Regular skew polyhedron

In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar faces or vertex figures.

These polyhedra have two forms: infinite polyhedra that span 3-space, and finite polyhedra that close into 4-space.

History

According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to regular skew polyhedra.

Coxeter offered a modified Schläfli symbol {l,m|n} for these figures, with {l,m} implying the vertex figure, m l-gons around a vertex, and n-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.

The regular skew polyhedra, reresented by {l,m|n}, follow this equation:

• 2*sin(π/l)*sin(π/m)=cos(π/n)

Infinite regular skew polyhedra

There are 3 regular skew polyhedra, the first two being duals:

1. {4,6|4}: 6 squares on a vertex (related to cubic honeycomb, constructed by cubic cells, removing two opposite faces from each, and linking sets of six together around a faceless cube.)
2. {6,4|4}: 4 hexagons on a vertex (related to bitruncated cubic honeycomb, constructed by truncated octahedron with their square faces removed and linking hole pairs of holes together.)
3. {6,6|3}: 6 hexagons on a vertex (related to quarter cubic honeycomb, constructed by truncated tetrahedron cells, removing triangle faces, and linking sets of four around a faceless tetrahedron.)

Also solutions to the equation above are the Euclidean regular tilings {3,6}, {6,3}, {4,4}, represented as {3,6|6}, {6,3|6}, and {4,4|∞}.

Here are some partial representations, vertical projected views of their skew vertex figures, and partial corresponding uniform honeycombs.

Partial polyhedra
{4,6|4}	{6,4|4}	{6,6|3}
Vertex figures
{4,6}	{6,4}	{6,6}
Related convex uniform honeycombs
Runcinated cubic honeycomb t0,3{4,3,4}	Bitruncated cubic t1,2{4,3,4}	quarter cubic honeycomb t0,1{3[4]}

Finite regular skew polyhedra of 4-space

A4 Coxeter plane projections
{4, 6 | 3}	{6, 4 | 3}
Runcinated 5-cell (60 edges, 20 vertices)	Bitruncated 5-cell (60 edges, 30 vertices)
F4 Coxeter plane projections
{4, 8 | 3}	{8, 4 | 3}
Runcinated 24-cell (576 edges, 144 vertices)	Bitruncated 24-cell (576 edges, 288 vertices)
Some of the 4-dimensional regular skew polyhedra fits inside the uniform polychora as shown in these projections.

Coxeter also enumerated the a larger set of finite regular polyhedra in his paper "regular skew polyhedra in three and four dimensions, and their topological analogues".

Just like the infinite skew polyhedra represent manifold surfaces between the cells of the convex uniform honeycombs, the finite forms all represent manifold surfaces within the cells of the uniform polychora.

A first form, {l, m | n}, repeats the five convex Platonic solids, and one nonconvex Kepler-Poinsot solid:

{l, m | n}	Faces	Edges	Vertices	p	Polyhedron	Symmetry order
{3,3| 3} = {3,3}	4	6	4	0	Tetrahedron	12
{3,4| 4} = {3,4}	8	12	6	0	Octahedron	24
{4,3| 4} = {4,3}	6	12	8	0	Cube	24
{3,5| 5} = {3,5}	20	30	12	0	Icosahedron	60
{5,3| 5} = {5,3}	12	30	20	0	Dodecahedron	60
{5,5| 3} = {5,5/2}	12	30	12	4	Great dodecahedron	60

The remaining solutions of the first form, {l, m | n} exist in 4-space. Polyhedra of the form {l, m | n} have a cyclic Coxeter group symmetry of [(l/2,n,m/2,n)], which reduces to the linear [n,l/2,n] when m is 4, and [n,m,/2,n] when l=4. {4,4|n} produces a double n-prism, or n-n duoprism, and specifically {4,4|4} fits inside of a {4}x{4} tesseract. {a,4|b} is represented by the {a} faces of the bitruncated {b,a/2,b} uniform polychoron, and {4,a|b} is represented by square faces of the runcinated {b,a/2,b}.

Even ordered solutions

{l, m | n}	Faces	Edges	Vertices	p	Structure	Symmetry	Order	Related uniform polychora
{4,4| 3}	9	18	9	1	D3xD3	[3,2,3]	18	3-3 duoprism
{4,4| 4}	16	32	16	1	D4xD4	[4,2,4]	32	4-4 duoprism or tesseract
{4,4| 5}	25	50	25	1	D5xD5	[5,2,5]	50	5-5 duoprism
{4,4| 6}	36	72	36	1	D6xD6	[6,2,6]	72	6-6 duoprism
{4,4| n}	n2	2n2	n2	1	DnxDn	[n,2,n]	2n2	n-n duoprism
{4,6| 3}	30	60	20	6	S5	[3,3,3]	120	Runcinated 5-cell
{6,4| 3}	20	60	30	6	S5	[3,3,3]	120	Bitruncated 5-cell
{4,8| 3}	288	576	144	73		[3,4,3]	1152	Runcinated 24-cell
{8,4| 3}	144	576	288	73		[3,4,3]	1152	Bitruncated 24-cell

pentagramic solutions

{l, m | n}	Faces	Edges	Vertices	p	Structure	Symmetry	Order	Related uniform polychora
{4,5| 5}	90	180	72	10	A6	[5,5/2,5]	360	Runcinated grand stellated 120-cell
{5,4| 5}	72	180	90	10	A6	[5,5/2,5]	360	Bitruncated grand stellated 120-cell

{l, m | n}	Faces	Edges	Vertices	p	Structure	Order
{4,5| 4}	40	80	32	5	?	160
{5,4| 4}	32	80	40	5	?	160
{4,7| 3}	42	84	24	10	LF(2,7)	168
{7,4| 3}	24	84	42	10	LF(2,7)	168
{5,5| 4}	72	180	72	19	A6	360
{6,7| 3}	182	546	156	105	LF(2,13)	1092
{7,6| 3}	156	546	182	105	LF(2,13)	1092
{7,7| 3}	156	546	156	118	LF(2,13)	1092
{4,9| 3}	612	1224	272	171	LF(2,17)	2448
{9,4| 3}	272	1224	612	171	LF(2,17)	2448
{7,8| 3}	1536	5376	1344	1249	?	10752
{8,7| 3}	1344	5376	1536	1249	?	10752

A final set is based on Coxeter's further extended form {q1,m|q2,q3...} or with q2 unspecified: {l, m |, q}.

{l, m |, q}	Faces	Edges	Vertices	p	Structure	Order
{3,6|,q}	2q2	3q2	q2	1	?	2q2
{3,2q|,3}	2q2	3q2	3q	(q-1)*(q-2)/2	?	2q2
{3,7|,4}	56	84	24	3	LF(2,7)	168
{3,8|,4}	112	168	42	8	PGL(2,7)	336
{4,6|,3}	84	168	56	15	PGL(2,7)	336
{3,7|,6}	364	546	156	14	LF(2,13)	1092
{3,7|,7}	364	546	156	14	LF(2,13)	1092
{3,8|,5}	720	1080	270	46	?	2160
{3,10|,4}	720	1080	216	73	?	2160
{4,6|,2}	12	24	8	3	S4xS2	48
{5,6|,2}	24	60	20	9	A5xS2	120
{3,11|,4}	2024	3036	552	231	LF(2,23)	6072
{3,7|,8}	3584	5376	1536	129	?	10752
{3,9|,5}	12180	18270	4060	1016	LF(2,29)xA3	36540

See also

• Skew polygon
• Infinite skew polyhedron

References

• Coxeter, Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
  • (Paper 2) H.S.M. Coxeter, "The Regular Sponges, or Skew Polyhedra", Scripta Mathematica 6 (1939) 240-244.
• 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.)
  • Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937.
• Garner, C. W. L. Regular Skew Polyhedra in Hyperbolic Three-Space. Canad. J. Math. 19, 1179-1186, 1967.