Regular skew polyhedron

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The Petrial cube

There are 48 non-degenerate non-dense 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]

Petrial tetrahedronTetrahedronCubePetrial cubeOctahedronPetrial octahedronIcosahedronPetrial icosahedronPetrial small stellated dodecahedronSmall stellated dodecahedronPetrial great dodecahedronGreat dodecahedronGreat icosahedronPetrial great icosahedronGreat stellated dodecahedronPetrial great stellated dodecahedronPetrial dodecahedronDodecahedronPetrialPetrialPetrialPetrialPetrialPetrialPetrialPetrialPetrialDual polytopeDual polytopeΚΚΚΚΚΚΚΚΚDual polytopeDual polytope
The finite polyhedra of full rank and their relations

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 non-skew 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:

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 Kepler-Poinsot polyhedra, but with a different pairing.

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 2-dimensions. 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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A section of the blended triangular tiling, with the edges of one face highlighted in red.

We can also create new regular skew polyhedra by taking the 6 2-dimensional polyhedra (the tilings of the Euclidean plane and their Petrials) and blending them with a 1-dimensional polygon, either a digon or an apeirogon[note 3]

Blended apeirohedra
Tiling Blended by ...


Triangular tiling Blended triangular tiling Helical triangular tiling
Square tiling Blended square tiling Helical square tiling

Hexagonal tiling

Blended hexagonal tiling Helical hexagonal tiling
Petrial triangular tiling Petrial blended triangular tiling Petrial helical triangular tiling
Petrial square tiling Petrial blended square tiling Petrial helical square tiling
Petrial hexagonal tiling Petrial blended hexagonal tiling Petrial helical hexagonal tiling

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.

6 squares around a vertex. Concave edges are colored red and convex edges are colored yellow.

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,

These three regular polyhedra are called the Petrie-Coxeter 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,

Helical faces[edit | edit source]

Skewed muoctahedronPetrial mucubeMuoctahedronMucubePetrial muoctahedronHalved mucbePetrial halved mucubeSkewed Petrial muoctahedronMutetrahedronPetrial mutetrahedronTrihelical square tilingTetrahelical triangular tilingSkewingSkewingRectificationRectificationHalvingHalvingPetrie dualPetrie dualPetrie dualPetrie dualPetrie dualPetrie dualPetrie dualDualDualSecond-order facettingSecond-order facettingSecond-order facettingSecond-order facettingSecond-order facettingSecond-order facettingPetrial cubePetrial tetrahedronTetrahedronCube
Some relationships between the 12 pure apeirohedra in 3D Euclidean space

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:

  • Petrial mucube
  • Petrial muoctahedron
  • Petrial mutetrahedron

There is one polyhedron obtained from skewing the muoctahedron that is self-Petrie:

  • Skewed muoctahedron

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 ad-hoc Schläfli symbols.

Regular pure apeirohedra in 3D hyperbolic space[edit | edit source]

{6,6∣4} is a compact apeirohedron in 3D hyperbolic space.

The 3 pure apeirohedra with planar faces each consist of faces taken from a uniform honeycomb:

Upon careful observation we can see that the polyhedron {2r,2qp} 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.

Regular pure apeirohedra in 3D hyperbolic space
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 self-intersections 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 self-intersection) is currently incomplete.

4-dimensional 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]

Projection of a blended cube rotating about a plane. Vertices are two colored such that vertices of the same color lie on the same 3-plane with both planes being parallel.

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 2-dimensional Euclidean tiling with a flat 1-dimensional polygon, either a digon or a apeirogon. However there are two ways to achieve a 4-dimensional blend:

  • Blend a 3-dimensional polyhedron with a flat 1-dimensional polygon
  • Blend a flat 2-dimensional Euclidean tiling with a 2-dimensional 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 non-degenerate Euclidean tilings (See § Euclidean tilings) and their Petrials. Another 12 of them are non-trivial blends (See § Blends), since blending is associative these will be covered in the next section. This leaves 30 pure or non-skew regular polyhedra in 3-space. Each of these can be blended with either the digon or an apeirogon.

Blends of 3-dimnesional polyhedra with 1-dimensional polygons
Polyhedron blended with... Petrial Petrial and blended with ...
Digon Apierogon Digon Apeirogon
Tetrahedron Blended tetrahedron Helical tetrahedron Petrial tetrahedron Blended Petrial tetrahedron Helical Petrial tetrahedron
Cube Blended cube[t 1] Helical cube Petrial cube Blended Petrial cube[t 1] Helical Petrial cube
Octahedron Blended octahedron Helical octahedron Petrial octahedron Blended Petrial octahedron Helical Petrial octahedron
Dodecahedron Blended dodecahedron Helical dodecahedron Petrial dodecahedron Blended Petrial dodecahedron Helical Petrial dodecahedron
Icosahedron Blended icosahedron Helical icosahedron Petrial icosahedron Blended Petrial icosahedron Helical Petrial icosahedron
Great dodecahedron Blended great dodecahedron Helical great dodecahedron Petrial great dodecahedron Blended Petrial great dodecahedron Helical Petrial great dodecahedron
Small stellated dodecahedron Blended small stellated dodecahedron Helical small stellated dodecahedron Petrial small stellated dodecahedron Blended Petrial small stellated dodecahedron Helical Petrial small stellated dodecahedron
Great icosahedron Blended great icosahedron Helical great icosahedron Petrial great icosahedron Blended Petrial great icosahedron Helical Petrial great icosahedron
Great stellated dodecahedron Blended great stellated dodecahedron Helical great stellated dodecahedron Petrial great stellated dodecahedron Blended Petrial great stellated dodecahedron Helical Petrial great stellated dodecahedron
Mucube Blended mucube[t 1] Helical mucube Petrial mucube Blended Petrial mucube[t 1] Helical Petrial mucube
Muoctahedron Blended muoctahedron[t 1] Helical muoctahedron Petrial muoctahedron Blended Petrial muoctahedron[t 1] Helical Petrial muoctahedron
Mutetrahedron Blended mutetrahedron Helical mutetrahedron Petrial mutetrahedron Blended Petrial mutetrahedron Helical Petrial mutetrahedron
Halved mucube Blended halved mucube[t 1] Helical halved mucube Petrial halved mucube Blended Petrial halved mucube[t 1] Helical Petrial halved mucube
Trihelical square tiling Blended trihelical square tiling Helical trihelical square tiling Tetrahelical triangular tiling Blended tetrahelical triangular tiling Helical tetrahelical triangular tiling
Skewed muoctahedron Blended skewed muoctahedron Helical skewed muoctahedron
Skewed Petrial muoctahedron Blended skewed Petrial muoctahedron[t 1] Helical skewed Petrial muoctahedron
  1. 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/non-skew polyhedron it is made from.

2+2 dimensional blends[edit | edit source]

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4-dimensional blended polyhedra can also be made by blending the 2-dimensional apeirohedra with other 2-dimensional polytopes. The 6 honeycombs can be blended 15 different ways with each other. Additionally any of the honeycombs can be blended with a 2-dimensional polygon, producing infinitely many blends.

It is also possible for the blend of two dense honeycombs to result in a discrete apeirohedron in 4-space. This class has not been fully enumerated. Some known examples are:

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]

A 3D projection of the pentagonal duocomb.

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 2-dimensional polygons is a 4-dimensional 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]

The fundamental domain of the halved triangular duocomb. Dotted edges are identified across the square. A single face is highlighted in orange.

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:

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:

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 4-dimensional space. The halved square duocomb is self-Petrial.

Non-prismatic 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.

5-dimensional 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 non-trivial 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 5-dimensional space:

6-dimensional skews[edit | edit source]

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In 4-dimensional 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 6-dimensional 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 3-holes. For example from the square triocomb the polyhedron can be formed.

This gives an infinite family of pure polyhedra in 6-dimensional 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 2-dimensional 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 3-dimensional Euclidean space, but they were the only skew polyhedra in 3-space as there Coxeter showed there were no finite cases.

In 1967[11] Garner investigated regular skew apeirohedra in hyperbolic 3-space 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 3-dimensional Euclidean space and 3 in 2-dimensional space which are skew by virtue of their faces. 12 of Grünbaum's polyhedra were formed using the blending operation on 2-dimensional apeirohedra, and the other 11 were pure, i.e. could not be formed by a non-trivial 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 3-space, and proved that with this additional skew apeirohedron the list was complete.

External links[edit | edit source]

Notes[edit | edit source]

  1. 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.
  2. 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.
  3. Depending on the definition of blending, a line segment may be used as a blend component instead of a digon.
  4. 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]

  1. McMullen & Schulte (2002)
  2. Grünbaum (1975)
  3. McMullen (2004)
  4. Grünbaum (1975)
  5. Dress (1985)
  6. McMullen & Schulte (1997)
  7. McMullen (2004)
  8. McMullen (2004)
  9. Garner (1967)
  10. 10.0 10.1 10.2 McMullen & Schulte (1997:449-450)
  11. Garner (1967)
  12. Grünbaum (1977)
  13. Dress (1985)

Bibliography[edit | edit source]

  • Garner, Cyril (1967), "Regular skew polyhedra in hyperbolic three-space" (PDF), Canadian Journal of Mathematics, 19, doi:10.4153/CJM-1967-106-9
  • 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 0-521-81496-0.
  • McMullen, Peter (2004). "Regular Polytopes of Full Rank" (PDF). Discrete Computational Geometry.
  • McMullen, Peter (2007). "Four-Dimensional Regular Polyhedra" (PDF). Discrete & Computational Geometry.