Regular skew polyhedron
There are 48 nondegenerate nondense regular polyhedra in 3D Euclidean space, 36 of which are skew.^{[note 1]} There are also skew polyhedra in other spaces such has hyperbolic space and higher dimensional Euclidean space.
Finite skew polyhedra[edit  edit source]
Petrie duals[edit  edit source]
The Petrie dual, or Petrial, of a polytope can take any regular polyhedron and transform it into one sharing edges and vertices with the original, but with skew faces. Because of this, there is a Petrie dual to every finite nonskew regular polyhedron. Applying the Petrie dual twice gives back the original polyhedron.
Skew polyhedron  Schläfli symbol  Faces  Image  Petrie dual  Kappa  

Petrial tetrahedron  {3,3}^{π}  {4,3}_{3}  3 skew squares  Tetrahedron  Cube  
Petrial cube  {4,3}^{π}  {6,3}_{4}  4 skew hexagons  Cube  Tetrahedron  
Petrial octahedron  {3,4}^{π}  {6,4}_{3}  4 skew hexagons  Octahedron  Octahedron  
Petrial dodecahedron  {5,3}^{π}  {10,3}_{5}  6 skew decagons  Dodecahedron  Great stellated dodecahedron  
Petrial icosahedron  {3,5}^{π}  {10,5}_{3}  6 skew decagons  Icosahedron  Small stellated dodecahedron  
Petrial great dodecahedron  {5,5/2}^{π}  {6,5/2}_{5}  10 skew hexagons  Great dodecahedron  Great icosahedron  
Petrial small stellated dodecahedron  {5/2,5}^{π}  {6,5}_{5/2}  10 skew hexagons  Small stellated dodecahedron  Icosahedron  
Petrial great icosahedron  {3,5/2}^{π}  {10/3,5/2}_{3}  4 skew decagrams  Great icosahedron  Great dodecahedron  
Petrial great stellated dodecahedron  {5/2,3}^{π}  {10/3,3}_{5/2}  4 skew decagrams  Great stellated dodecahedron  Dodecahedron 
There are multiple extensions to Schläfli symbols which allow the Petrials to be represented:
 The Petrie dual of a base polytope 𝓟 operation can be represented by adding a superscript π to a polyhedron, 𝓟π
 We also may write {p , q }_{r } where:
 {p} is the face of the polytope.^{[note 2]}
 {q} is the vertex figure of the polytope.
 {r} is the Petrie polygon of the polytope, i.e. the face of the base polytope of which it is the Petrie dual.
Kappas[edit  edit source]
An alternative alternative way to construct this set is via the kappa operation. The kappa operation produces the same set of skew polyhedra, when applied to the KeplerPoinsot polyhedra, but with a different pairing.
 Tetrahedron <> Petrial cube
 Cube <> Petrial tetrahedron
 Octahedron <> Petrial octahedron
 Dodecahedron <> Petrial great stellated dodecahedron
 Icosahedron <> Petrial small stellated dodecahedron
 Great stellated dodecahedron <> Petrial dodecahedron
 Small stellated dodecahedron <> Petrial icosahedron
 Great dodecahedron <> Petrial great icosahedron
 Great icosahedron <> Petrial great dodecahedron
The kappa operation is less intuitive that the Petrial, however it generalizes better. The only finite planar polytope with a Petrial beyond rank 3 is the tesseract, while kappa polytopes of full rank come in pairs in every rank. And with the exception of rank 4 the applying the kappa to the finite planar polytopes is sufficient to produce all the finite polytopes of full rank.^{[3]}
Proof of completeness[edit  edit source]
This list can be shown complete using the same method used in the proof there are 9 finite planar regular polyhedra, the initial lemma applies to finite skews as well, so to modify the proof we simply don't discard skew polygons in the search.
Skew honeycombs[edit  edit source]
There are 3 regular skew apeirohedra of full rank, also called regular skew honeycombs. These are the regular skew apeirohedra in 2dimensions. As with the finite skew polyhedra of full rank, all three of these can be obtained by applying the Petrie dual to planar polytopes, in this case the three regular tilings.^{[4]}^{[5]}^{[6]}
Alternatively they can be constructed using the apeir operation on regular polygons.^{[7]} While the Petrial is used the classical construction, it does not generalize well to higher ranks. In contrast, the apeir operation is used to construct higher rank skew honeycombs.^{[8]}
Skew honeycomb  Schläfli symbol  Faces  Image  Petrie dual  Apeir of  

Petrial square tiling  {4,4}^{π}  {∞,4}_{4}  ∞ zigzags  Square tiling  Square  
Petrial triangular tiling  {3,6}^{π}  {∞,6}_{3}  ∞ zigzags  Triangular tiling  Hexagon  
Petrial hexagonal tiling  {6,3}^{π}  {∞,3}_{6}  ∞ zigzags  Hexagonal tiling  Triangle 
The apeir of any other planar polygon produces a dense skew honeycomb.
Blends[edit  edit source]
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We can also create new regular skew polyhedra by taking the 6 2dimensional polyhedra (the tilings of the Euclidean plane and their Petrials) and blending them with a 1dimensional polygon, either a digon or an apeirogon^{[note 3]}
Pure apeirohedra[edit  edit source]
In Euclid's proof a polyhedron with 6 square faces at each vertex is ruled out because its angular defect, 3π , would make it hyperbolic. However it is possible to arrange 6 squares at a vertex such that adjacent squares meet at the same angle.
This configuration alternates between "concave" and "convex" edges. In order for this to be the vertex of a regular polyhedron there has to be a symmetry taking the each convex edge to each concave edge. For a traditional polytope this is impossible, one side must be the "inside" and one side must be the "outside", and thus you can't bring a convex edge to a concave edge or vice versa. However for skew polyhedra there is no such requirement. If we take this vertex and repeat it indefinitely we indeed get a regular skew polyhedron, the mucube, .
In total, there are 3 possible configurations of planar polygons that work:
 Schläfli type {4,6} gives the mucube,
 Schläfli type {6,4} gives the muoctahedron,
 Schläfli type {6,6} gives the mutetrahedron,

A section of the mucube

A section of the muoctahedron

A section of the mutetrahedron
These three regular polyhedra are called the PetrieCoxeter polyhedra. The first two polyhedra were discovered by John Flinders Petrie and Coxeter found the final polyhedron shortly after.
Using finite skew polygons as faces gives three more apeirohedra:
 Schläfli type gives the halved mucube,
 Schläfli type gives the Petrial halved mucube,
 Schläfli type gives the skewed Petrial muoctahedron,

6 skew hexagons in a hexagonal arrangement form the vertex of the halved mucube.

6 skew squares in a hexagonal arrangement form the vertex of the Petrial halved mucube.

4 skew hexagons in a square arrangement form the vertex of the skewed Petrial muoctahedron.
Helical faces[edit  edit source]
In addition to the above there are 6 polyhedra with helical faces. Three of them are formed as Petrie duals of the pure apeirohedra with flat faces:
There is one polyhedron obtained from skewing the muoctahedron that is selfPetrie:
And the remaining two are mutually Petrial:
Unlike the other regular polyhedra these last two cannot be expressed as a simple identification of points along holes and zigzags, so they are given adhoc Schläfli symbols.
Regular pure apeirohedra in 3D hyperbolic space[edit  edit source]
The 3 pure apeirohedra with planar faces each consist of faces taken from a uniform honeycomb:
 Mucube from the cubic honeycomb
 Muoctahedron from the bitruncated cubic honeycomb
 Mutetrahedron form the cyclotruncated tetrahedraloctahedral honeycomb
Upon careful observation we can see that the polyhedron {2r,2q∣p} consists of faces from . For the muoctahedron and mutetrahedron it is exactly the 2r gonal faces of the honeycomb. For the the mucube there are two ways to get square faces from , selecting only those created by gives the faces of the mucube.
These three uniform tilings are the only uniform tilings in Euclidean space fitting this form, however there are more in hyperbolic space. If we generalize the process, we find 31 additional regular skew apeirohedra with planar faces with compact and paracompact symmetry.
Symmetry  Compactness  Schläfli symbol  φ 2 

Compact  {4,6∣5}  {5,3}  
{6,4∣5}  {5,2}  
Compact  {6,10∣3}  {3,5}  
{10,6∣3}  {3,3}  
Compact  {8,10∣3}  {3,5}  
{10,8∣3}  {3,4}  
Compact  {10,4∣3}  {3,2}  
{4,10∣3}  {3,5}  
Compact  {6,8∣3}  {3,4}  
{8,6∣3}  {3,3}  
Compact  {6,6∣4}  {4,3}  
{8,8∣3}  {3,4}  
Compact  {6,6∣5}  {5,3}  
{10,10∣3}  {3,5}  
Paracompact  {4,6∣6}  {6,3}  
{6,4∣6}  {6,2}  
Paracompact  {4,8∣4}  {4,4}  
{8,4∣4}  {4,2}  
Paracompact  {4,12∣3}  {3,6}  
{12,4∣3}  {3,2}  
Paracompact  {6,8∣4}  {4,4}  
{8,6∣4}  {4,3}  
Paracompact  {6,12∣3}  {3,6}  
{12,6∣3}  {3,3}  
Paracompact  {8,12∣3}  {3,6}  
{12,8∣3}  {3,4}  
Paracompact  {10,12∣3}  {3,6}  
{12,10∣3}  {3,5}  
Paracompact  {6,6∣6}  {6,3}  
{12,12∣3}  {3,6}  
Paracompact  {8,8∣4}  {4,4} 
This list completely enumerates the regular skew apeirohedra with planar faces and without selfintersections that have compact or paracompact hyperbolic symmetry.^{[9]} There are an infinite number of noncompact regular skew apeirohedra meeting this requirement.
The classification of the regular skew polyhedra in 3D hyperbolic space (including skew faces and selfintersection) is currently incomplete.
4dimensional skew polyhedra[edit  edit source]
Just as we can make skew polygons whose vertices lie in 3D space, polyhedra can be made with points lying in 4D space. There are an infinite number of distinct skew polyhedra in 4D space.
Blends[edit  edit source]
With four dimensions it is possible to create a much wider variety of blends than three. In three dimensions the blend has to be made of two polytopes whose spanning dimensions add up to 3, this can only be achieved by blending a flat 2dimensional Euclidean tiling with a flat 1dimensional polygon, either a digon or a apeirogon. However there are two ways to achieve a 4dimensional blend:
 Blend a 3dimensional polyhedron with a flat 1dimensional polygon
 Blend a flat 2dimensional Euclidean tiling with a 2dimensional polygon
3+1 dimensional blends[edit  edit source]
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Of the 48 regular polyhedra in 3 dimensional space, 6 of them have vertices spanning only two dimensional space. The nondegenerate Euclidean tilings (See § Euclidean tilings) and their Petrials. Another 12 of them are nontrivial blends (See § Blends), since blending is associative these will be covered in the next section. This leaves 30 pure or nonskew regular polyhedra in 3space. Each of these can be blended with either the digon or an apeirogon.
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} ^{1.5} ^{1.6} ^{1.7} ^{1.8} This blend is abstractly equivalent to the pure/nonskew polyhedron it is made from.
2+2 dimensional blends[edit  edit source]
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4dimensional blended polyhedra can also be made by blending the 2dimensional apeirohedra with other 2dimensional polytopes. The 6 honeycombs can be blended 15 different ways with each other. Additionally any of the honeycombs can be blended with a 2dimensional polygon, producing infinitely many blends.
It is also possible for the blend of two dense honeycombs to result in a discrete apeirohedron in 4space. This class has not been fully enumerated. Some known examples are:
 {8,8/3}#{8/3,8}
 {8,8/3}^{π}#{8/3,8}^{π}
 {10,5/2}#{10/3,5}
 {10,5/2}^{π}#{10/3,5}^{π}
 {5,10/3}#{5/2,10}
 {5,10/3}^{π}#{5/2,10}^{π}
 {12/5,12}#{12,12/5}
 {12/5,12}^{π}#{12,12/5}^{π}
The six blends made with the zigzag can also be seen as 3+1 dimensional blends, since the zigzag is the blend of the digon and the flat apeirogon and blending is associative. For example blending the square tiling with the zigzag is the same as blending the helical square tiling ({4,4}#{∞}) with a digon, or the blended square tiling ({4,4}#{}) with the flat apeirogon.
Duocombs[edit  edit source]
Within four dimensions we can take the comb product of any regular polygon with itself to produce a regular 4D skew polyhedron. These polyhedra are called duocombs and form an infinite family of regular skew polyhedra:
Alternatively these can be thought of as polyhedra with Schläfli symbols of the form where is some polygon. However, these symbols describe an abstract polytope and it can be hard to tell from a symbol alone what dimension is require to realize it. Since the comb product is additive on the dimensions of its constituents, we know that the comb product of two 2dimensional polygons is a 4dimensional polyhedron.
The square tiling is also a duocoumb of two apeirogons, although it is usually not considered part of this family as its vertices lie in two dimensions.
Halvings[edit  edit source]
Since each duocomb has square faces they can potentially be halved. These halved duocombs yield new regular 4D polyhedra. These form another infinite family of halved duocombs:
 Halved triangular duocomb
 Halved square duocomb
 Halved pentagonal duocomb
 Halved pentagrammic duocomb
 Halved hexagonal duocomb
 etc.
The halvings of odd duocombs are abstractly equivalent to their original versions. Even duocombs have bipartite skeletons and thus their halvings have half as many vertices, and thus cannot be abstractly equivalent.
These halvings also have square faces, and thus can be halved again. Halving a duocomb twice yields another duocomb. For powers of two repeated halving of the n duocomb will eventually reach the degenerate digonal duocomb, for all other numbers it will fall into a loop once it reaches an odd duocomb. Some examples of repeated halvings:
 Triangular duocomb → Halved triangular duocomb → Triangular duocomb → ...
 Square duocomb → Halved square duocomb → Digonal duocomb (degenerate)
 Pentagonal duocomb → Halved pentagonal duocomb → Pentagrammic duocomb → Halved pentagrammic duocomb → Pentagonal duocomb → ...
 Hexagonal duocomb → Halved hexagonal duocomb → Triangular duocomb → Halved triangular duocomb → Triangular duocomb → ...
Petrials[edit  edit source]
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Both the duocombs and their halvings have Petrials, with the exception of the halved square duocomb each of these constitutes new regular pure polyhedra in 4dimensional space. The halved square duocomb is selfPetrial.
Nonprismatic pure polyhedra[edit  edit source]
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Additionally there are skew 4D polyhedra which are not duocombs or trivial blends. These include:
For a complete list of 4D regular polyhedra, see this blog post.
5dimensional skews[edit  edit source]
Unsolved problem in mathematics: How many regular pure polyhedra are there in 5 dimensional Euclidean space? (more unsolved problems)

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For every dimension greater than 3 there are an infinite number of regular skew polyhedra formed by nontrivial blends. However it is unknown whether there are also an infinite number of pure polyhedra in 5 dimensions, as there are in 4 and 6 dimensions.
Examples of regular pure polyhedra in 5dimensional space:
6dimensional skews[edit  edit source]
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In 4dimensional space it was possible to form regular polyhedra as the comb product of a regular polygon with itself. Triocombs, formed by the comb product of three copies of a regular polygon, live in 6dimensional space, and form regular skew polychora. Just as the duocombs were abstractly quotients of the square tiling, triocombs are abstractly quotients of the cubic honeycomb. Just as the mucube can be formed by selecting faces from the the cubic honeycomb, new pure polyhedra can be formed by selecting from faces of the triocombs. These polyhedra are abstractly quotients of the mucube along its 3holes. For example from the square triocomb the polyhedron can be formed.
This gives an infinite family of pure polyhedra in 6dimensional space. These polyhedra additionally have duals, Petrials, halvings etc.
History[edit  edit source]
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In 1926 John Flinders Petrie took the concept of a regular skew polygons, polygons whose vertices are not all in the same plane, and extended it to polyhedra. While apeirohedra are typically required to tile the 2dimensional plane, Petrie considered cases where the faces were still convex but were not required to lie flat in the plane, they could have a skew polygon vertex figure.
Petrie discovered two regular skew apeirohedra, the mucube and the muoctahedron.^{[10]} Coxeter derived a third, the mutetrahedron, and proved that the these three were complete. Under Coxeter and Petrie's definition, requiring convex faces and allowing a skew vertex figure, the three were not only the only skew apeirohedra in 3dimensional Euclidean space, but they were the only skew polyhedra in 3space as there Coxeter showed there were no finite cases.
In 1967^{[11]} Garner investigated regular skew apeirohedra in hyperbolic 3space with Petrie and Coxeters definition, discovering 31^{[note 4]} regular skew apeirohedra with compact or paracompact symmetry.
In 1977^{[12]}^{[10]} Grünbaum generalized skew polyhedra to allow for skew faces as well. Grünbaum discovered an additional 32^{[note 1]} skew apeirohedra in 3dimensional Euclidean space and 3 in 2dimensional space which are skew by virtue of their faces. 12 of Grünbaum's polyhedra were formed using the blending operation on 2dimensional apeirohedra, and the other 11 were pure, i.e. could not be formed by a nontrivial blend. Grünbaum conjectured that this new list was complete for the parameters considered.
In 1985^{[13]}^{[10]} Dress found an additional pure regular skew apeirohedron in 3space, and proved that with this additional skew apeirohedron the list was complete.
External links[edit  edit source]
 jan Misali (2020). "there are 48 regular polyhedra".
 "Regular polyhedra".
 Wikipedia contributors. "Regular skew polyhedron".
Notes[edit  edit source]
 ↑ ^{1.0} ^{1.1} 12 of the regular polyhedra have a degree of freedom. Some authors^{[1]}^{[2]} consider these to be infinite families. For the purposes of counting this article considers two realizations to be equal if there is a bijective affine transformation between them.
 ↑ When the face is skew, as is the case for the Petrials, it is often projected to a planar polygon for simplicity. So {10/3} in context may mean the skew decagram.
 ↑ Depending on the definition of blending, a line segment may be used as a blend component instead of a digon.
 ↑ Garner mistakenly counts {8,8∣4} twice giving a count of 18 paracompact cases and 32 total, but only listing 17 paracompact and 31 total.
References[edit  edit source]
 ↑ McMullen & Schulte (2002)
 ↑ Grünbaum (1975)
 ↑ McMullen (2004)
 ↑ Grünbaum (1975)
 ↑ Dress (1985)
 ↑ McMullen & Schulte (1997)
 ↑ McMullen (2004)
 ↑ McMullen (2004)
 ↑ Garner (1967)
 ↑ ^{10.0} ^{10.1} ^{10.2} McMullen & Schulte (1997:449450)
 ↑ Garner (1967)
 ↑ Grünbaum (1977)
 ↑ Dress (1985)
Bibliography[edit  edit source]
 Garner, Cyril (1967), "Regular skew polyhedra in hyperbolic threespace" (PDF), Canadian Journal of Mathematics, 19, doi:10.4153/CJM19671069
 Grünbaum, Branko (1975), "Regular polyhedra  old and new" (PDF), Aequationes Mathematicae
 Dress, Andreas (1985). "A combinatorial theory of Grünbaum's new regular polyhedra, Part II: Complete enumeration". Aequationes mathematicae. 29: 222–243. doi:10.1007/BF02189831.
 McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF). Discrete Computational Geometry (47): 449–478. doi:10.1007/PL00009304.
 McMullen, Peter; Schulte, Egon (December 2002). Abstract Regular Polytopes. Cambridge University Press. ISBN 0521814960.
 McMullen, Peter (2004). "Regular Polytopes of Full Rank" (PDF). Discrete Computational Geometry.
 McMullen, Peter (2007). "FourDimensional Regular Polyhedra" (PDF). Discrete & Computational Geometry.