Omnitruncated simplectic honeycomb

In geometry an omnitruncated simplectic honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional uniform tessellation, based on the symmetry of the ${\tilde {A}}_{n}$ affine Coxeter group. Each is composed of omnitruncated simplex facets. The vertex figure for each is an irregular n-simplex.

The facets of an omnitruncated simplectic honeycomb are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).

n	${\tilde {A}}_{1+}$	Tessellation	Facets	Vertex figure	Facets per vertex figure	Vertices per vertex figure
1	${\tilde {A}}_{1}$	Apeirogon	Line segment	Line segment	1	2
2	${\tilde {A}}_{2}$	Hexagonal tiling	hexagon	Equilateral triangle	3 hexagons	3
3	${\tilde {A}}_{3}$	Bitruncated cubic honeycomb	Truncated octahedron	irr. tetrahedron	4 truncated octahedron	4
4	${{\tilde {A}}}_{4}$	Omnitruncated 4-simplex honeycomb	Omnitruncated 4-simplex	irr. 5-cell	5 omnitruncated 4-simplex	5
5	${\tilde {A}}_{5}$	Omnitruncated 5-simplex honeycomb	Omnitruncated 5-simplex	irr. 5-simplex	6 omnitruncated 5-simplex	6
6	${\tilde {A}}_{6}$	Omnitruncated 6-simplex honeycomb	Omnitruncated 6-simplex	irr. 6-simplex	7 omnitruncated 6-simplex	7
7	${\tilde {A}}_{7}$	Omnitruncated 7-simplex honeycomb	Omnitruncated 7-simplex	irr. 7-simplex	8 omnitruncated 7-simplex	8
8	${{\tilde {A}}}_{8}$	Omnitruncated 8-simplex honeycomb	Omnitruncated 8-simplex	irr. 8-simplex	9 omnitruncated 8-simplex	9

Projection by folding

The (2n-1)-simplex honeycombs can be projected into the n-dimensional omnitruncated hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

${\tilde {A}}_{3}$		${\tilde {A}}_{5}$		${\tilde {A}}_{7}$		${{\tilde {A}}}_{9}$		...
${\tilde {C}}_{2}$		${\tilde {C}}_{3}$		${{\tilde {C}}}_{4}$		${{\tilde {C}}}_{5}$		...

References

George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
Norman Johnson Uniform Polytopes, Manuscript (1991)
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E²	Uniform tiling	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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Omnitruncated simplectic honeycomb

Projection by folding

See also

References