5-demicube

Cartesian coordinates for the vertices of a demipenteract centered at the origin and edge length 2√2 are alternate halves of the penteract:

(±1,±1,±1,±1,±1)

with an odd number of plus signs.

This configuration matrix represents the 5-demicube. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[1][2]

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.[3]

D5		k-face	fk	f0	f1	f2	f3		f4		k-figure	notes(*)
A4		( )	f0	16	10	30	10	20	5	5	rectified 5-cell	D5/A4 = 16*5!/5! = 16
A2A1A1		{ }	f1	2	80	6	3	6	3	2	triangular prism	D5/A2A1A1 = 16*5!/3!/2/2 = 80
A2A1		{3}	f2	3	3	160	1	2	2	1	Isosceles triangle	D5/A2A1 = 16*5!/3!/2 = 160
A3A1		h{4,3}	f3	4	6	4	40	*	2	0	Segment { }	D5/A3A1 = 16*5!/4!/2 = 40
A3		{3,3}		4	6	4	*	80	1	1	Segment { }	D5/A3 = 16*5!/4! = 80
D4		h{4,3,3}	f4	8	24	32	8	8	10	*	Point ( )	D5/D4 = 16*5!/8/4! = 10
A4		{3,3,3}		5	10	10	0	5	*	16	Point ( )	D5/A4 = 16*5!/5! = 16

* = The number of elements (diagonal values) can be computed by the symmetry order D₅ divided by the symmetry order of the subgroup with selected mirrors removed.

It is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 23 Uniform 5-polytopes (uniform 5-polytopes) that can be constructed from the D₅ symmetry of the demipenteract, 8 of which are unique to this family, and 15 are shared within the penteractic family.

D5 polytopes
h{4,3,3,3}	h2{4,3,3,3}	h3{4,3,3,3}	h4{4,3,3,3}	h2,3{4,3,3,3}	h2,4{4,3,3,3}	h3,4{4,3,3,3}	h2,3,4{4,3,3,3}

The 5-demicube is third in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (5-simplices and 5-orthoplexes in the case of the 5-demicube). In Coxeter's notation the 5-demicube is given the symbol 1₂₁.

k21 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
En	3	4	5	6	7	8	9	10
Coxeter group	E3=A2A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8+	E10 = ${\displaystyle {\bar {T}}_{8}}$ = E8++
Coxeter diagram
Symmetry	[3−1,2,1]	[30,2,1]	[31,2,1]	[32,2,1]	[33,2,1]	[34,2,1]	[35,2,1]	[36,2,1]
Order	12	120	1,920	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	−121	021	121	221	321	421	521	621

1k2 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
n	3	4	5	6	7	8	9	10
Coxeter group	E3=A2A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8+	E10 = ${\displaystyle {\bar {T}}_{8}}$ = E8++
Coxeter diagram
Symmetry (order)	[3−1,2,1]	[30,2,1]	[31,2,1]	[[32,2,1]]	[33,2,1]	[34,2,1]	[35,2,1]	[36,2,1]
Order	12	120	1,920	103,680	2,903,040	696,729,600	∞
Graph							-	-
Name	1−1,2	102	112	122	132	142	152	162

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• H.S.M. Coxeter:
  • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • 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]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_n1)
• Klitzing, Richard. "5D uniform polytopes (polytera) x3o3o *b3o3o - hin".

• Olshevsky, George. "Demipenteract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Multi-dimensional Glossary