Aregular polyhedron is apolyhedron withregular andcongruent polygons asfaces. Itssymmetry group actstransitively on itsflags. A regular polyhedron is highly symmetrical, being all ofedge-transitive,vertex-transitive andface-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces arecongruent regular polygons which are assembled in the same way around eachvertex.
A regular polyhedron is identified by itsSchläfli symbol of the form {n,m}, wheren is the number of sides of each face andm the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (thePlatonic solids), and four regularstar polyhedra (theKepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra.
There are fiveconvex regular polyhedra, known as thePlatonic solids; four regularstar polyhedra, theKepler–Poinsot polyhedra; and five regular compounds of regular polyhedra:
 |  |  |  |  |
| Tetrahedron {3, 3} | Cube {4, 3} | Octahedron {3, 4} | Dodecahedron {5, 3} | Icosahedron {3, 5} |
| χ = 2 | χ = 2 | χ = 2 | χ = 2 | χ = 2 |
 |  |  |  |
| Small stellated dodecahedron {5/2, 5} | Great dodecahedron {5, 5/2} | Great stellated dodecahedron {5/2, 3} | Great icosahedron {3, 5/2} |
| χ = −6 | χ = −6 | χ = 2 | χ = 2 |
 |  |  |  |  |
| Two tetrahedra 2 {3, 3} | Five tetrahedra 5 {3, 3} | Ten tetrahedra 10 {3, 3} | Five cubes 5 {4, 3} | Five octahedra 5 {3, 4} |
| χ = 4 | χ = 10 | χ = 0 | χ = −10 | χ = 10 |
The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition:
A convex regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre:
The regular polyhedra are the mostsymmetrical of all the polyhedra. They lie in just threesymmetry groups, which are named after the Platonic solids:
Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry.
The five Platonic solids have anEuler characteristic of 2. This simply reflects that the surface is a topological 2-sphere, and so is also true, for example, ofany polyhedron which is star-shaped with respect to some interior point.
The sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point (this is an extension ofViviani's theorem.) However, the converse does not hold, not even fortetrahedra.[2]
In adual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, and vice versa.
The regular polyhedra show this duality as follows:
The Schläfli symbol of the dual is just the original written backwards, for example the dual of {5, 3} is {3, 5}.
Stones carved in shapes resembling clusters of spheres or knobs have been found inScotland and may be as much as 4,000 years old. Some of these stones show not only the symmetries of the five Platonic solids, but also some of the relations of duality amongst them (that is, that the centres of the faces of the cube gives the vertices of an octahedron). Examples of these stones are on display in the John Evans room of theAshmolean Museum atOxford University. Why these objects were made, or how their creators gained the inspiration for them, is a mystery. There is doubt regarding the mathematical interpretation of these objects, as many have non-platonic forms, and perhaps only one has been found to be a true icosahedron, as opposed to a reinterpretation of the icosahedron dual, the dodecahedron.[3]
It is also possible that theEtruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery nearPadua (in NorthernItaly) in the late 19th century of adodecahedron made ofsoapstone, and dating back more than 2,500 years (Lindemann, 1987).
The earliest knownwritten records of the regular convex solids originated from Classical Greece. When these solids were all discovered and by whom is not known, butTheaetetus (anAthenian) was the first to give a mathematical description of all five (Van der Waerden, 1954), (Euclid, book XIII).H.S.M. Coxeter (Coxeter, 1948, Section 1.9) creditsPlato (400 BC) with having made models of them, and mentions that one of the earlierPythagoreans,Timaeus of Locri, used all five in a correspondence between the polyhedra and the nature of the universe as it was then perceived – this correspondence is recorded in Plato's dialogueTimaeus. Euclid's reference to Plato led to their common description as thePlatonic solids.
One might characterise the Greek definition as follows:
This definition rules out, for example, thesquare pyramid (since although all the faces are regular, the square base is not congruent to the triangular sides), or the shape formed by joining two tetrahedra together (since although all faces of thattriangular bipyramid would be equilateral triangles, that is, congruent and regular, some vertices have 3 triangles and others have 4).
This concept of a regular polyhedron would remain unchallenged for almost 2000 years.
Regular star polygons such as thepentagram (star pentagon) were also known to the ancient Greeks – thepentagram was used by thePythagoreans as their secret sign, but they did not use them to construct polyhedra. It was not until the early 17th century thatJohannes Kepler realised that pentagrams could be used as the faces of regularstar polyhedra. Some of these star polyhedra may have been discovered by others before Kepler's time, but Kepler was the first to recognise that they could be considered "regular" if one removed the restriction that regular polyhedra be convex. Two hundred years laterLouis Poinsot also allowed starvertex figures (circuits around each corner), enabling him to discover two new regular star polyhedra along with rediscovering Kepler's. These four are the only regular star polyhedra, and have come to be known as theKepler–Poinsot polyhedra. It was not until the mid-19th century, several decades after Poinsot published, that Cayley gave them their modern English names: (Kepler's)small stellated dodecahedron andgreat stellated dodecahedron, and (Poinsot's)great icosahedron andgreat dodecahedron.
The Kepler–Poinsot polyhedra may be constructed from the Platonic solids by a process calledstellation. The reciprocal process to stellation is calledfacetting (or faceting). Every stellation of one polyhedron isdual, or reciprocal, to some facetting of the dual polyhedron. The regular star polyhedra can also be obtained by facetting the Platonic solids. This was first done by Bertrand around the same time that Cayley named them.
By the end of the 19th century there were therefore nine regular polyhedra – five convex and four star.
Each of the Platonic solids occurs naturally in one form or another.
The tetrahedron, cube, and octahedron all occur ascrystals. These by no means exhaust the numbers of possible forms of crystals (Smith, 1982, p212), of which there are 48. Neither theregular icosahedron nor theregular dodecahedron are amongst them, but crystals can have the shape of apyritohedron, which is visually almost indistinguishable from a regular dodecahedron. Truly icosahedral crystals may be formed byquasicrystalline materials which are very rare in nature but can be produced in a laboratory.
A more recent discovery is of a series of new types ofcarbon molecule, known as thefullerenes (see Curl, 1991). Although C60, the most easily produced fullerene, looks more or less spherical, some of the larger varieties (such as C240, C480 and C960) are hypothesised to take on the form of slightly rounded icosahedra, a few nanometres across.
Regular polyhedra appear in biology as well. ThecoccolithophoreBraarudosphaera bigelowii has a regular dodecahedral structure, about 10micrometres across.[4] In the early 20th century,Ernst Haeckel described a number of species ofradiolarians, some of whose shells are shaped like various regular polyhedra.[5] Examples includeCircoporus octahedrus,Circogonia icosahedra,Lithocubus geometricus andCircorrhegma dodecahedra; the shapes of these creatures are indicated by their names.[5] The outer protein shells of manyviruses form regular polyhedra. For example,HIV is enclosed in a regular icosahedron, as is the head of a typicalmyovirus.[6][7]
In ancient times thePythagoreans believed that there was a harmony between the regular polyhedra and the orbits of theplanets. In the 17th century,Johannes Kepler studied data on planetary motion compiled byTycho Brahe and for a decade tried to establish the Pythagorean ideal by finding a match between the sizes of the polyhedra and the sizes of the planets' orbits. His search failed in its original objective, but out of this research came Kepler's discoveries of the Kepler solids as regular polytopes, the realisation that the orbits of planets are not circles, andthe laws of planetary motion for which he is now famous. In Kepler's time only five planets (excluding the earth) were known, nicely matching the number of Platonic solids. Kepler's work, and the discovery since that time ofUranus andNeptune, have invalidated the Pythagorean idea.
Around the same time as the Pythagoreans, Plato described a theory of matter in which the five elements (earth, air, fire, water and spirit) each comprised tiny copies of one of the five regular solids. Matter was built up from a mixture of these polyhedra, with each substance having different proportions in the mix. Two thousand years laterDalton's atomic theory would show this idea to be along the right lines, though not related directly to the regular solids.
The 20th century saw a succession of generalisations of the idea of a regular polyhedron, leading to several new classes.
In the first decades, Coxeter and Petrie allowed "saddle" vertices with alternating ridges and valleys, enabling them to construct three infinite folded surfaces which they calledregular skew polyhedra.[8] Coxeter offered a modifiedSchläfli symbol {l,m|n} for these figures, with {l,m} implying thevertex figure, withm regularl-gons around a vertex. Then definesn-gonalholes. Their vertex figures areregular skew polygons, vertices zig-zagging between two planes.
| Infinite regular skew polyhedra in 3-space (partially drawn) | ||
|---|---|---|
 {4,6|4} |  {6,4|4} |  {6,6|3} |
Finite regular skew polyhedra exist in 4-space. These finite regular skew polyhedra in 4-space can be seen as a subset of the faces ofuniform 4-polytopes. They have planarregular polygon faces, butregular skew polygonvertex figures.
Two dual solutions are related to the5-cell, two dual solutions are related to the24-cell, and an infinite set of self-dualduoprisms generate regular skew polyhedra as {4, 4 | n}. In the infinite limit these approach aduocylinder and look like atorus in theirstereographic projections into 3-space.
| OrthogonalCoxeter plane projections | Stereographic projection | |||
|---|---|---|---|---|
| A4 | F4 | |||
 |  |  |  |  |
| {4, 6 | 3} | {6, 4 | 3} | {4, 8 | 3} | {8, 4 | 3} | {4, 4 | n} |
| 30{4} faces 60 edges 20 vertices | 20{6} faces 60 edges 30 vertices | 288 {4} faces 576 edges 144 vertices | 144{8} faces 576 edges 288 vertices | n2 {4} faces 2n2 edges n2 vertices |
Studies ofnon-Euclidean (hyperbolic andelliptic) and other spaces such ascomplex spaces, discovered over the preceding century, led to the discovery of more new polyhedra such ascomplex polyhedra which could only take regular geometric form in those spaces.
In H3hyperbolic space,paracompact regular honeycombs have Euclidean tilingfacets andvertex figures that act like finite polyhedra. Such tilings have anangle defect that can be closed by bending one way or the other. If the tiling is properly scaled, it willclose as anasymptotic limit at a singleideal point. These Euclidean tilings are inscribed in ahorosphere just as polyhedra are inscribed in a sphere (which contains zero ideal points). The sequence extends when hyperbolic tilings are themselves used as facets of noncompact hyperbolic tessellations, as in theheptagonal tiling honeycomb {7,3,3}; they are inscribed in an equidistant surface (a 2-hypercycle), which has two ideal points.
Another group of regular polyhedra comprise tilings of thereal projective plane. These include thehemi-cube,hemi-octahedron,hemi-dodecahedron, andhemi-icosahedron. They are (globally)projective polyhedra, and are the projective counterparts of thePlatonic solids. The tetrahedron does not have a projective counterpart as it does not have pairs of parallel faces which can be identified, as the other four Platonic solids do.
 Hemi-cube {4,3} |  Hemi-octahedron {3,4} |  Hemi-dodecahedron {3,5} |  Hemi-icosahedron {5,3} |
These occur as dual pairs in the same way as the original Platonic solids do. Their Euler characteristics are all 1.
By now, polyhedra were firmly understood as three-dimensional examples of more generalpolytopes in any number of dimensions. The second half of the century saw the development of abstract algebraic ideas such asPolyhedral combinatorics, culminating in the idea of anabstract polytope as apartially ordered set (poset) of elements. The elements of an abstract polyhedron are its body (the maximal element), its faces, edges, vertices and thenull polytope or empty set. These abstract elements can be mapped into ordinary space orrealised as geometrical figures. Some abstract polyhedra have well-formed orfaithful realisations, others do not. Aflag is a connected set of elements of each dimension – for a polyhedron that is the body, a face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to beregular if its combinatorial symmetries are transitive on its flags – that is to say, that any flag can be mapped onto any other under a symmetry of the polyhedron. Abstract regular polytopes remain an active area of research.
Five such regular abstract polyhedra, which can not be realised faithfully, were identified byH. S. M. Coxeter in his bookRegular Polytopes (1977) and again byJ. M. Wills in his paper "The combinatorially regular polyhedra of index 2" (1987). All five have C2×S5 symmetry but can only be realised with half the symmetry, that is C2×A5 or icosahedral symmetry.[9][10][11] They are all topologically equivalent totoroids. Their construction, by arrangingn faces around each vertex, can be repeated indefinitely as tilings of thehyperbolic plane. In the diagrams below, the hyperbolic tiling images have colors corresponding to those of the polyhedra images.
| Polyhedron |  Medial rhombic triacontahedron |  Dodecadodecahedron |  Medial triambic icosahedron |  Ditrigonal dodecadodecahedron |  Excavated dodecahedron |
|---|---|---|---|---|---|
| Type | Dual {5,4}6 | {5,4}6 | Dual of {5,6}4 | {5,6}4 | {6,6}6 |
| (v,e,f) | (24,60,30) | (30,60,24) | (24,60,20) | (20,60,24) | (20,60,20) |
| Vertex figure | {5}, {5/2}  | (5.5/2)2 | {5}, {5/2} | (5.5/3)3 |  |
| Faces | 30 rhombi | 12 pentagons 12 pentagrams | 20 hexagons | 12 pentagons 12 pentagrams | 20 hexagrams |
| Tiling |  {4, 5} |  {5, 4} |  {6, 5} |  {5, 6} |  {6, 6} |
| χ | −6 | −6 | −16 | −16 | −20 |
ThePetrie dual of a regular polyhedron is aregular map whose vertices and edges correspond to the vertices and edges of the original polyhedron, and whose faces are the set ofskewPetrie polygons.[12]
| Name | Petrial tetrahedron | Petrial cube | Petrial octahedron | Petrial dodecahedron | Petrial icosahedron |
|---|---|---|---|---|---|
| Symbol | {3,3}π | {4,3}π | {3,4}π | {5,3}π | {3,5}π |
| (v,e,f),χ | (4,6,3),χ = 1 | (8,12,4),χ = 0 | (6,12,4),χ = −2 | (20,30,6),χ = −4 | (12,30,6),χ = −12 |
| Faces | 3 skew squares | 4 skew hexagons | 6 skew decagons | ||
 |  |  |  | ||
| Image |  |  |  |  |  |
| Animation |  |  |  |  |  |
| Related figures | {4,3}3 ={4,3}/2 = {4,3}(2,0) |  {6,3}3 = {6,3}(2,0) |  {6,4}3 = {6,4}(4,0) | {10,3}5 | {10,5}3 |
The usual five regular polyhedra can also be represented as spherical tilings (tilings of thesphere):
 Tetrahedron {3,3} |  Cube {4,3} |  Octahedron {3,4} |  Dodecahedron {5,3} |  Icosahedron {3,5} |
 Small stellated dodecahedron {5/2,5} |  Great dodecahedron {5,5/2} |  Great stellated dodecahedron {5/2,3} |  Great icosahedron {3,5/2} |
For a regular polyhedron whose Schläfli symbol is {m, n}, the number of polygonal faces may be found by:
ThePlatonic solids known to antiquity are the only integer solutions form ≥ 3 andn ≥ 3. The restrictionm ≥ 3 enforces that the polygonal faces must have at least three sides.
When considering polyhedra as aspherical tiling, this restriction may be relaxed, sincedigons (2-gons) can be represented as spherical lunes, having non-zeroarea. Allowingm = 2 admits a new infinite class of regular polyhedra, which are thehosohedra. On a spherical surface, the regular polyhedron {2, n} is represented asn abutting lunes, with interior angles of 2π/n. All these lunes share two common vertices.[13]
A regulardihedron, {n, 2}[13] (2-hedron) in three-dimensionalEuclidean space can be considered adegenerateprism consisting of two (planar)n-sidedpolygons connected "back-to-back", so that the resulting object has no depth, analogously to how a digon can be constructed with twoline segments. However, as aspherical tiling, a dihedron can exist as nondegenerate form, with twon-sided faces covering the sphere, each face being ahemisphere, and vertices around agreat circle. It isregular if the vertices are equally spaced.
 Digonal dihedron {2,2} |  Trigonal dihedron {3,2} |  Square dihedron {4,2} |  Pentagonal dihedron {5,2} |  Hexagonal dihedron {6,2} | ... | {n,2} |
Digonal hosohedron {2,2} |  Trigonal hosohedron {2,3} |  Square hosohedron {2,4} |  Pentagonal hosohedron {2,5} |  Hexagonal hosohedron {2,6} | ... | {2,n} |
The hosohedron {2,n} is dual to the dihedron {n,2}. Note that whenn = 2, we obtain the polyhedron {2,2}, which is both a hosohedron and a dihedron. All of these have Euler characteristic 2.