is ...

, tetrahemihexahedron ;

Hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices.. Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used ...

Truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncation (geometry), truncating all 4 vertices of ...

, Tetragonal trapezohedron ;

Enneahedron In geometry, an enneahedron (or nonahedron) is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, and face connections. None of them are regular. Examples The most familiar ...

* Octagonal pyramid, Heptagonal prism ;

Decahedron In geometry, a decahedron is a polyhedron with ten faces. There are 32300 topologically distinct decahedra, and none are regular, so this name does not identify a specific type of polyhedron except for the number of faces. Some decahedra have ...

Octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by rectangular sides and two regular octagon caps. If faces are all regular, it is a semiregular polyhedron. Symmetry Images The octagonal prism can also ...

Square antiprism In geometry, the square antiprism is the second in an infinite family of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an ''anticube''. If all its faces are regular, it is a sem ...

Square cupola In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids (). It can be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in t ...

Pentagonal dipyramid 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 Plat ...

, Augmented pentagonal prism ;

Pentagonal antiprism In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for ...

Decagonal prism In geometry, the decagonal prism is the eighth in the infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra. The decagonal prism has 12 faces, 30 edges, an ...

Pentagonal cupola In geometry, the pentagonal cupola is one of the Johnson solids (). It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon. Formulae The ...

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 vert ...

, Elongated square dipyramid,

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, replacing two sets of five triangular faces of the ...

Hexagonal bipyramid A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles. Although it is face-transitiv ...

Hexagonal trapezohedron In geometry, a hexagonal trapezohedron or deltohedron is the fourth in an infinite series of trapezohedra which are dual polyhedra to the antiprisms. It has twelve faces which are congruence (geometry), congruent kite (geometry), kites. It can be ...

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 ...

, Hendecagonal pyramid,

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 ...

Archimedean solids

; Archimedean solid *

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,

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 In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges. Geometric relations This polyhedron can be formed from a regular dodecahedron by tr ...

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 pen ...

, Truncated icosidodecahedron, Snub dodecahedron

Prisms and antiprisms

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Prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...

Pentagonal prism In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. As a semiregular (or uniform) polyhedron If faces are all regular, the pentagonal prism is ...

, Heptagonal prism,

, Enneagonal prism,

Hendecagonal prism In geometry, the hendecagonal prism is one in an infinite set of convex prisms formed by square sides and two regular polygon caps, in this case two hendecagon In geometry, a hendecagon (also undecagon or endecagon) or 11-gon is an eleven-sided ...

Dodecagonal prism In geometry, the dodecagonal prism is the tenth in an infinite set of prisms, formed by square sides and two regular dodecagon caps. If faces are all regular, it is a uniform polyhedron In geometry, a uniform polyhedron has regular polygons ...

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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 ...

Hexagonal antiprism In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. Antiprisms are similar to prisms except the bases are twisted relative to each oth ...

Heptagonal antiprism In geometry, the heptagonal antiprism is the fifth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. Antiprisms are similar to prisms except the bases are twisted relative to each ot ...

, Octagonal antiprism, Enneagonal antiprism, Decagonal antiprism, Dodecagonal antiprism

Catalan solids

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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 s ...

Triakis octahedron In geometry, a triakis octahedron (or trigonal trisoctahedron or kisoctahedronConway, Symmetries of things, p. 284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube. It can be seen as an octahedron with triangula ...

, 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 p ...

, Pentakis dodecahedron, Deltoidal hexecontahedron, Disdyakis triacontahedron, Pentagonal hexecontahedron

Bipyramids and Trapezohedron

* Bipyramid **

, Octagonal bipyramid, Decagonal bipyramid *

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 ...

Uniform star polyhedra

; Uniform star polyhedron * 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 In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by , and . The ed ...

* Great cubicuboctahedron *

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 * Great ditrigonal dodecicosidodecahedron * Great ditrigonal icosidodecahedron *

Great dodecahemicosahedron In geometry, the great dodecahemicosahedron (or small dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U65. It has 22 faces (12 pentagons and 10 hexagons), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilater ...

* Great dodecahemidodecahedron *

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 * Great icosidodecahedron * Great icosihemidodecahedron * Great inverted snub icosidodecahedron * Great retrosnub icosidodecahedron * Great rhombidodecahedron * Great rhombihexahedron *

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 * 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 *

Nonconvex great rhombicuboctahedron In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48 edges, and 24 vertices. It is represented by the Schläfli symbol rr and Coxeter-Dynkin di ...

* Octahemioctahedron * 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 cubicuboctahedro ...

* 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 In geometry, the small dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U33. It has 44 faces (20 triangles, 12 pentagons, and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a ...

* Small icosicosidodecahedron * Small icosihemidodecahedron * Small retrosnub icosicosidodecahedron *

Small rhombidodecahedron In geometry, the small rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U39. It has 42 faces (30 squares and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. Related polyhedra It shares ...

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 In geometry, the small stellated truncated dodecahedron (or quasitruncated small stellated dodecahedron or small stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U58. It has 24 faces (12 pentagons and 12 decagrams), 90 ...

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 * Tetrahemihexahedron *

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

Uniform prismatic star polyhedra

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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 ...

Pentagrammic prism In geometry, the pentagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams. It is a special case of a right prism with a pentagram as base, which in gene ...

Pentagrammic antiprism In geometry, the pentagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams. It has 12 faces, 20 edges and 10 vertices. This polyhedron is iden ...

, Pentagrammic crossed-antiprism *Heptagrammic antiprism (7/2), Heptagrammic antiprism (7/3) * Enneagrammic antiprism (9/2). Enneagrammic antiprism (9/4) * Enneagrammic crossed-antiprism, Enneagrammic prism (9/2), Enneagrammic prism (9/4) * Decagrammic prism, Decagrammic antiprism

Johnson solids

; Johnson solid # Augmented dodecahedron # Augmented hexagonal prism # Augmented pentagonal prism # Augmented sphenocorona # Augmented triangular prism #

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 the ...

# Augmented truncated cube # Augmented truncated dodecahedron #

Augmented truncated tetrahedron In geometry, the augmented truncated tetrahedron is one of the Johnson solids (). It is created by attaching a triangular cupola () to one hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a s ...

# 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 In geometry, the bilunabirotunda is one of the Johnson solids (). Geometry 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 a str ...

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

# Elongated triangular gyrobicupola # Elongated triangular orthobicupola # Elongated triangular pyramid # Gyrate bidiminished rhombicosidodecahedron # Gyrate rhombicosidodecahedron #

Gyrobifastigium In geometry, the gyrobifastigium is the 26th Johnson solid (). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile ...

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 In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

# 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 In geometry, the gyroelongated pentagonal rotunda is one of the Johnson solids (''J''25). As the name suggests, it can be constructed by gyroelongating a pentagonal rotunda (''J''6) by attaching a decagonal antiprism to its base. It can also be ...

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 In geometry, the gyroelongated square bipyramid, heccaidecadeltahedron, or tetrakis square antiprism is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating an octahedron (square bipyramid) by inserting a s ...

# Gyroelongated square cupola # Gyroelongated square pyramid #

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 In geometry, the metabiaugmented hexagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by doubly augmenting a hexagonal prism by attaching square pyramids () to two of its nonadjacent, nonparallel equatorial ...

# Metabiaugmented truncated dodecahedron #

# Metabidiminished rhombicosidodecahedron # Metabigyrate rhombicosidodecahedron # Metagyrate diminished rhombicosidodecahedron # Parabiaugmented dodecahedron #

Parabiaugmented hexagonal prism In geometry, the parabiaugmented hexagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by doubly augmenting a hexagonal prism by attaching square pyramids () to two of its nonadjacent, parallel (opposite) eq ...

# 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 Plat ...

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

# 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 als ...

Pentagonal orthocupolarotunda In geometry, the pentagonal orthocupolarotunda is one of the Johnson solids (). As the name suggests, it can be constructed by joining a pentagonal cupola () and a pentagonal rotunda () along their decagonal bases, matching the pentagonal fac ...

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 square antiprism #

Sphenocorona In geometry, the sphenocorona 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. Johnson uses the prefix ''spheno-'' to ref ...

# Sphenomegacorona #

# Square gyrobicupola # Square orthobicupola # Square pyramid #

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, i ...

# Triangular cupola # Triangular hebesphenorotunda #

Triangular orthobicupola In geometry, the triangular orthobicupola is one of the Johnson solids (). As the name suggests, it can be constructed by attaching two triangular cupolas () along their bases. It has an equal number of squares and triangles at each vertex; howev ...

# Triaugmented dodecahedron # Triaugmented hexagonal prism #

Triaugmented triangular prism The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The sa ...

# Triaugmented truncated dodecahedron #

Tridiminished icosahedron In geometry, the tridiminished icosahedron is one of the Johnson solids (). The name refers to one way of constructing it, by removing three pentagonal pyramids () from a regular icosahedron, which replaces three sets of five triangular faces fro ...

# Tridiminished rhombicosidodecahedron #

Trigyrate rhombicosidodecahedron In geometry, the trigyrate rhombicosidodecahedron is one of the Johnson solids (). It contains 20 triangles, 30 squares and 12 pentagons. It is also a Midsphere#Canonical polyhedron, canonical polyhedron. It can be constructed as a rhombicosi ...

Dual uniform star polyhedra

* Great complex icosidodecahedron * Great deltoidal hexecontahedron * Great deltoidal icositetrahedron *

Great dirhombicosidodecacron In geometry, the great dirhombicosidodecacron is a nonconvex isohedral polyhedron. It is the dual of the great dirhombicosidodecahedron. In Magnus Wenninger's ''Dual Models'', it is represented with intersecting infinite prisms passing through t ...

* 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 * 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, isotoxal polyhedron. It is the dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombic faces, 60 edges and 32 vertices (also ...

* Great rhombidodecacron * Great rhombihexacron *

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 In geometry, the great triambic icosahedron and medial triambic icosahedron (or midly triambic icosahedron) are visually identical dual uniform polyhedra. The exterior surface also represents the De2f2 stellation of the icosahedron. These fig ...

* 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 dua ...

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 *

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 * 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 In geometry, the small rhombihexacron (or small dipteral disdodecahedron) is the dual of the small rhombihexahedron. It is visually identical to the small hexacronic icositetrahedron. Its faces are antiparallelograms formed by pairs of coplanar ...

* 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 ...

Honeycombs

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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 ...

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 ...

* 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 t ...

* 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 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 re ...

* Omnitruncated cubic honeycomb *

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 c ...

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 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 ...

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 *

Truncated square prismatic 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 re ...

Rhombitriangular-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 ...

* Omnitruncated triangular-hexagonal prismatic honeycomb * 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 Hor ...

Rhombic dodecahedral honeycomb The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal s ...

;Others * Trapezo-rhombic dodecahedral honeycomb *

Weaire–Phelan structure In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution ...

; Convex uniform honeycombs in hyperbolic space *

Order-4 dodecahedral honeycomb In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular polytope, regular space-filling tessellations (or honeycomb (geometry), honeycombs) of hyperbolic 3-space. With Schläfli symbol it has four regular dodeca ...

* Order-5 cubic honeycomb * Order-5 dodecahedral honeycomb * Icosahedral honeycomb

Other

Apeirogonal prism In geometry, an apeirogonal prism or infinite prism is the arithmetic limit of the family of prisms; it can be considered an infinite polyhedron or a tiling of the plane.Conway (2008), p.263 Thorold Gosset called it a ''2-dimensional semi-check ...

* Apeirohedron * Bicupola *

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, fro ...

* Bifrustum *

Boerdijk–Coxeter helix The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices. There are ...

* Császár polyhedron *

Flexible polyhedron In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such ...

Heronian tetrahedron A Heronian tetrahedron (also called a Heron tetrahedron or perfect pyramid) is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles. Every Heronian tetrahedron can be arranged ...

* Hexagonal bifrustum * Hexagonal truncated trapezohedron *

Hill tetrahedron In geometry, the Hill tetrahedra are a family of space-filling tetrahedra. They were discovered in 1896 by M. J. M. Hill, a professor of mathematics at the University College London, who showed that they are scissor-congruent to a cube. Con ...

* Holyhedron * Infinite skew polyhedron * Jessen's icosahedron * Near-miss Johnson solid * Parallelepiped * Pentagonal bifrustum * Polytetrahedron *

Pyritohedron 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 pentago ...

* Rhombic enneacontahedron *

Rhombic icosahedron The rhombic icosahedron is a polyhedron shaped like an oblate sphere. Its 20 faces are congruent golden rhombi; 3, 4, or 5 faces meet at each vertex. It has 5 faces (green on top figure) meeting at each of its 2 poles; these 2 vertices lie on it ...

* Scalenohedron * Schönhardt polyhedron * Square bifrustum * Square truncated trapezohedron * Szilassi polyhedron *

Tetradecahedron 240px, A tetradecahedron with ''D2d'' symmetry, existing in the Weaire–Phelan structure A tetradecahedron is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely wi ...

* Tetradyakis hexahedron * Tetrated dodecahedron * Triangular bifrustum *

* Truncated rhombic dodecahedron * Truncated trapezohedron * Truncated triakis tetrahedron * Tridyakis icosahedron *

* Regular skew polyhedron * Waterman polyhedron *

Wedge A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...

Regular and uniform compound polyhedra

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Polyhedral compound In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connect ...

and

Uniform polyhedron compound In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts tran ...

Compound of cube and octahedron The compound of cube and octahedron is a polyhedron which can be seen as either a polyhedral stellation or a compound. Construction The 14 Cartesian coordinates of the vertices of the compound are. : 6: (±2, 0, 0), ( 0, ±2, 0), ( 0, 0, ±2) : ...

Compound of dodecahedron and icosahedron In geometry, this polyhedron can be seen as either a polyhedral stellation or a compound. As a compound It can be seen as the compound of an icosahedron and dodecahedron. It is one of four compounds constructed from a Platonic solid or Ke ...

* Compound of eight octahedra with rotational freedom * Compound of eight triangular prisms *

Compound of five cubes The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876. It is one of five regular compounds, and dual to the compound of five octahedra. It can be seen as a faceting of a regu ...

* 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 The compound of five octahedra is one of the five regular polyhedron compounds. This polyhedron can be seen as either a polyhedral stellation or a compound. This compound was first described by Edmund Hess in 1876. It is unique among the regula ...

* 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 The compound of five tetrahedra is one of the five regular polyhedral compounds. This compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hess in 1876. It can be seen as a faceting of a regular d ...

* 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 The compound of four octahedra is a uniform polyhedron compound. It's composed of a symmetric arrangement of 4 octahedron, octahedra, considered as triangular antiprisms. It can be constructed by superimposing four identical octahedra, and then r ...

* Compound of four octahedra with rotational freedom * Compound of four tetrahedra * Compound of four triangular prisms *

Compound of great icosahedron and great stellated dodecahedron There are two different compounds of great icosahedron and great stellated dodecahedron: one is a dual compound and a stellation of the great icosidodecahedron, the other is a stellation of the icosidodecahedron. Dual compound It can be seen as ...

* 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 This uniform polyhedron compound is a chiral symmetric arrangement of 6 pentagrammic prisms, aligned with the axes of fivefold rotational symmetry of a dodecahedron. Related polyhedra This compound shares its vertex arrangement with four uni ...

Compound of six tetrahedra The compound of six tetrahedra is a uniform polyhedron compound. It's composed of a symmetric arrangement of 6 tetrahedra. It can be constructed by inscribing a stella octangula within each cube in the compound of three cubes, or by stellatin ...

* Compound of six tetrahedra with rotational freedom *

Compound of small stellated dodecahedron and great dodecahedron The compound of small stellated dodecahedron and great dodecahedron is a polyhedron compound where the great dodecahedron is internal to its dual, the small stellated dodecahedron. This can be seen as one of the two three-dimensional equivalen ...

* Compound of ten hexagonal prisms * Compound of ten octahedra *

Compound of ten tetrahedra The compound of ten tetrahedra is one of the five regular polyhedral compounds. This polyhedron can be seen as either a stellation of the icosahedron or a compound. This compound was first described by Edmund Hess in 1876. It can be seen as a ...

* 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 This uniform polyhedron compound is a symmetric arrangement of 12 pentagrammic prisms, aligned in pairs with the axes of fivefold rotational symmetry of a dodecahedron. It results from composing the two Chirality (mathematics), enantiomorphs of t ...

Compound of twelve tetrahedra with rotational freedom This uniform polyhedron compound is a symmetric arrangement of 12 tetrahedra, considered as antiprisms. It can be constructed by superimposing six identical copies of the stella octangula, and then rotating them in pairs about the three axes that ...

Compound of twenty octahedra The compound of twenty octahedra is a uniform polyhedron compound. It's composed of a symmetric arrangement of 20 octahedra (considered as triangular antiprisms). It is a special case of the compound of 20 octahedra with rotational freedom, in ...

* Compound of twenty octahedra with rotational freedom *

Compound of twenty tetrahemihexahedra This uniform polyhedron compound is a symmetric arrangement of 20 tetrahemihexahedra. It is chiral with icosahedral symmetry (I). John Skilling notes, in his enumeration of uniform compounds of uniform polyhedra, that this compound of 20 tetr ...

* 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 This uniform polyhedron compound is a composition of 2 icosahedra. It has octahedral symmetry ''Oh''. As a holosnub, it is represented by Schläfli symbol β and Coxeter diagram . The triangles in this compound decompose into two orbits unde ...

* 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

Four dimensions

; Four-dimensional space

4-polytope In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...

– general term for a four dimensional polytope ;

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 i ...

Tesseract 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 e ...

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 ...

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, hec ...

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 ...

;

Abstract regular 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 ...

; Regular star 4-polytope * 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, Grand stellated 120-cell, Great grand 120-cell, Great icosahedral 120-cell, Grand 600-cell, Great grand stellated 120-cell ;

Uniform 4-polytope In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform 4-polytopes. Th ...

Rectified 5-cell In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In t ...

, Truncated 5-cell,

Cantellated 5-cell In four-dimensional geometry, a cantellated 5-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation, up to edge-planing) of the regular 5-cell. Cantellated 5-cell The cantellated 5-cell or small rhombated pentach ...

Runcinated 5-cell In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to face-planing) of the regular 5-cell. There are 3 unique degrees of runcinations of the 5-cell, including with pe ...

Rectified tesseract In geometry, the rectified tesseract, rectified 8-cell is a uniform 4-polytope (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It has half the vertices of a runcinated tesseract, with its construction, called a ru ...

Truncated tesseract In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract. There are three truncations, including a bitruncation, and a tritruncation, which creates the ''truncated 16-cell''. Truncated tesserac ...

, Cantellated tesseract, Runcinated tesseract * Rectified 16-cell,

Truncated 16-cell In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract. There are three truncations, including a bitruncation, and a tritruncation, which creates the ''truncated 16-cell''. Truncated tesserac ...

Rectified 24-cell In geometry, the rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by rectification of the 24-cell, reducing ...

, Truncated 24-cell, Cantellated 24-cell,

Runcinated 24-cell In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 24-cell. There are 3 unique degrees of runcinations of the 24-cell including with permutations truncati ...

Snub 24-cell In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces ...

* Rectified 120-cell, Truncated 120-cell, Cantellated 120-cell,

Runcinated 120-cell In four-dimensional geometry, a runcinated 120-cell (or ''runcinated 600-cell'') is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 120-cell. There are 4 degrees of runcinations of the 120-cell including w ...

Rectified 600-cell In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icos ...

, Truncated 600-cell,

Cantellated 600-cell In four-dimensional geometry, a cantellated 120-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular 120-cell. There are four degrees of cantellations of the 120-cell including with permutations trunca ...

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Prismatic uniform 4-polytope In four-dimensional geometry, a prismatic uniform 4-polytope is a uniform 4-polytope with a nonconnected Coxeter diagram symmetry group. These figures are analogous to the set of prisms and antiprism uniform polyhedra, but add a third category cal ...

Grand antiprism In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. It is an anomalous, non-Wythoffian uniform 4-polytope ...

Duoprism In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, wher ...

Tetrahedral prism In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vertices. It is one of 18 u ...

Truncated tetrahedral prism In geometry, a truncated tetrahedral prism is a convex uniform polychoron (four-dimensional polytope). This polychoron has 10 polyhedral cells: 2 truncated tetrahedra connected by 4 triangular prisms and 4 hexagonal prisms. It has 24 faces: 8 tr ...

* Truncated cubic prism,

Truncated octahedral prism In 4-dimensional geometry, a truncated octahedral prism or omnitruncated tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 16 cells (2 truncated octahedra connected by 6 cubes, 8 hexagonal prisms.) It has 64 faces (48 squares ...

Cuboctahedral prism In geometry, a cuboctahedral prism is a convex uniform 4-polytope. This 4-polytope has 16 polyhedral cells: 2 cuboctahedra connected by 8 triangular prisms and 6 cubes. It is one of 18 uniform polyhedral prisms created by using uniform prisms to ...

, 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 In 4-dimensional geometry, a uniform antiprismatic prism or antiduoprism is a uniform 4-polytope with two uniform antiprism cells in two parallel 3-space hyperplanes, connected by uniform prisms cells between pairs of faces. The symmetry of a ' ...

Honeycombs

Tesseractic honeycomb In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol , and constructed by a 4-dimensional packing of tesseract facets. Its ver ...

* 24-cell honeycomb *

Snub 24-cell honeycomb In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation (or honeycomb) by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 ...

* Rectified 24-cell honeycomb *

Truncated 24-cell honeycomb In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells. It has a uniform alte ...

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 around ...

* 5-cell honeycomb *

Omnitruncated 5-cell honeycomb In Four-dimensional space, four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb (geometry), honeycomb. It is composed of 5-cells and recti ...

* Truncated 5-cell honeycomb *

Omnitruncated 5-simplex honeycomb In five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 5-simplex facets. The facets of all omn ...

Five dimensions

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Five-dimensional space A five-dimensional space is a space with five dimensions. In mathematics, a sequence of ''N'' numbers can represent a location in an ''N''-dimensional space. If interpreted physically, that is one more than the usual three spatial dimensions ...

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 ...

and

uniform 5-polytope In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets. The complete set of convex uniform 5-polytopes has not been deter ...

5-simplex In five-dimensional geometry, a 5- simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°. The 5 ...

Rectified 5-simplex In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex. There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the '' ...

, Truncated 5-simplex, Cantellated 5-simplex, Runcinated 5-simplex, Stericated 5-simplex *

5-demicube In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' (penteract) with alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only semiregular 5- ...

, Truncated 5-demicube, Cantellated 5-demicube, Runcinated 5-demicube *

5-cube In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts, ...

, Rectified 5-cube,

, Truncated 5-cube,

Cantellated 5-cube In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube. There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the ...

Runcinated 5-cube In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube. There are 8 unique degrees of runcinations of the 5-cube, along with permutations of truncation ...

Stericated 5-cube In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube. There are eight degrees of sterication for the 5-cube, including permutations of runcination, c ...

5-orthoplex In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces. It has two constructed forms, the first being regular with ...

Rectified 5-orthoplex In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex. There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and ...

Truncated 5-orthoplex In five-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex. There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs o ...

Cantellated 5-orthoplex In five-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex. There are 6 cantellation for the 5-orthoplex, including truncations. Some of them are more easily construct ...

Runcinated 5-orthoplex In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation ( runcination) of the regular 5-orthoplex. There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantell ...

truncated 24-cell prism Truncation is the term used for limiting the number of digits right of the decimal point by discarding the least significant ones. Truncation may also refer to: Mathematics * Truncation (statistics) refers to measurements which have been cut of ...

, cantellated 24-cell prism, runcinated 24-cell prism, bitruncated 24-cell prism, cantitruncated 24-cell prism, runcitruncated 24-cell prism, omnitruncated 24-cell prism, snub 24-cell prism * 120-cell prism, Rectified 120-cell prism, Truncated 120-cell prism, Cantellated 120-cell prism, Runcinated 120-cell prism, Bitruncated 120-cell prism, Cantitruncated 120-cell prism, Runcitruncated 120-cell prism, Omnitruncated 120-cell prism * 600-cell prism, Rectified 600-cell prism, Truncated 600-cell prism, Cantellated 600-cell prism, Cantitruncated 600-cell prism, Runcitruncated 600-cell prism * Grand antiprism prism

Honeycombs

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

5-demicubic honeycomb The 5-demicube honeycomb (or demipenteractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb. It is the first tessellation in the demih ...

Six dimensions

; Six-dimensional space, 6-polytope and

uniform 6-polytope In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes. The complete set of convex uniform 6-polytopes has not been determined, bu ...

6-simplex In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°. Alte ...

Rectified 6-simplex In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex. There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the ''rect ...

Truncated 6-simplex In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex. There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the ...

, Cantellated 6-simplex, Runcinated 6-simplex, Stericated 6-simplex,

Pentellated 6-simplex In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex. There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations ...

* 6-demicube, Truncated 6-demicube, Cantellated 6-demicube, Runcinated 6-demicube,

Stericated 6-demicube In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope. There are 8 pentic forms of the 6-cube. Pentic 6-cube The ''pentic 6-cube'', , has half of the vertices of a pentellated 6-cube, . Alternate names * Stericated 6-d ...

6-cube In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It can ...

, Rectified 6-cube,

Truncated 6-cube In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube. There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the ...

Cantellated 6-cube In six-dimensional geometry, a cantellated 6-cube is a convex uniform 6-polytope, being a cantellation of the regular 6-cube. There are 8 cantellations for the 6-cube, including truncations. Half of them are more easily constructed from the dual ...

, Runcinated 6-cube, Stericated 6-cube,

Pentellated 6-cube In six-dimensional geometry, a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube. There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcina ...

6-orthoplex In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell ''4-faces'', and 64 ''5-faces''. It has two constructed forms, the first being regular wi ...

, Rectified 6-orthoplex, Truncated 6-orthoplex,

Cantellated 6-orthoplex In six-dimensional geometry, a cantellated 6-orthoplex is a convex uniform 6-polytope, being a cantellation of the regular 6-orthoplex. There are 8 cantellation for the 6-orthoplex including truncations. Half of them are more easily constructed f ...

Runcinated 6-orthoplex In six-dimensional geometry, a runcinated 6-orthplex is a convex uniform 6-polytope with 3rd order truncations ( runcination) of the regular 6-orthoplex. There are 12 unique runcinations of the 6-orthoplex with permutations of truncations, and c ...

Stericated 6-orthoplex In six-dimensional geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-orthoplex. There are 16 unique sterications for the 6-orthoplex with permutations of trunca ...

* 1₂₂ polytope, 2₂₁ polytope

Honeycombs

6-cubic 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 ...

* 6-simplex honeycomb *

6-demicubic honeycomb The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 6-space. It is constructed as an Alternation (geometry), alternation of the regular 6-cube honeycomb. I ...

* 2₂₂ honeycomb

Seven dimensions

;

Seven-dimensional space In mathematics, a sequence of ''n'' real numbers can be understood as a location in ''n''-dimensional space. When ''n'' = 7, the set of all such locations is called 7-dimensional space. Often such a space is studied as a vector space, without any n ...

uniform 7-polytope In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose ...

*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 *Simplex *Hypercube *Cross-polytope ;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 *Hypercubic honeycomb *Alternated hypercubic honeycomb

Geometric operators

*Rectification (geometry) *Truncation (geometry) *Bitruncation *Cantellation *Runcination *Sterication *Omnitruncation *Expansion (geometry) *Snub (geometry) *Alternation (geometry) *Dual polyhedron *Gyration (geometry) *Elongation (geometry) *Augmentation (geometry) *Diminishment (geometry) *Greatening (geometry) *Aggrandizement (geometry) *Stellation *Kleetope *Conway polyhedron notation

Polytope elements

Polygon (2-polytope)

Polyhedron (3-polytope)

4-polytope

5-polytope

Other

Two dimensional (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 to ...s)

Star polygons

Families

Tilings

Three dimensional (polyhedra 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 on ...)

Regular

Archimedean solids

Prisms and antiprisms

Catalan solids

Bipyramids and Trapezohedron

Uniform star polyhedra

Uniform prismatic star polyhedra

Johnson solids

Dual uniform star polyhedra

Honeycombs

Other

Regular and uniform compound polyhedra

Four dimensions

Honeycombs

Five dimensions

Honeycombs

Six dimensions

Honeycombs

Seven dimensions

Honeycombs

Eight dimension

Honeycombs

Nine dimensions

Hyperbolic honeycombs

Ten dimensions

Dimensional families

Geometric operators

See also

Two dimensional (
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 to ...
s)

Three dimensional (
polyhedra 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 on ...
)