Regular skew polyhedron

The Petrial cube

There are 48 non-degenerate non-dense regular polyhedra in 3D Euclidean space, 36 of which are skew. There are also skew polyhedra in other spaces such has hyperbolic space and higher dimensional Euclidean space.

Finite skew polyhedra

Petrie duals

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:
  • {p} is the face of the polytope.
  • {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

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.

• 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.

Proof of completeness

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

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.

Alternatively they can be constructed using the apeir operation on regular polygons. 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.

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

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

Blended apeirohedra

Tiling	Blended by ...
	Digon	Apeirogon
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

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, $\left\{4,6\mid 4\right\}$.

In total, there are 3 possible configurations of planar polygons that work:

• Schläfli type {4,6} gives the mucube, $\left\{4,6\mid 4\right\}$
• Schläfli type {6,4} gives the muoctahedron, $\left\{6,4\mid 4\right\}$
• Schläfli type {6,6} gives the mutetrahedron, $\left\{6,6\mid 3\right\}$

• 
  A section of the mucube
• 
  A section of the muoctahedron
• 
  A section of 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 $\left\{{\dfrac {6}{1,3}},6\right\}$ gives the halved mucube, $\left\{{\dfrac {6}{1,3}},6:{\dfrac {4}{1,2}}\right\}$
• Schläfli type $\left\{{\dfrac {4}{1,2}},6\right\}$ gives the Petrial halved mucube, $\left\{{\dfrac {4}{1,2}},6:{\dfrac {6}{1,3}}\right\}$
• Schläfli type $\left\{{\dfrac {6}{1,3}},4\right\}$ gives the skewed Petrial muoctahedron, $\left\{{\dfrac {6}{1,3}},4:{\dfrac {6}{1,3}}\right\}$

• 
  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

Some relationships between the 12 pure apeirohedra in 3D Euclidean space

• π  represents the Petrial
• δ  represents the dual
• η  represents halving
• φ  represents facetting
• σ  represents skewing
• r  represents rectification

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:

• $\left\{{\dfrac {3}{0,1}},{\dfrac {6}{1,3}}\right\}_{4,4}$ Petrial mucube
• $\left\{{\dfrac {3}{0,1}},{\dfrac {4}{1,2}}\right\}_{6,4}$ Petrial muoctahedron
• $\left\{{\dfrac {3}{0,1}},{\dfrac {6}{1,3}}\right\}_{6,3}$ Petrial mutetrahedron

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

• $\left\{{\dfrac {3}{0,1}},4\right\}_{\cdot ,*3}$ Skewed muoctahedron

And the remaining two are mutually Petrial:

• $\{\infty ,3\}^{(a)}$ Trihelical square tiling
• $\{\infty ,3\}^{(b)}$ Tetrahelical triangular tiling

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

{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:

• Mucube from the cubic honeycomb
• Muoctahedron from the bitruncated cubic honeycomb
• Mutetrahedron form the cyclotruncated tetrahedral-octahedral 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.

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. 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

See also: Category:Regular skew polyhedra in 4 dimensions

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

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

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	Helical cube	Petrial cube	Blended Petrial cube	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	Helical mucube	Petrial mucube	Blended Petrial mucube	Helical Petrial mucube
Muoctahedron	Blended muoctahedron	Helical muoctahedron	Petrial muoctahedron	Blended Petrial muoctahedron	Helical Petrial muoctahedron
Mutetrahedron	Blended mutetrahedron	Helical mutetrahedron	Petrial mutetrahedron	Blended Petrial mutetrahedron	Helical Petrial mutetrahedron
Halved mucube	Blended halved mucube	Helical halved mucube	Petrial halved mucube	Blended Petrial halved mucube	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	Helical skewed Petrial muoctahedron

2+2 dimensional blends

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:

• {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

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:

• $\{4,4\mid 3\}$ - Triangular duocomb
• $\{4,4\mid 4\}$ - Square duocomb
• $\{4,4\mid 5\}$ - Pentagonal duocomb
• $\{4,4\mid 5/2\}$ - Pentagrammic duocomb
• etc.

Alternatively these can be thought of as polyhedra with Schläfli symbols of the form $\{4,4\mid n\}$ where $\{n\}$ 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

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:

• 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

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

Additionally there are skew 4D polyhedra which are not duocombs or trivial blends. These include:

• {4,6∣3}
• {6,4∣3}
• {4,8∣3}
• {8,4∣3}
• {4,8/3∣3}
• {8/3,4∣3}
• {8,8/3∣3}
• {8/3,8∣3}

• 
  A projection of {4,6∣3}
• 
  A projection of {6,4∣3} with a face removed for visibility

For a complete list of 4D regular polyhedra, see this blog post.

5-dimensional skews

Unsolved problem in mathematics: How many regular pure polyhedra are there in 5 dimensional Euclidean space? (more unsolved problems)

See also: Category:Regular skew polyhedra in 5 dimensions

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:

• Hemiicosahedron
• Hemidodecahedron

6-dimensional skews

See also: Category:Regular skew polyhedra in 6 dimensions

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 $\{4,6\mid 4,4\}$ can be formed.

This gives an infinite family of pure polyhedra in 6-dimensional space. These polyhedra additionally have duals, Petrials, halvings etc.

History

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. 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 Garner investigated regular skew apeirohedra in hyperbolic 3-space with Petrie and Coxeters definition, discovering 31 regular skew apeirohedra with compact or paracompact symmetry.

In 1977 Grünbaum generalized skew polyhedra to allow for skew faces as well. Grünbaum discovered an additional 32 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 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

• jan Misali (2020). "there are 48 regular polyhedra".
• "Regular polyhedra".
• Wikipedia contributors. "Regular skew polyhedron".

Bibliography

• 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.