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A '''regular polyhedron''' is a [[polyhedron]] whose faces are [[Congruence (geometry)|congruent]] [[regular polygons]] which are assembled in the same way around each [[vertex (geometry)|vertex]]. A regular polyhedron is highly symmetrical, being all of [[edge-transitive]], [[vertex-transitive]] and [[face-transitive]] - i.e. it is transitive on its [[Flag (geometry)|flag]]s. This last alone is a sufficient definition.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


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 regular polyhedra, which are called the [[Platonic solid]]s, the self-dual [[tetrahedron]] {3,3}, dual-pair [[cube]]/[[octahedron]] {4,3}, and dual pair [[dodecahedron]]/[[icosahedron]] {5,3}.
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
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== The regular polyhedra ==
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There are  five [[Convex polygon|convex]] regular polyhedra, known as the '''[[Platonic solid]]s''', and four regular [[star polyhedron|star polyhedra]], the '''[[Kepler-Poinsot polyhedra]]''':
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


=== Platonic solids ===
<!--'''PNG'''  (currently default in production)
{{Main|Platonic solid}}
:<math forcemathmode="png">E=mc^2</math>


:{| class="wikitable"
'''source'''
|- align=center
:<math forcemathmode="source">E=mc^2</math> -->
|[[Image:Tetrahedron.png|100px]]
|[[Image:Hexahedron.png|100px]]
|[[Image:Octahedron.png|100px]]
|[[Image:Dodecahedron.png|100px]]
|[[Image:Icosahedron.png|100px]]
|- align=center
|[[Tetrahedron]] {3,&nbsp;3}||[[Cube]] {4,&nbsp;3}||[[Octahedron]] {3,&nbsp;4}||[[Dodecahedron]] {5,&nbsp;3}||[[Icosahedron]] {3,&nbsp;5}
|}


=== Kepler-Poinsot polyhedra ===
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


{{Main|Kepler-Poinsot polyhedra}}
==Demos==


:{| class="wikitable"
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
|- align=center
|[[Image:Small stellated dodecahedron.png|100px]]
|[[Image:Great dodecahedron.png|100px]]
|[[Image:Great stellated dodecahedron.png|100px]]
|[[Image:Great icosahedron.png|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}
|}


== 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:
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


*The vertices of the polyhedron all lie on a [[sphere]].
==Test pages ==
*All the [[dihedral angle]]s of the polyhedron are equal.
*All the [[vertex figure]]s of the polyhedron are [[regular polygon]]s.
*All the [[solid angle]]s of the polyhedron are congruent. (Cromwell, 1997)


===Concentric spheres===
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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*[[Help:Formula]]


A regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre:
*[[Inputtypes|Inputtypes (private Wikis only)]]
* An [[insphere]], tangent to all faces.
*[[Url2Image|Url2Image (private Wikis only)]]
* An intersphere or [[midsphere]], tangent to all edges.
==Bug reporting==
* A [[circumsphere]], tangent to all vertices.
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
 
===Symmetry===
 
The regular polyhedra are the most [[symmetry|symmetrical]] of all the polyhedra. They lie in just three [[symmetry group]]s, which are named after them:
*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. Some of the regular stars have a [[Kepler-Poinsot polyhedra#The Euler characteristic|different value]].
 
===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", ''[[The College Mathematics Journal]]'' 37(5), 2006, pp. 390–391.</ref>
 
== Duality of the regular polyhedra ==
 
The regular polyhedra come in natural pairs, with each twin being [[dual polyhedron|dual]] to the other (i.e. the vertices of one polyhedron correspond to the faces of the other, and vice versa):
* The [[tetrahedron]] is self-dual, i.e. it pairs with itself.
* The [[cube]] and [[octahedron]] are dual to each other.
* The [[icosahedron]] and [[dodecahedron]] are dual to each other.
* The [[small stellated dodecahedron]] and [[great dodecahedron]] are dual to each other.
* The [[great stellated dodecahedron]] and [[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}.
 
For further information please see the individual articles or the general [[polyhedron]] article.
 
==History==<!-- This section is linked from [[Polyhedron]] -->
 
'''See also''' [[Regular polytope#History of discovery|''Regular polytope: History of discovery'']].
 
=== 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/index.php/the-scottish-solids-hoax.html The Scottish Solids Hoax], </ref> 
 
It is also possible that the [[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]] made of [[soapstone]], and dating back more than 2,500 years (Lindemann, 1987). [[#Polyhedra|Pyritohedric crystals]] are dodecahedral (though not quite regular) and may be found in northern Italy{{Citation needed|date=February 2007}}.
 
=== Greeks ===
 
The earliest known ''written'' records of the regular convex solids originated from Classical Greece. When these solids were all discovered and who by is not known, but [[Theaetetus (mathematician)|Theaetetus]], (an [[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 [[Plato]] (400 BC) with having made models of them, and mentions that one of the earlier [[Pythagoreans]], [[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 [[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]] (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]] 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]] (star pentagon) were also known to the ancient Greeks - the [[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 [[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]]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]] and [[great stellated dodecahedron]], and (Poinsot's) [[great icosahedron]] and [[great dodecahedron]].
 
The Kepler-Poinsot polyhedra may be constructed from the Platonic solids by a process called [[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 [[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]] nor the [[regular dodecahedron]] are amongst them, although one of the forms, called the [[pyritohedron]], 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.
 
[[File:Circogoniaicosahedra ekw.jpg|left|frame|Circogonia icosahedra, a species of [[Radiolaria]].]]
 
Polyhedra appear in biology as well. In the early 20th century, [[Ernst Haeckel]] described a number of species of [[Radiolaria]], some of whose skeletons are shaped like various regular polyhedra. (Haeckel, 1904) Examples include ''Circoporus octahedrus'', ''Circogonia icosahedra'', ''Lithocubus geometricus'' and ''Circorrhegma dodecahedra''; the shapes of these creatures are indicated by their names. The outer protein shells of many [[virus]]es form regular polyhedra. For example, [[HIV]] is enclosed in a regular icosahedron.
 
A more recent discovery is of a series of new types of [[carbon]] molecule, known as the [[fullerene]]s (see (Curl, 1991) for an exposition of this discovery). 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.
 
In ancient times the [[Pythagoreanism|Pythagoreans]] believed that there was a harmony between the regular polyhedra and the orbits of the [[planet]]s. In the 17th century, [[Johannes Kepler]] studied data on planetary motion compiled by [[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 [[Uranus]] and [[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 polyhedra ===
 
{{Main|Regular skew polyhedron}}
 
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]], with ''m'' regular ''l''-gons around a vertex. The ''n'' defines ''n''-gonal ''holes''. Their vertex figures are [[regular skew polygon]]s, vertices zig-zagging between two planes.
 
{| class="wikitable"
!colspan=3| Infinite regular skew polyhedra in 3-space (partially drawn)
|-
|align=center|[[Image:Six-square skew polyhedron.png|210px]]<br>{4,6|4}
|align=center|[[Image:Four-hexagon skew polyhedron.png|210px]]<br>{6,4|4}
|align=center|[[Image:Six-hexagon skew polyhedron.png|210px]]<br>{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 of [[uniform polychoron]]s. Two dual solutions are related to the [[5-cell]], two dual solutions are related to the [[24-cell]], and an infinite set of self-dual [[duoprism]]s generate regular skew polyhedra as {4, 4 &#124; n}. In the infinite limit these approach a [[duocylinder]] and look like a [[torus]] in their [[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}]]
![[Bitruncated 5-cell#Related skew polyhedron|{6, 4 &#124; 3}]]
![[Runcinated 24-cell#Related skew polyhedron|{4, 8 &#124; 3}]]
![[Bitruncated 24-cell#Related skew polyhedron|{8, 4 &#124; 3}]]
![[Duoprism#Related polytopes|{4, 4 &#124; n}]]
|}
 
=== Regular polyhedra in non-Euclidean spaces ===
Studies of non-Euclidean [[hyperbolic space|hyperbolic]], [[elliptic space|elliptic]] and [[complex space|complex]] spaces, discovered over the preceding century, led to the  discovery of more new polyhedra such as [[complex polytope|complex polyhedra]] which could only take regular geometric form in those spaces.
 
=== Abstract regular polyhedra ===
{{Main|Abstract regular polyhedron}}
 
By now, polyhedra were firmly understood as three-dimensional examples of more general ''[[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]] as a [[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).<ref>[http://homepages.wmich.edu/~drichter/regularpolyhedra.htm The Regular Polyhedra (of index two)], David A. Richter</ref> They are all topologically equivalent to [[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 (except for the [[ditrigonal dodecadodecahedron]], where the correct image is not available yet).
 
:{| 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
![[Vertex figure]]
|{5}, {5/2}<br>[[File:Regular polygon 5.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 polygon 5.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.png|60px]]
|- align=center valign=top
!Faces
|30 rhombi<br>[[File:Rhombus definition2.svg|60px]]
|12 pentagons<br>12 pentagram<br>[[File:Regular polygon 5.svg|40px]][[File:Pentagram green.svg|40px]]
|20 hexagons<br>[[File:Medial triambic icosahedron face.png|60px]]
|12 pentagons<br>12 pentagrams<br>[[File:Regular polygon 5.svg|40px]][[File:Pentagram green.svg|40px]]
|20 hexagons<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>{6, 5}
|[[Image:Uniform tiling 56-t0.png|100px]]<br>{5, 6}
|[[Image:Uniform tiling 66-t0.png|100px]]<br>{6, 6}
|}
 
These occur as dual pairs as follows:
*The [[medial rhombic triacontahedron]] and [[dodecadodecahedron]] are dual to each other.
*The [[medial triambic icosahedron]] and [[ditrigonal dodecadodecahedron]] are dual to each other.
*The [[excavated dodecahedron]] is self-dual.
 
==== Tilings of the real projective plane ====
Another group of regular abstract polyhedra comprise tilings of the [[real projective plane]]. These include the [[Hemi-cube (geometry)|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 parallel faces which can be identified as one, as the other four Platonic solids do.
 
{| class=wikitable
|- align=center
|[[File:Hemicube2.PNG|150px]]<br>[[Hemicube (geometry)|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.
 
=== Spherical polyhedra ===
{{main|Spherical polyhedron}}
 
The usual nine regular polyhedra can also be represented as spherical tilings (tilings of the [[sphere]]):
 
{| class=wikitable
|- align=center
|[[File:Uniform tiling 332-t0-1-.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>[[Icosahedron]]<br>{3,5}
|}
{| class=wikitable
|- align=center
|[[File:Small stellated dodecahedron tiling.png|100px]]<br>[[Small stellated dodecahedron]]<br>{5/2,5}
|[[File:Great dodecahedron tiling.png|100px]]<br>[[Great dodecahedron]]<br>{5,5/2}
|[[File:Great stellated dodecahedron tiling.png|100px]]<br>[[Great stellated dodecahedron]]<br>{5/2,3}
|[[File:Great icosahedron tiling.png|100px]]<br>[[Great icosahedron]]<br>{3,5/2}
|}
 
==== Regular polyhedra that can only exist as spherical polyhedra ====
{{see also|Hosohedron|Dihedron}}
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]]s (2-gons) can be represented as spherical lunes, having non-zero [[area (geometry)|area]]. Allowing ''m'' = 2 admits a new infinite class of regular polyhedra, which are the [[hosohedron|hosohedra]]. On a spherical surface, the polyhedron {2,&nbsp;''n''} is represented as n abutting lunes, with interior angles of 2π/''n''. All these lunes share two common vertices.
 
A [[dihedron]] (2-hedron) in three-dimensional [[Euclidean space]] can be considered a [[degeneracy (mathematics)|degenerate]] [[Prism (geometry)|prism]] consisting of two (planar) ''n''-sided [[polygon]]s connected "back-to-back", so that the resulting object has no depth. 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]]. It is ''regular'' if the vertices are equally spaced.
 
{| class=wikitable
|- align=center
|[[File:Hengonal dihedron.png|100px]]<br>[[Henagon]]al dihedron<br>{1,2}
|[[File:Digonal dihedron.png|100px]]<br>[[Digon]]al dihedron<br>{2,2}
|[[File:Trigonal dihedron.png|100px]]<br>[[Triangle|Trigon]] dihedron<br>{3,2}
|...
|[[File:Hexagonal dihedron.png|100px]]<br>[[Hexagon]]al dihedron<br>{6,2}
|...
|{''n'',2}
|- align=center
|[[File:Digonal hosohedron.png|100px]]<br>Henagonal hosohedron<br>{2,1}
|[[File:Digonal dihedron.png|100px]]<br>Digonal hosohedron<br>{2,2}
|[[File:Trigonal hosohedron.png|100px]]<br>Trigonal hosohedron<br>{2,3}
|...
|[[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.
 
== See also ==
* [[Quasiregular polyhedron]]
* [[Semiregular polyhedron]]
* [[Uniform polyhedron]]
* [[Regular polytope]]
 
== References ==
{{reflist}}
 
* [[Joseph Louis François Bertrand|Bertrand, J.]] (1858). Note sur la théorie des polyèdres réguliers, ''Comptes rendus des séances de l'Académie des Sciences'', '''46''', pp.&nbsp;79–82.
*{{cite book
  | last = Cromwell | first = Peter R. | title = Polyhedra | publisher = Cambridge University Press | year = 1997 | page = 77 | isbn = 0-521-66405-5 }}
* Haeckel, E. (1904). ''[[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.
* [[Duncan MacLaren Young Sommerville|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.
 
== External links ==
* {{MathWorld |urlname=RegularPolyhedron |title=Regular Polyhedron}}
 
{{DEFAULTSORT:Regular Polyhedron}}
[[Category:Regular polyhedra| ]]
 
[[ca:Políedre regular]]
[[el:Κανονικό πολύεδρο]]
[[et:Korrapärane hulktahukas]]
[[es:Poliedro regular]]
[[eo:Regula pluredro]]
[[fr:Polyèdre régulier]]
[[no:Regulært polyeder]]
[[sq:Poliedri i rregullt]]
[[fi:Säännöllinen monitahokas]]
[[th:ทรงหลายหน้าปรกติ]]
[[vi:Khối đa diện đều]]

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