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{{Short description|Polyhedron with regular congruent polygons as faces}}
A '''regular polyhedron''' is a [[Polyhedron|polyhedron]] whose [[Symmetry group|symmetry group]] acts transitively on its [[Flag (geometry)|flag]]s. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are [[Congruence (geometry)|congruent]] [[Regular polygon|regular polygon]]s which are assembled in the same way around each [[Vertex (geometry)|vertex]].

A regular polyhedron is identified by its [[Schläfli symbol]] of the form {''n'', ''m''}, where ''n'' is the number of sides of each face and ''m'' the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the [[Platonic solid]]s), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra.

== The regular polyhedra ==

There are five [[Convex polygon|convex]] regular polyhedra, known as the '''[[Platonic solid]]s'''; four regular [[Star polyhedron|star polyhedra]], the '''Kepler–Poinsot polyhedra'''; and five regular compounds of regular polyhedra:

=== Platonic solids ===
{{Main|Platonic solid}}

{| class="wikitable" width=640
|- align=center
|[[Image:Tetrahedron.jpg|100px]]
|[[Image:Hexahedron.jpg|100px]]
|[[Image:Octahedron.jpg|100px]]
|[[Image:Dodecahedron.jpg|100px]]
|[[Image:Icosahedron.jpg|100px]]
|- align=center
|[[Tetrahedron]]<br>{3,&nbsp;3}||[[Cube]]<br>{4,&nbsp;3}||[[Octahedron]]<br>{3,&nbsp;4}||[[Regular dodecahedron|Dodecahedron]]<br>{5,&nbsp;3}||[[Regular icosahedron|Icosahedron]]<br>{3,&nbsp;5}
|- align=center
|[[Euler characteristic|''χ'']] = 2||''χ'' = 2||''χ'' = 2||''χ'' = 2||''χ'' = 2
|}

=== Kepler–Poinsot polyhedra ===


{| class="wikitable" width=640
|- align=center
|[[Image:SmallStellatedDodecahedron.jpg|100px]]
|[[Image:GreatDodecahedron.jpg|100px]]
|[[Image:GreatStellatedDodecahedron.jpg|100px]]
|[[Image:GreatIcosahedron.jpg|100px]]
|- align=center
|[[Small stellated dodecahedron]]<br>{5/2,&nbsp;5}
||[[Great dodecahedron]]<br>{5,&nbsp;5/2}
||[[Great stellated dodecahedron]]<br>{5/2,&nbsp;3}
||[[Great icosahedron]]<br>{3,&nbsp;5/2}
|- align=center
|''χ'' = −6||''χ'' = −6||''χ'' = 2||''χ'' = 2
|}

=== Regular compounds ===

{| class="wikitable" style="width:640px; text-align:center;"
|-
| [[Image:Compound of two tetrahedra.png|100px]]
| [[Image:Compound of five tetrahedra.png|100px]]
| [[Image:Compound of ten tetrahedra.png|100px]]
| [[Image:Compound of five cubes.png|100px]]
| [[Image:Compound of five octahedra.png|100px]]
|-
| [[Stellated octahedron|Two tetrahedra]]<br>2 {3,&nbsp;3}
| [[Compound of five tetrahedra|Five tetrahedra]]<br>5 {3,&nbsp;3}
| [[Compound of ten tetrahedra|Ten tetrahedra]]<br>10 {3,&nbsp;3}
| [[Compound of five cubes|Five cubes]]<br>5 {4,&nbsp;3}
| [[Compound of five octahedra|Five octahedra]]<br>5 {3,&nbsp;4}
|-
| ''χ'' = 4 || ''χ'' = 10 || ''χ'' = 0 || ''χ'' = −10 || ''χ'' = 10
|}

== Characteristics ==
===Equivalent properties===

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:

*The vertices of a convex regular polyhedron all lie on a [[Sphere|sphere]].
*All the [[Dihedral angle|dihedral angle]]s of the polyhedron are equal
*All the [[Vertex figure|vertex figure]]s of the polyhedron are [[Regular polygon|regular polygon]]s.
*All the [[Solid angle|solid angle]]s of the polyhedron are congruent.<ref>{{cite book
  | last = Cromwell | first = Peter R. | title = Polyhedra | title-link = Polyhedra (book) | publisher = Cambridge University Press | year = 1997 | page = 77 | isbn = 0-521-66405-5 }}</ref>

===Concentric spheres===

A convex regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre:
* An insphere, tangent to all faces.
* An intersphere or [[Midsphere|midsphere]], tangent to all edges.
* A circumsphere, tangent to all vertices.

===Symmetry===

The regular polyhedra are the most [[Symmetry|symmetrical]] of all the polyhedra. They lie in just three [[Symmetry group|symmetry group]]s, which are named after the Platonic solids:
*Tetrahedral
*Octahedral (or cubic)
*Icosahedral (or dodecahedral)

Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry.

===Euler characteristic===

The five Platonic solids have an [[Euler characteristic]] of 2. This simply reflects that the surface is a topological 2-sphere, and so is also true, for example, of
any polyhedron which is star-shaped with respect to some interior point.

===Interior points===

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 of [[Viviani's theorem]].) However, the converse does not hold, not even for tetrahedra.<ref>Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", ''[[Organization:The College Mathematics Journal|The College Mathematics Journal]]'' 37(5), 2006, pp. 390–391.</ref>

== Duality of the regular polyhedra ==
In a [[Dual polyhedron|dual]] 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 [[Tetrahedron|tetrahedron]] is self-dual, i.e. it pairs with itself.
* The [[Cube|cube]] and [[Octahedron|octahedron]] are dual to each other.
* The [[Regular icosahedron|icosahedron]] and [[Dodecahedron|dodecahedron]] are dual to each other.
* The [[Small stellated dodecahedron|small stellated dodecahedron]] and [[Great dodecahedron|great dodecahedron]] are dual to each other.
* The [[Great stellated dodecahedron|great stellated dodecahedron]] and [[Great icosahedron|great icosahedron]] are dual to each other.

The Schläfli symbol of the dual is just the original written backwards, for example the dual of {5, 3} is {3, 5}.

==History==<!-- This section is linked from [[Polyhedron]] -->


=== Prehistory ===

Stones carved in shapes resembling clusters of spheres or knobs have been found in Scotland 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 the Ashmolean Museum at Oxford 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.<ref>[http://www.neverendingbooks.org/the-scottish-solids-hoax The Scottish Solids Hoax].</ref>

It is also possible that the [[Place:Etruscan civilization|Etruscans]] preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua (in Northern Italy) in the late 19th century of a [[Dodecahedron|dodecahedron]] made of [[Earth:Soapstone|soapstone]], and dating back more than 2,500 years (Lindemann, 1987).

=== Greeks ===

The earliest known ''written'' records of the regular convex solids originated from Classical Greece. When these solids were all discovered and by whom is not known, but [[Biography:Theaetetus (mathematician)|Theaetetus]] (an [[Place:Athens|Athenian]]) 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) credits [[Biography:Plato|Plato]] (400 BC) with having made models of them, and mentions that one of the earlier Pythagoreans, [[Biography:Timaeus of Locri|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 dialogue [[Astronomy:Timaeus (dialogue)|''Timaeus'']]. Euclid's reference to Plato led to their common description as the ''Platonic solids''.

One might characterise the Greek definition as follows:
*A regular polygon is a ([[Convex polygon|convex]]) planar figure with all edges equal and all corners equal.
*A regular polyhedron is a solid (convex) figure with all faces being congruent regular polygons, the same number arranged all alike around each vertex.

This definition rules out, for example, the [[Square pyramid|square 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 that [[Triangular bipyramid|triangular 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 polyhedra ===

Regular star polygons such as the [[Pentagram|pentagram]] (star pentagon) were also known to the ancient Greeks – the [[Pentagram|pentagram]] was used by the Pythagoreans as their secret sign, but they did not use them to construct polyhedra. It was not until the early 17th century that [[Biography:Johannes Kepler|Johannes Kepler]] realised that pentagrams could be used as the faces of regular [[Star polyhedron|star 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 later Louis Poinsot also allowed star [[Vertex figure|vertex figure]]s (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 the Kepler–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|small stellated dodecahedron]] and [[Great stellated dodecahedron|great stellated dodecahedron]], and (Poinsot's) [[Great icosahedron|great icosahedron]] and [[Great dodecahedron|great dodecahedron]].

The Kepler–Poinsot polyhedra may be constructed from the Platonic solids by a process called [[Stellation|stellation]]. The reciprocal process to stellation is called facetting (or faceting). Every stellation of one polyhedron is [[Dual polyhedron|dual]], 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.

== Regular polyhedra in nature ==
Each of the Platonic solids occurs naturally in one form or another.

The tetrahedron, cube, and octahedron all occur as [[Earth:Crystal|crystal]]s. These by no means exhaust the numbers of possible forms of crystals (Smith, 1982, p212), of which there are 48. Neither the [[Regular icosahedron|regular icosahedron]] nor the [[Regular dodecahedron|regular dodecahedron]] are amongst them, but crystals can have the shape of a [[Pyritohedron|pyritohedron]], which is visually almost indistinguishable from a regular dodecahedron. Truly icosahedral crystals may be formed by [[Physics:Quasicrystal|quasicrystalline 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 of [[Chemistry:Carbon|carbon]] molecule, known as the [[Chemistry:Fullerene|fullerene]]s (see Curl, 1991). Although C<sub>60</sub>, the most easily produced fullerene, looks more or less spherical, some of the larger varieties (such as C<sub>240</sub>, C<sub>480</sub> and C<sub>960</sub>) are hypothesised to take on the form of slightly rounded icosahedra, a few nanometres across.

Regular polyhedra appear in biology as well. The [[Biology:Coccolithophore|coccolithophore]] ''Braarudosphaera bigelowii'' has a regular dodecahedral structure, about 10 [[Micrometre|micrometre]]s across.<ref name=Hagino2013>Hagino, K., Onuma, R., Kawachi, M. and Horiguchi, T. (2013) "Discovery of an endosymbiotic nitrogen-fixing cyanobacterium UCYN-A in ''Braarudosphaera bigelowii'' (Prymnesiophyceae)". ''PLoS One'', '''8'''(12): e81749. {{doi|10.1371/journal.pone.0081749}}.</ref> In the early 20th century, [[Biography:Ernst Haeckel|Ernst Haeckel]] described a number of species of radiolarians, some of whose shells are shaped like various regular polyhedra.<ref name=Haeckel1904>Haeckel, E. (1904). ''[[Biology:Kunstformen der Natur|Kunstformen der Natur]]''. Available as Haeckel, E. ''Art forms in nature'', Prestel USA (1998), {{isbn|3-7913-1990-6}}. Online version at [http://www.biolib.de/haeckel/kunstformen/index.html Kurt Stüber's Biolib] (in german)</ref> Examples include ''Circoporus octahedrus'', ''Circogonia icosahedra'', ''Lithocubus geometricus'' and ''Circorrhegma dodecahedra''; the shapes of these creatures are indicated by their names.<ref name=Haeckel1904 /> The outer protein shells of many [[Biology:Virus|virus]]es form regular polyhedra. For example, [[Biology:HIV|HIV]] is enclosed in a regular icosahedron, as is the head of a typical myovirus.<ref>{{cite book | title=Virus Taxonomy | chapter=Myoviridae | publisher=Elsevier | year=2012 | pages=46–62 | doi=10.1016/b978-0-12-384684-6.00002-1 | isbn=9780123846846 | ref={{sfnref | Elsevier | 2012}}}}</ref><ref>{{cite book | last1=STRAUSS | first1=JAMES H. | last2=STRAUSS | first2=ELLEN G. | title=Viruses and Human Disease | chapter=The Structure of Viruses | publisher=Elsevier | year=2008 | pages=35–62 | doi=10.1016/b978-0-12-373741-0.50005-2| isbn=9780123737410 | s2cid=80803624 }}</ref>

<gallery mode=packed style=float:left heights=180px>
File:Braarudosphaera bigelowii.jpg| The [[Biology:Coccolithophore|coccolithophore]] ''Braarudosphaera bigelowii'' has a regular dodecahedral structure
File:Circogonia icosahedra.jpg| The radiolarian ''[[Biology:Circoporidae|Circogonia icosahedra]]'' has a regular icosahedral structure
File:Structure of a Myoviridae bacteriophage 2.jpg| A myovirus typically has a regular icosahedral [[Biology:Capsid|capsid]] (head) about 100 nanometers across.
</gallery>
{{clear}}

In ancient times the [[Philosophy:Pythagoreanism|Pythagoreans]] believed that there was a harmony between the regular polyhedra and the orbits of the [[Astronomy:Planet|planet]]s. In the 17th century, [[Biography:Johannes Kepler|Johannes Kepler]] studied data on planetary motion compiled by [[Biography:Tycho Brahe|Tycho 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, and [[Kepler's laws of planetary motion|the 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 of [[Astronomy:Uranus|Uranus]] and [[Astronomy:Neptune|Neptune]], 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 later Dalton's atomic theory would show this idea to be along the right lines, though not related directly to the regular solids.

== Further generalisations ==
The 20th century saw a succession of generalisations of the idea of a regular polyhedron, leading to several new classes.

=== Regular skew apeirohedra ===

{{Main|Regular skew apeirohedron}}

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 called [[Regular skew polyhedron|regular skew polyhedra]].<ref>Coxeter, ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, {{isbn|0-486-40919-8}} (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.)</ref> Coxeter offered a modified [[Schläfli symbol]] {l,m|n} for these figures, with {l,m} implying the [[Vertex figure|vertex figure]], with ''m'' regular ''l''-gons around a vertex. The ''n'' defines ''n''-gonal ''holes''. Their vertex figures are regular skew polygons, vertices zig-zagging between two planes.

{| class="wikitable"
!colspan=3| Infinite regular skew polyhedra in 3-space (partially drawn)
|- align=center
||[[Image:mucube external.png|150px]]<br>{4,6|4}
|[[Image:muoctahedron external.png|150px]]<br>{6,4|4}
|[[Image:mutetrahedron external.png|150px]]<br>{6,6|3}
|}

=== Regular skew polyhedra ===
{{Main|Regular skew polyhedron}}

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 of uniform 4-polytopes. They have planar [[Regular polygon|regular polygon]] faces, but regular skew polygon [[Vertex figure|vertex figure]]s.

Two dual solutions are related to the [[5-cell|5-cell]], two dual solutions are related to the [[24-cell|24-cell]], and an infinite set of self-dual [[Duoprism|duoprism]]s generate regular skew polyhedra as {4, 4 &#124; n}. In the infinite limit these approach a [[Duocylinder|duocylinder]] and look like a [[Torus|torus]] in their [[Stereographic projection|stereographic projection]]s into 3-space.

{| class=wikitable
|+ Finite regular skew polyhedra in 4-space
|-
!colspan=4|Orthogonal Coxeter plane projections
!rowspan=2|[[Stereographic projection]]
|-
!colspan=2| A<sub>4</sub>
!colspan=2| F<sub>4</sub>
|-
|[[File:4-simplex t03.svg|150px]]
|[[File:4-simplex t12.svg|150px]]
|[[File:24-cell t03 F4.svg|150px]]
|[[File:24-cell t12 F4.svg|150px]]
|[[File:Clifford-torus.gif|150px]]
|-
![[Runcinated 5-cell#Related skew polyhedron|{4, 6 &#124; 3}]]
![[Truncated 5-cell#Related skew polyhedron|{6, 4 &#124; 3}]]
!{4, 8 &#124; 3}
![[Truncated 24-cells#Related regular skew polyhedron|{8, 4 &#124; 3}]]
![[Duoprism#Related polytopes|{4, 4 &#124; n}]]
|-
!30 [[Square|{4}]] faces<BR>60 edges<BR>20 vertices
!20 [[Hexagon|{6}]] faces<BR>60 edges<BR>30 vertices
!288 {4} faces<BR>576 edges<BR>144 vertices
!144 [[Octagon|{8}]] faces<BR>576 edges<BR>288 vertices
!{{math|''n''<sup>2</sup>}} {4} faces<BR>{{math|2''n''<sup>2</sup>}} edges<BR>{{math|''n''<sup>2</sup>}} vertices
|}

=== Regular polyhedra in non-Euclidean and other spaces ===
Studies of non-Euclidean ([[Hyperbolic space|hyperbolic]] and elliptic) and other spaces such as [[Complex affine space|complex spaces]], discovered over the preceding century, led to the discovery of more new polyhedra such as complex polyhedra which could only take regular geometric form in those spaces.

==== Regular polyhedra in hyperbolic space ====

[[File:633 honeycomb one cell horosphere.png|thumb|The [[Hexagonal tiling honeycomb|hexagonal tiling honeycomb]], {6,3,3}, has [[Hexagonal tiling|hexagonal tiling]], {6,3}, facets with vertices on a [[Horosphere|horosphere]]. One such facet is shown in as seen in this [[Poincaré disk model|Poincaré disk model]].]]
In H<sup>3</sup> [[Hyperbolic space|hyperbolic space]], paracompact regular honeycombs have Euclidean tiling [[Facet (geometry)|facets]] and [[Vertex figure|vertex figure]]s that act like finite polyhedra. Such tilings have an angle defect that can be closed by bending one way or the other. If the tiling is properly scaled, it will ''close'' as an [[Asymptote|asymptotic limit]] at a single [[Ideal point|ideal point]]. These Euclidean tilings are inscribed in a [[Horosphere|horosphere]] 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 the [[Heptagonal tiling honeycomb|heptagonal tiling honeycomb]] {7,3,3}; they are inscribed in an equidistant surface (a 2-hypercycle), which has two ideal points.

==== Regular tilings of the real projective plane ====
Another group of regular polyhedra comprise tilings of the [[Real projective plane|real projective plane]]. These include the hemi-cube, hemi-octahedron, hemi-dodecahedron, and hemi-icosahedron. They are (globally) projective polyhedra, and are the projective counterparts of the [[Platonic solid]]s. 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.

{| class=wikitable
|- align=center
|[[File:Hemicube.svg|150px]]<br>Hemi-cube<br>{4,3}
|[[File:Hemioctahedron.png|150px]]<br>Hemi-octahedron<br>{3,4}
|[[File:Hemi-Dodecahedron2.PNG|150px]]<br>Hemi-dodecahedron<br>{3,5}
|[[File:Hemi-icosahedron.png|150px]]<br>Hemi-icosahedron<br>{5,3}
|}

These occur as dual pairs in the same way as the original Platonic solids do. Their Euler characteristics are all 1.

=== Abstract regular polyhedra ===
{{See|Abstract regular polytope}}

By now, polyhedra were firmly understood as three-dimensional examples of more general ''[[Polytope|polytope]]s'' in any number of dimensions. The second half of the century saw the development of abstract algebraic ideas such as [[Polyhedral combinatorics]], culminating in the idea of an [[Abstract polytope|abstract polytope]] as a [[Partially ordered set|partially ordered set]] (poset) of elements. The elements of an abstract polyhedron are its body (the maximal element), its faces, edges, vertices and the ''null polytope'' or empty set. These abstract elements can be mapped into ordinary space or ''realised'' as geometrical figures. Some abstract polyhedra have well-formed or ''faithful'' realisations, others do not. A ''flag'' 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 be ''regular'' 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 by H. S. M. Coxeter in his book ''Regular Polytopes'' (1977) and again by J. M. Wills in his paper "The combinatorially regular polyhedra of index 2" (1987). All five have C<sub>2</sub>×S<sub>5</sub> symmetry but can only be realised with half the symmetry, that is C<sub>2</sub>×A<sub>5</sub> or icosahedral symmetry.<ref>[http://homepages.wmich.edu/~drichter/regularpolyhedra.htm The Regular Polyhedra (of index two)], David A. Richter</ref><ref>[https://arxiv.org/abs/1005.4911 Regular Polyhedra of Index Two, I] Anthony M. Cutler, Egon Schulte, 2010</ref><ref>[https://www.researchgate.net/publication/225386108_Regular_Polyhedra_of_Index_Two_II Regular Polyhedra of Index Two, II]  Beitrage zur Algebra und Geometrie 52(2):357–387 · November 2010, Table 3, p.27</ref> They are all topologically equivalent to [[Toroid|toroid]]s. Their construction, by arranging ''n'' faces around each vertex, can be repeated indefinitely as tilings of the [[Hyperbolic geometry#Models of the hyperbolic plane|hyperbolic plane]]. In the diagrams below, the hyperbolic tiling images have colors corresponding to those of the polyhedra images.

:{| class="wikitable" width=400
|- align=center
! Polyhedron
|[[Image:DU36 medial rhombic triacontahedron.png|100px]]<br>Medial rhombic triacontahedron
|[[Image:Dodecadodecahedron.png|100px]]<br>[[Dodecadodecahedron]]
|[[Image:DU41 medial triambic icosahedron.png|100px]]<br>Medial triambic icosahedron
|[[Image:Ditrigonal dodecadodecahedron.png|100px]]<br>[[Ditrigonal dodecadodecahedron]]
|[[Image:Excavated dodecahedron.png|100px]]<br>Excavated dodecahedron
|- align=center
!Type
||Dual {5,4}<sub>6</sub> ||{5,4}<sub>6</sub> ||Dual of {5,6}<sub>4</sub> ||{5,6}<sub>4</sub> || {6,6}<sub>6</sub>
|- align=center
!(''v'',''e'',''f'')
|(24,60,30) ||(30,60,24) ||(24,60,20) ||(20,60,24) ||(20,60,20)
|- align=center
![[Vertex figure]]
|{5}, {5/2}<br>[[File:Regular pentagon.svg|40px]][[File:Pentagram green.svg|40px]]
|(5.5/2)<sup>2</sup><br>[[File:Dodecadodecahedron vertfig.png|60px]]
|{5}, {5/2}<br>[[File:Regular pentagon.svg|40px]][[File:Pentagram green.svg|40px]]
|(5.5/3)<sup>3</sup><br>[[File:Ditrigonal dodecadodecahedron vertfig.png|60px]]
|[[File:Medial triambic icosahedron face.svg|60px]]
|- align=center valign=top
!Faces
|30 rhombi<br>[[File:Rhombus definition2.svg|60px]]
|12 pentagons<br>12 pentagrams<br>[[File:Regular pentagon.svg|40px]][[File:Pentagram green.svg|40px]]
|20 hexagons<br>[[File:Medial triambic icosahedron face.svg|60px]]
|12 pentagons<br>12 pentagrams<br>[[File:Regular pentagon.svg|40px]][[File:Pentagram green.svg|40px]]
|20 hexagrams<br>[[File:Star hexagon face.png|60px]]
|- align=center
! Tiling
|[[Image:Uniform tiling 45-t0.png|100px]]<br>[[Order-5 square tiling|{4, 5}]]
|[[Image:Uniform tiling 552-t1.png|100px]]<br>[[Order-4 pentagonal tiling|{5, 4}]]
|[[Image:Uniform tiling 65-t0.png|100px]]<br>[[Order-5 hexagonal tiling|{6, 5}]]
|[[Image:Uniform tiling 553-t1.png|100px]]<br>[[Order-6 pentagonal tiling|{5, 6}]]
|[[Image:Uniform tiling 66-t2.png|100px]]<br>[[Order-6 hexagonal tiling|{6, 6}]]
|- align=center
! [[Euler characteristic|χ]]
| −6
| −6
| −16
| −16
| −20
|}

==== Petrie dual====
{{main|Petrie dual}}
The [[Petrie dual]] of a regular polyhedron is a [[Regular map (graph theory)|regular map]] whose vertices and edges correspond to the vertices and edges of the original polyhedron, and whose faces are the set of [[Skew polygon|skew]] [[Petrie polygon]]s.<ref>{{citation|title=Abstract Regular Polytopes|volume=92|series=Encyclopedia of Mathematics and its Applications|first1=Peter|last1=McMullen|first2=Egon|last2=Schulte|publisher=Cambridge University Press|year=2002|isbn=9780521814966|page=192|url=https://books.google.com/books?id=JfmlMYe6MJgC&pg=PA192}}</ref>

{| class=wikitable
|+ Regular petrials
!Name
!Petrial tetrahedron<BR>
!Petrial cube
!Petrial octahedron
!Petrial dodecahedron
!Petrial icosahedron
|- align=center
!Symbol
|{3,3}<sup>{{pi}}</sup>
|{4,3}<sup>{{pi}}</sup>
|{3,4}<sup>{{pi}}</sup>
|{5,3}<sup>{{pi}}</sup>
|{3,5}<sup>{{pi}}</sup>
|- align=center
!(''v'',''e'',''f''), [[Euler characteristic|''&chi;'']]
|(4,6,3), ''&chi;'' = 1||(8,12,4), ''&chi;'' = 0||(6,12,4), ''&chi;'' = −2||(20,30,6), ''&chi;'' = −4||(12,30,6), ''&chi;'' = −12
|- align=center
!rowspan=2|Faces
|rowspan=2|3 skew squares<br/>[[File:Face_of_petrial_tetrahedron.gif|120px]]
|colspan=2 style="border-bottom-style:none;"|4 skew hexagons
|colspan=2 style="border-bottom-style:none;"|6 skew decagons
|- align=center
|style="border-top-style:none;"|[[File:Face_of_petrial_cube.gif|120px]]
|style="border-top-style:none;"|[[File:Face_of_petrial_octahedron.gif|120px]]
|style="border-top-style:none;"|[[File:Face_of_petrial_dodecahedron.gif|120px]]
|style="border-top-style:none;"|[[File:Face_of_petrial_icosahedron.gif|120px]]
|- align=center
!Image
|[[File:Tetrahedron_3_petrie_polygons.png|120px]]
|[[File:Cube_4_petrie_polygons.png|120px]]
|[[File:Octahedron_4_petrie_polygons.png|120px]]
|[[File:Petrial_dodecahedron.png|120px]]
|[[File:petrial_icosahedron.png|120px]]
|- align=center
!Animation
|[[File:Petrial_tetrahedron.gif|120px]]
|[[File:Petrial_cube.gif|120px]]
|[[File:Petrial octahedron.gif|120px]]
|[[File:Petrial_dodecahedron.gif|120px]]
|[[File:petrial_icosahedron.gif|120px]]
|- align=center valign=bottom
!Related<BR>figures
|[[File:Hemicube.svg|120px]]<BR>{4,3}<sub>3</sub> = {4,3}/2 = {4,3}<sub>(2,0)</sub>
|[[File:Regular map 6-3 2-0.png|120px]]<BR>{6,3}<sub>3</sub> = {6,3}<sub>(2,0)</sub>
|[[File:Regular_map_6_4-3_pattern.png|120px]]<BR>{6,4}<sub>3</sub> = {6,4}<sub>(4,0)</sub>
|{10,3}<sub>5</sub>
|{10,5}<sub>3</sub>
|}

=== Spherical polyhedra ===
{{main|Spherical polyhedron}}

The usual five regular polyhedra can also be represented as spherical tilings (tilings of the [[Sphere|sphere]]):

{| class=wikitable width=640
|- align=center
|[[File:Uniform tiling 332-t2.png|100px]]<br>[[Tetrahedron]]<br>{3,3}
|[[File:Uniform tiling 432-t0.png|100px]]<br>[[Cube]]<br>{4,3}
|[[File:Uniform tiling 432-t2.png|100px]]<br>[[Octahedron]]<br>{3,4}
|[[File:Uniform tiling 532-t0.png|100px]]<br>[[Dodecahedron]]<br>{5,3}
|[[File:Uniform tiling 532-t2.png|100px]]<br>[[Regular icosahedron|Icosahedron]]<br>{3,5}
|}
{| class=wikitable width=640
|- align=center
|[[File:Small stellated dodecahedron tiling.png|100px]]<br>[[Small stellated dodecahedron]]<br>{5/2,5}
|[[File:Great dodecahedron tiling.svg|100px]]<br>[[Great dodecahedron]]<br>{5,5/2}
|[[File:Great stellated dodecahedron tiling.svg|100px]]<br>[[Great stellated dodecahedron]]<br>{5/2,3}
|[[File:Great icosahedron tiling.svg|100px]]<br>[[Great icosahedron]]<br>{3,5/2}
|}

==== Regular polyhedra that can only exist as spherical polyhedra ====
For a regular polyhedron whose Schläfli symbol is {''m'',&nbsp;''n''}, the number of polygonal faces may be found by:

:<math>N_2=\frac{4n}{2m+2n-mn}</math>

The [[Platonic solid]]s known to antiquity are the only integer solutions for ''m'' ≥ 3 and ''n'' ≥ 3. The restriction ''m'' ≥ 3 enforces that the polygonal faces must have at least three sides.

When considering polyhedra as a spherical tiling, this restriction may be relaxed, since [[Digon|digon]]s (2-gons) can be represented as spherical lunes, having non-zero area. Allowing ''m'' = 2 admits a new infinite class of regular polyhedra, which are the [[Hosohedron|hosohedra]]. On a spherical surface, the regular polyhedron {2,&nbsp;''n''} is represented as ''n'' abutting lunes, with interior angles of 2{{pi}}/''n''. All these lunes share two common vertices.<ref name=cox>Coxeter, ''Regular polytopes'', p. 12</ref>

A regular [[Dihedron|dihedron]], {''n'',&nbsp;2}<ref name=cox /> (2-hedron) in three-dimensional [[Euclidean space]] can be considered a [[Degeneracy (mathematics)|degenerate]] [[Prism (geometry)|prism]] consisting of two (planar) ''n''-sided [[Polygon|polygon]]s connected "back-to-back", so that the resulting object has no depth, analogously to how a digon can be constructed with two [[Line segment|line segment]]s. However, as a spherical tiling, a dihedron can exist as nondegenerate form, with two ''n''-sided faces covering the sphere, each face being a [[Sphere|hemisphere]], and vertices around a [[Great circle|great circle]]. It is ''regular'' if the vertices are equally spaced.

{| class=wikitable
|- align=center valign=bottom
|[[File:Digonal dihedron.png|100px]]<br>[[Digon]]al dihedron<br>{2,2}
|[[File:Trigonal dihedron.png|100px]]<br>[[Triangle|Trigonal]] dihedron<br>{3,2}
|[[File:Tetragonal dihedron.png|100px]]<br>[[Square]] dihedron<br>{4,2}
|[[File:Pentagonal dihedron.png|100px]]<br>[[Pentagon]]al dihedron<br>{5,2}
|[[File:Hexagonal dihedron.png|100px]]<br>[[Hexagon]]al dihedron<br>{6,2}
|...
|{''n'',2}
|- align=center valign=bottom
|[[File:Digonal dihedron.png|100px]]<br>Digonal hosohedron<br>{2,2}
|[[File:Trigonal hosohedron.png|100px]]<br>Trigonal hosohedron<br>{2,3}
|[[File:Spherical square hosohedron.svg|100px]]<br>Square hosohedron<br>{2,4}
|[[File:Spherical pentagonal hosohedron.svg|100px]]<br>Pentagonal hosohedron<br>{2,5}
|[[File:Hexagonal hosohedron.png|100px]]<br>Hexagonal hosohedron<br>{2,6}
|...
|{2,''n''}
|}

The hosohedron {2,''n''} is dual to the dihedron {''n'',2}. Note that when ''n'' = 2, we obtain the polyhedron {2,2}, which is both a hosohedron and a dihedron. All of these have Euler characteristic 2.

== See also ==
* [[Quasiregular polyhedron]]
* [[Semiregular polyhedron]]
* [[Uniform polyhedron]]
* [[Regular polytope]]

== References ==
{{reflist}}
* Bertrand, J. (1858). Note sur la théorie des polyèdres réguliers, ''Comptes rendus des séances de l'Académie des Sciences'', '''46''', pp.&nbsp;79–82.
* Haeckel, E. (1904). ''[[Biology:Kunstformen der Natur|Kunstformen der Natur]]''. Available as Haeckel, E. ''Art forms in nature'', Prestel USA (1998), {{isbn|3-7913-1990-6}}, or online at http://caliban.mpiz-koeln.mpg.de/~stueber/haeckel/kunstformen/natur.html
*Smith, J. V. (1982). ''Geometrical And Structural Crystallography''. John Wiley and Sons.
* Sommerville, D. M. Y. (1930). ''An Introduction to the Geometry of n Dimensions.'' E. P. Dutton, New York. (Dover Publications edition, 1958). Chapter X: The Regular Polytopes.
*Coxeter, H.S.M.; Regular Polytopes (third edition). Dover Publications Inc. {{isbn|0-486-61480-8}}
*[https://www.youtube.com/watch?v=_hjRvZYkAgA 'there are 48 regular polyhedra' jan Misali]

== External links ==
* {{MathWorld |urlname=RegularPolyhedron |title=Regular Polyhedron}}

{{DEFAULTSORT:Regular Polyhedron}}
[[Category:Regular polyhedra| ]]

{{Sourceattribution|Regular polyhedron}}