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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a Platonic solid is a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
,
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 ...
in three-dimensional Euclidean space. Being a regular polyhedron means that the
faces 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 affe ...
are
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
(identical in shape and size)
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 ...
s (all
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s congruent and all
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 ...
s congruent), and the same number of faces meet at each vertex. There are only five such polyhedra:
Geometer A geometer is a mathematician whose area of study is geometry. Some notable geometers and their main fields of work, chronologically listed, are: 1000 BCE to 1 BCE * Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra * ...
s have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
who hypothesized in one of his dialogues, the '' Timaeus'', that the
classical element Classical elements typically refer to earth, water, air, fire, and (later) aether which were proposed to explain the nature and complexity of all matter in terms of simpler substances. Ancient cultures in Greece, Tibet, and India had simi ...
s were made of these regular solids.


History

The Platonic solids have been known since antiquity. It has been suggested that certain
carved stone balls Carved stone balls are petrospheres dated from the late Neolithic, to possibly as late as the Iron Age, mainly found in Scotland, but also elsewhere in Britain and Ireland. They are usually round and rarely oval, and of fairly uniform size ...
created by the late Neolithic people of
Scotland Scotland (, ) is a country that is part of the United Kingdom. Covering the northern third of the island of Great Britain, mainland Scotland has a border with England to the southeast and is otherwise surrounded by the Atlantic Ocean to ...
represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the numbers of knobs frequently differed from the numbers of vertices of the Platonic solids, there is no ball whose knobs match the 20 vertices of the dodecahedron, and the arrangement of the knobs was not always symmetric. The
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cult ...
studied the Platonic solids extensively. Some sources (such as
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophe ...
) credit
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His poli ...
with their discovery. Other evidence suggests that he may have only been familiar with the tetrahedron, cube, and dodecahedron and that the discovery of the octahedron and icosahedron belong to
Theaetetus Theaetetus (Θεαίτητος) is a Greek name which could refer to: * Theaetetus (mathematician) (c. 417 BC – 369 BC), Greek geometer * ''Theaetetus'' (dialogue), a dialogue by Plato, named after the geometer * Theaetetus (crater) Theaetetus ...
, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist. The Platonic solids are prominent in the philosophy of
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
, their namesake. Plato wrote about them in the dialogue ''Timaeus'' 360 B.C. in which he associated each of the four
classical element Classical elements typically refer to earth, water, air, fire, and (later) aether which were proposed to explain the nature and complexity of all matter in terms of simpler substances. Ancient cultures in Greece, Tibet, and India had simi ...
s (
earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
,
air The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing f ...
,
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
, and
fire Fire is the rapid oxidation of a material (the fuel) in the exothermic chemical process of combustion, releasing heat, light, and various reaction products. At a certain point in the combustion reaction, called the ignition point, flames ...
) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly nonspherical solid, the hexahedron (cube) represents "earth". These clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cube's being the only regular solid that tessellates
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, "...the god used tfor arranging the constellations on the whole heaven".
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
added a fifth element, aithēr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
completely mathematically described the Platonic solids in the ''Elements'', the last book (Book XIII) of which is devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the chief goal of the deductive system canonized in the ''Elements''. Much of the information in Book XIII is probably derived from the work of Theaetetus. In the 16th century, the German
astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either ...
Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
attempted to relate the five extraterrestrial
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s known at that time to the five Platonic solids. In ''
Mysterium Cosmographicum ''Mysterium Cosmographicum'' (lit. ''The Cosmographic Mystery'', alternately translated as ''Cosmic Mystery'', ''The Secret of the World'', or some variation) is an astronomy book by the German astronomer Johannes Kepler, published at Tübingen i ...
'', published in 1596, Kepler proposed a model of the
Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres. Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids enclosed within a sphere that represented the orbit of
Saturn Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with an average radius of about nine and a half times that of Earth. It has only one-eighth the average density of Earth; h ...
. The six spheres each corresponded to one of the planets ( Mercury,
Venus Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never f ...
,
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
,
Mars Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury. In the English language, Mars is named for the Roman god of war. Mars is a terrestrial planet with a thin at ...
,
Jupiter Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but slightly less than one-thousand ...
, and Saturn). The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube, thereby dictating the structure of the solar system and the distance relationships between the planets by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came his three laws of orbital dynamics, the first of which was that the orbits of planets are ellipses rather than circles, changing the course of physics and astronomy. He also discovered the Kepler solids, which are two ''nonconvex'' regular polyhedra.


Cartesian coordinates

For Platonic solids centered at the origin, simple
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
of the vertices are given below. The Greek letter ''φ'' is used to represent the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
≈ 1.6180. The coordinates for the tetrahedron, dodecahedron, and icosahedron are given in two positions such that each can be deduced from the other: in the case of the tetrahedron, by changing all coordinates of sign ( central symmetry), or, in the other cases, by exchanging two coordinates (
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
with respect to any of the three diagonal planes). These coordinates reveal certain relationships between the Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as or , one of two sets of 4 vertices in dual positions, as h or . Both tetrahedral positions make the compound
stellated octahedron The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depic ...
. The coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform
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 ...
, t or , also called a '' snub octahedron'', as s or , and seen in the compound of two icosahedra. Eight of the vertices of the dodecahedron are shared with the cube. Completing all orientations leads to the
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 regul ...
.


Combinatorial properties

A convex polyhedron is a Platonic solid if and only if # all its faces are
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
convex
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s, # none of its faces intersect except at their edges, and # the same number of faces meet at each of its vertices. Each Platonic solid can therefore be denoted by a symbol where : ''p'' is the number of edges (or, equivalently, vertices) of each face, and : ''q'' is the number of faces (or, equivalently, edges) that meet at each vertex. The symbol , called the
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...
, gives a
combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below. All other combinatorial information about these solids, such as total number of vertices (''V''), edges (''E''), and faces (''F''), can be determined from ''p'' and ''q''. Since any edge joins two vertices and has two adjacent faces we must have: : pF = 2E = qV.\, The other relationship between these values is given by
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
: : V - E + F = 2.\, This can be proved in many ways. Together these three relationships completely determine ''V'', ''E'', and ''F'': : V = \frac,\quad E = \frac,\quad F = \frac. Swapping ''p'' and ''q'' interchanges ''F'' and ''V'' while leaving ''E'' unchanged. For a geometric interpretation of this property, see .


As a configuration

The elements of a polyhedron can be expressed in a configuration matrix. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole polyhedron. The nondiagonal numbers say how many of the column's element occur in or at the row's element. Dual pairs of polyhedra have their configuration matrices rotated 180 degrees from each other.


Classification

The classical result is that only five convex regular polyhedra exist. Two common arguments below demonstrate no more than five Platonic solids can exist, but positively demonstrating the existence of any given solid is a separate question—one that requires an explicit construction.


Geometric proof

The following geometric argument is very similar to the one given by
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
in the ''Elements'':


Topological proof

A purely topological proof can be made using only combinatorial information about the solids. The key is Euler's observation that ''V'' − ''E'' + ''F'' = 2, and the fact that ''pF'' = 2''E'' = ''qV'', where ''p'' stands for the number of edges of each face and ''q'' for the number of edges meeting at each vertex. Combining these equations one obtains the equation : \frac - E + \frac = 2. Simple algebraic manipulation then gives : + = + . Since ''E'' is strictly positive we must have : \frac + \frac > \frac. Using the fact that ''p'' and ''q'' must both be at least 3, one can easily see that there are only five possibilities for : : , , , , .


Geometric properties


Angles

There are a number of
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s associated with each Platonic solid. The
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...
is the interior angle between any two face planes. The dihedral angle, ''θ'', of the solid is given by the formula : \sin\left(\frac\right) = \frac. This is sometimes more conveniently expressed in terms of the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
by : \tan\left(\frac\right) = \frac. The quantity ''h'' (called the
Coxeter number In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there ...
) is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively. The
angular deficiency In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess. Classically the def ...
at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2. The defect, ''δ'', at any vertex of the Platonic solids is : \delta = 2\pi - q\pi\left(1 - \right). By a theorem of Descartes, this is equal to 4 divided by the number of vertices (i.e. the total defect at all vertices is 4). The 3-dimensional analog of a plane angle is a
solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poi ...
. The solid angle, ''Ω'', at the vertex of a Platonic solid is given in terms of the dihedral angle by : \Omega = q\theta - (q - 2)\pi.\, This follows from the
spherical excess Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are g ...
formula for a
spherical polygon Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are g ...
and the fact that the
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 ...
of the polyhedron is a regular ''q''-gon. The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4 steradians) divided by the number of faces. This is equal to the angular deficiency of its dual. The various angles associated with the Platonic solids are tabulated below. The numerical values of the solid angles are given in
steradian The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radian ...
s. The constant ''φ'' = is the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
.


Radii, area, and volume

Another virtue of regularity is that the Platonic solids all possess three concentric spheres: * the
circumscribed sphere In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle' ...
that passes through all the vertices, * the
midsphere In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point. Not every polyhedron has a midsphere, but for every convex po ...
that is tangent to each edge at the midpoint of the edge, and * the inscribed sphere that is tangent to each face at the center of the face. The radii of these spheres are called the ''circumradius'', the ''midradius'', and the ''inradius''. These are the distances from the center of the polyhedron to the vertices, edge midpoints, and face centers respectively. The circumradius ''R'' and the inradius ''r'' of the solid with edge length ''a'' are given by : \begin R &= \frac \tan\left(\frac\right)\tan\left(\frac\right) \\ pt r &= \frac \cot\left(\frac\right)\tan\left(\frac\right) \end where ''θ'' is the dihedral angle. The midradius ''ρ'' is given by : \rho = \frac \cos\left(\frac\right)\,\biggl(\frac\biggr) where ''h'' is the quantity used above in the definition of the dihedral angle (''h'' = 4, 6, 6, 10, or 10). The ratio of the circumradius to the inradius is symmetric in ''p'' and ''q'': : \frac = \tan\left(\frac\right) \tan\left(\frac\right) = \frac. The
surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ...
, ''A'', of a Platonic solid is easily computed as area of a regular ''p''-gon times the number of faces ''F''. This is: : A = \biggl(\frac\biggr)^2 Fp\cot\left(\frac\right). The
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
is computed as ''F'' times the volume of the
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
whose base is a regular ''p''-gon and whose height is the inradius ''r''. That is, : V = \frac rA. The following table lists the various radii of the Platonic solids together with their surface area and volume. The overall size is fixed by taking the edge length, ''a'', to be equal to 2. The constants ''φ'' and ''ξ'' in the above are given by : \varphi = 2\cos = \frac,\qquad \xi = 2\sin = \sqrt = \sqrt. Among the Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest to that of a sphere of the same size (i.e. either the same surface area or the same volume.) The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most.


Point in space

For an arbitrary point in the space of a Platonic solid with circumradius ''R'', whose distances to the centroid of the Platonic solid and its ''n'' vertices are ''L'' and ''di'' respectively, and :S^_= \frac 1n\sum_^n d_i^, we have :\begin S^_ = S^_ = S^_= S^_= S^_ &= R^2+L^2, \\ pxS^_ = S^_ = S^_= S^_= S^_ &= \left(R^2+L^2\right)^2 + \frac 43 R^2L^2, \\ pxS^_ = S^_ = S^_= S^_&= \left(R^2+L^2\right)^3 + 4R^2L^2 \left(R^2+L^2\right), \\ pxS^_ = S^_ &= \left(R^2+L^2\right)^4 + 8R^2L^2 \left(R^2+L^2\right)^2+\frac R^4L^4, \\ pxS^_ = S^_ &= \left(R^2+L^2\right)^5 +\frac R^2L^2\left(R^2+L^2\right)^3+16R^4L^4\left(R^2+L^2\right). \end For all five Platonic solids, we have :S^_+\frac R^4= \left(S^_+ \frac 23R^2\right)^2. If ''di'' are the distances from the ''n'' vertices of the Platonic solid to any point on its circumscribed sphere, then :4\left(\sum_^n d_i^2\right)^2=3n \sum_^n d_i^4.


Rupert property

A polyhedron ''P'' is said to have the ''Rupert'' property if a polyhedron of the same or larger size and the same shape as ''P'' can pass through a hole in ''P''. All five Platonic solids have this property.


Symmetry


Dual polyhedra

Every polyhedron has a dual (or "polar") polyhedron with faces and vertices interchanged. The dual of every Platonic solid is another Platonic solid, so that we can arrange the five solids into dual pairs. * The tetrahedron is
self-dual In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...
(i.e. its dual is another tetrahedron). * The cube and the octahedron form a dual pair. * The dodecahedron and the icosahedron form a dual pair. If a polyhedron has Schläfli symbol , then its dual has the symbol . Indeed, every combinatorial property of one Platonic solid can be interpreted as another combinatorial property of the dual. One can construct the dual polyhedron by taking the vertices of the dual to be the centers of the faces of the original figure. Connecting the centers of adjacent faces in the original forms the edges of the dual and thereby interchanges the number of faces and vertices while maintaining the number of edges. More generally, one can dualize a Platonic solid with respect to a sphere of radius ''d'' concentric with the solid. The radii (''R'', ''ρ'', ''r'') of a solid and those of its dual (''R''*, ''ρ''*, ''r''*) are related by : d^2 = R^\ast r = r^\ast R = \rho^\ast\rho. Dualizing with respect to the midsphere (''d'' = ''ρ'') is often convenient because the midsphere has the same relationship to both polyhedra. Taking ''d''2 = ''Rr'' yields a dual solid with the same circumradius and inradius (i.e. ''R''* = ''R'' and ''r''* = ''r'').


Symmetry groups

In mathematics, the concept of
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
is studied with the notion of a mathematical group. Every polyhedron has an associated
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
, which is the set of all transformations ( Euclidean isometries) which leave the polyhedron invariant. The
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between the ''full symmetry group'', which includes reflections, and the ''proper symmetry group'', which includes only rotations. The symmetry groups of the Platonic solids are a special class of three-dimensional point groups known as polyhedral groups. The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of the symmetry group, as are the edges and faces. One says the action of the symmetry group is transitive on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is ''regular'' if and only if it is vertex-uniform,
edge-uniform In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given tw ...
, and
face-uniform In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its Face (geometry), faces are the same. More specifically, all faces must be not ...
. There are only three symmetry groups associated with the Platonic solids rather than five, since the symmetry group of any polyhedron coincides with that of its dual. This is easily seen by examining the construction of the dual polyhedron. Any symmetry of the original must be a symmetry of the dual and vice versa. The three polyhedral groups are: * the tetrahedral group ''T'', * the octahedral group ''O'' (which is also the symmetry group of the cube), and * the icosahedral group ''I'' (which is also the symmetry group of the dodecahedron). The orders of the proper (rotation) groups are 12, 24, and 60 respectively – precisely twice the number of edges in the respective polyhedra. The orders of the full symmetry groups are twice as much again (24, 48, and 120). See (Coxeter 1973) for a derivation of these facts. All Platonic solids except the tetrahedron are ''centrally symmetric,'' meaning they are preserved under reflection through the origin. The following table lists the various symmetry properties of the Platonic solids. The symmetry groups listed are the full groups with the rotation subgroups given in parenthesis (likewise for the number of symmetries). Wythoff's kaleidoscope construction is a method for constructing polyhedra directly from their symmetry groups. They are listed for reference Wythoff's symbol for each of the Platonic solids.


In nature and technology

The tetrahedron, cube, and octahedron all occur naturally in
crystal structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pattern ...
s. These by no means exhaust the numbers of possible forms of crystals. However, neither the regular icosahedron nor the regular dodecahedron are amongst them. One of the forms, called the
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 ...
(named for the group of
minerals In geology and mineralogy, a mineral or mineral species is, broadly speaking, a solid chemical compound with a fairly well-defined chemical composition and a specific crystal structure that occurs naturally in pure form.John P. Rafferty, ed ...
of which it is typical) has twelve pentagonal faces, arranged in the same pattern as the faces of the regular dodecahedron. The faces of the pyritohedron are, however, not regular, so the pyritohedron is also not regular.
Allotropes of boron Boron can be prepared in several crystalline and amorphous forms. Well known crystalline forms are α-rhombohedral (α-R), β-rhombohedral (β-R), and β-tetragonal (β-T). In special circumstances, boron can also be synthesized in the form of ...
and many boron compounds, such as
boron carbide Boron carbide (chemical formula approximately B4C) is an extremely hard boron– carbon ceramic, a covalent material used in tank armor, bulletproof vests, engine sabotage powders, as well as numerous industrial applications. With a Vickers ...
, include discrete B12 icosahedra within their crystal structures.
Carborane acid Carborane acids (X, Y, Z = H, Alk, F, Cl, Br, CF3) are a class of superacids, some of which are estimated to be at least one million times stronger than 100% pure sulfuric acid in terms of their Hammett acidity function values (''H''0 ≤ – ...
s also have molecular structures approximating regular icosahedra. In the early 20th century,
Ernst Haeckel Ernst Heinrich Philipp August Haeckel (; 16 February 1834 – 9 August 1919) was a German zoologist, naturalist, eugenicist, philosopher, physician, professor, marine biologist and artist. He discovered, described and named thousands of new s ...
described (Haeckel, 1904) a number of species of
Radiolaria The Radiolaria, also called Radiozoa, are protozoa of diameter 0.1–0.2 mm that produce intricate mineral skeletons, typically with a central capsule dividing the cell into the inner and outer portions of endoplasm and ectoplasm. The el ...
, some of whose skeletons are shaped like various regular polyhedra. Examples include ''Circoporus octahedrus'', ''Circogonia icosahedra'', ''Lithocubus geometricus'' and ''Circorrhegma dodecahedra''. The shapes of these creatures should be obvious from their names. Many
virus A virus is a submicroscopic infectious agent that replicates only inside the living cells of an organism. Viruses infect all life forms, from animals and plants to microorganisms, including bacteria and archaea. Since Dmitri Ivanovsk ...
es, such as the
herpes Herpes simplex is a viral infection caused by the herpes simplex virus. Infections are categorized based on the part of the body infected. Oral herpes involves the face or mouth. It may result in small blisters in groups often called cold ...
virus, have the shape of a regular icosahedron. Viral structures are built of repeated identical
protein Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, res ...
subunits and the icosahedron is the easiest shape to assemble using these subunits. A regular polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral
genome In the fields of molecular biology and genetics, a genome is all the genetic information of an organism. It consists of nucleotide sequences of DNA (or RNA in RNA viruses). The nuclear genome includes protein-coding genes and non-coding ...
. In
meteorology Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...
and
climatology Climatology (from Greek , ''klima'', "place, zone"; and , '' -logia'') or climate science is the scientific study of Earth's climate, typically defined as weather conditions averaged over a period of at least 30 years. This modern field of stu ...
, global numerical models of atmospheric flow are of increasing interest which employ
geodesic grid A geodesic grid is a spatial grid based on a geodesic polyhedron or Goldberg polyhedron. Construction A geodesic grid is a global Earth reference that uses triangular tiles based on the subdivision of a polyhedron (usually the icosahedron, a ...
s that are based on an icosahedron (refined by
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...
) instead of the more commonly used
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
/
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
grid. This has the advantage of evenly distributed spatial resolution without singularities (i.e. the poles) at the expense of somewhat greater numerical difficulty. Geometry of
space frame In architecture and structural engineering, a space frame or space structure ( 3D truss) is a rigid, lightweight, truss-like structure constructed from interlocking struts in a geometric pattern. Space frames can be used to span large areas w ...
s is often based on platonic solids. In the MERO system, Platonic solids are used for naming convention of various space frame configurations. For example, O+T refers to a configuration made of one half of octahedron and a tetrahedron. Several
Platonic hydrocarbons In organic chemistry, a Platonic hydrocarbon is a hydrocarbon ( molecule) whose structure matches one of the five Platonic solids, with carbon atoms replacing its vertices, carbon–carbon bonds replacing its edges, and hydrogen atoms as nee ...
have been synthesised, including
cubane Cubane () is a synthetic hydrocarbon compound that consists of eight carbon atoms arranged at the corners of a cube, with one hydrogen atom attached to each carbon atom. A solid crystalline substance, cubane is one of the Platonic hydrocarbons an ...
and
dodecahedrane Dodecahedrane is a chemical compound, a hydrocarbon with formula , whose carbon atoms are arranged as the vertices (corners) of a regular dodecahedron. Each carbon is bound to three neighbouring carbon atoms and to a hydrogen atom. This compound ...
and not
tetrahedrane Tetrahedrane is a hypothetical platonic hydrocarbon with chemical formula and a tetrahedral structure. The molecule would be subject to considerable angle strain and has not been synthesized as of 2021. However, a number of derivatives have b ...
. Image:Tetrahedrane-3D-balls.png ,
Tetrahedrane Tetrahedrane is a hypothetical platonic hydrocarbon with chemical formula and a tetrahedral structure. The molecule would be subject to considerable angle strain and has not been synthesized as of 2021. However, a number of derivatives have b ...
Image:Cubane-3D-balls.png ,
Cubane Cubane () is a synthetic hydrocarbon compound that consists of eight carbon atoms arranged at the corners of a cube, with one hydrogen atom attached to each carbon atom. A solid crystalline substance, cubane is one of the Platonic hydrocarbons an ...
Image:Dodecahedrane-3D-balls.png,
Dodecahedrane Dodecahedrane is a chemical compound, a hydrocarbon with formula , whose carbon atoms are arranged as the vertices (corners) of a regular dodecahedron. Each carbon is bound to three neighbouring carbon atoms and to a hydrogen atom. This compound ...
Platonic solids are often used to make
dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing ...
, because dice of these shapes can be made
fair A fair (archaic: faire or fayre) is a gathering of people for a variety of entertainment or commercial activities. Fairs are typically temporary with scheduled times lasting from an afternoon to several weeks. Types Variations of fairs incl ...
. 6-sided dice are very common, but the other numbers are commonly used in
role-playing game A role-playing game (sometimes spelled roleplaying game, RPG) is a game in which players assume the roles of characters in a fictional setting. Players take responsibility for acting out these roles within a narrative, either through literal ac ...
s. Such dice are commonly referred to as d''n'' where ''n'' is the number of faces (d8, d20, etc.); see
dice notation Dice notation (also known as dice algebra, common dice notation, RPG dice notation, and several other titles) is a system to represent different combinations of dice in wargames and tabletop role-playing games using simple algebra-like notation suc ...
for more details. These shapes frequently show up in other games or puzzles. Puzzles similar to a
Rubik's Cube The Rubik's Cube is a Three-dimensional space, 3-D combination puzzle originally invented in 1974 by Hungarians, Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the Magic Cube, the puzzle was licensed by Rubik t ...
come in all five shapes – see magic polyhedra.


Liquid crystals with symmetries of Platonic solids

For the intermediate material phase called
liquid crystal Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. Th ...
s, the existence of such symmetries was first proposed in 1981 by H. Kleinert and K. Maki. In aluminum the icosahedral structure was discovered three years after this by
Dan Shechtman Dan Shechtman ( he, דן שכטמן; born January 24, 1941)Dan Shechtman
. (PDF). Retri ...
, which earned him the
Nobel Prize in Chemistry ) , image = Nobel Prize.png , alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then "M ...
in 2011.


Related polyhedra and polytopes


Uniform polyhedra

There exist four regular polyhedra that are not convex, called Kepler–Poinsot polyhedra. These all have
icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...
and may be obtained as
stellation In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specif ...
s of the dodecahedron and the icosahedron. The next most regular convex polyhedra after the Platonic solids are the
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 ...
, which is a
rectification Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Recti ...
of the cube and the octahedron, and the
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 ...
, which is a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron is a regular octahedron). These are both ''quasi-regular'', meaning that they are vertex- and edge-uniform and have regular faces, but the faces are not all congruent (coming in two different classes). They form two of the thirteen
Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are compose ...
s, which are the convex
uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also ...
with polyhedral symmetry. Their duals, the
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 ...
and
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 ...
, are edge- and face-transitive, but their faces are not regular and their vertices come in two types each; they are two of the thirteen
Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan so ...
s. The uniform polyhedra form a much broader class of polyhedra. These figures are vertex-uniform and have one or more types of regular or
star polygon In geometry, a star polygon is a type of non- convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operatio ...
s for faces. These include all the polyhedra mentioned above together with an infinite set of
prisms 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 sedimentar ...
, an infinite set of
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 ...
s, and 53 other non-convex forms. 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 polygon, or that the same polygons join around each vertex. An example of a Johnso ...
s are convex polyhedra which have regular faces but are not uniform. Among them are five of the eight convex
deltahedra In geometry, a deltahedron (plural ''deltahedra'') is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle. There are infinitely many delt ...
, which have identical, regular faces (all equilateral triangles) but are not uniform. (The other three convex deltahedra are the Platonic tetrahedron, octahedron, and icosahedron.)


Regular tessellations

The three
regular tessellation Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his '' Harmonices Mundi'' ( Latin: ''The Harmony of the World'', 1619). Notation of Eu ...
s of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as regular tessellations of the
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
. This is done by projecting each solid onto a concentric sphere. The faces project onto regular
spherical polygon Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are g ...
s which exactly cover the sphere. Spherical tilings provide two infinite additional sets of regular tilings, the hosohedra, with 2 vertices at the poles, and lune faces, and the dual dihedra, with 2 hemispherical faces and regularly spaced vertices on the equator. Such tesselations would be degenerate in true 3D space as polyhedra. One can show that every regular tessellation of the sphere is characterized by a pair of integers with  +  > . Likewise, a regular tessellation of the plane is characterized by the condition  +  = . There are three possibilities: In a similar manner, one can consider regular tessellations of the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ' ...
. These are characterized by the condition  +  < . There is an infinite family of such tessellations.


Higher dimensions

In more than three dimensions, polyhedra generalize to
polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
s, with higher-dimensional convex
regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
s being the equivalents of the three-dimensional Platonic solids. In the mid-19th century the Swiss mathematician
Ludwig Schläfli Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional space ...
discovered the four-dimensional analogues of the Platonic solids, called
convex regular 4-polytope In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star reg ...
s. There are exactly six of these figures; five are analogous to the Platonic solids :
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 ...
as ,
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 ...
as ,
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 ...
as ,
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 ...
as , and
120-cell In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, he ...
as , and a sixth one, the self-dual
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 ...
, . In all dimensions higher than four, there are only three convex regular polytopes: the
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
as , the
hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...
as , and the
cross-polytope In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahe ...
as . In three dimensions, these coincide with the tetrahedron as , the cube as , and the octahedron as .


See also


References


Sources

* * * * *
Gardner, Martin Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis ...
(1987). ''The 2nd Scientific American Book of Mathematical Puzzles & Diversions'', University of Chicago Press, Chapter 1: The Five Platonic Solids, * Haeckel, Ernst, E. (1904). ''Kunstformen der Natur''. Available as Haeckel, E. (1998);
Art forms in nature
', Prestel USA. . * Kepler. Johannes ''Strena seu de nive sexangula (On the Six-Cornered Snowflake)'', 1611 paper by Kepler which discussed the reason for the six-angled shape of the snow crystals and the forms and symmetries in nature. Talks about platonic solids. * * * * * Wildberg, Christian (1988)
''John Philoponus' Criticism of Aristotle's Theory of Aether.''
Walter de Gruyter. pp. 11–12.


External links


''Platonic solids'' at Encyclopaedia of Mathematics
* *

of Euclid's ''Elements''.

in Java

in Visual Polyhedra

is an interactive 3D polyhedron viewer which allows you to save the model in svg, stl or obj format.

in Java
Paper models of the Platonic solids
created using nets generated by Stella software
Platonic Solids
Free paper models(nets) *

student-created models

teacher instructions for making models

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


How to make four platonic solids from a cube
{{DEFAULTSORT:Platonic Solid Multi-dimensional geometry