Demienneract (9-demicube)
Petrie polygon Petrie polygon In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets...
Type	Uniform 9-polytope 9-polytope In nine-dimensional geometry, a polyyotton is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets....
Family	demihypercube
Coxeter symbol	161
Schläfli symbol	{31,6,1} h{4,3,3,3,3,3,3,3} s{2,2,2,2,2,2,2,2}
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
8-faces	274	18 {31,5,1} Demiocteract In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices deleted...   256 {37}
7-faces	2448	144 {31,4,1} Demihepteract In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube with alternated vertices deleted...   2304 {36}
6-faces	9888	672 {31,3,1} Demihexeract In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube with alternate vertices deleted. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes....   9216 {35}
5-faces	23520	2016 {31,2,1} Demipenteract In five dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube with alternated vertices deleted.It was discovered by Thorold Gosset...   21504 {34}
4-faces	36288	4032 {31,1,1} 16-cell In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century....   32256 {33}
Cells	37632	5376 {31,0,1} Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...   32256 {3,3} Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
Faces	21504	{3} Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
Edges	4608
Vertices	256
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...	Rectified 8-simplex Rectified 8-simplex In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.There are unique 3 degrees of rectifications. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified...
Symmetry group Coxeter notation In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M...	D9, [36,1,1] = [1+,4,37] [28]+
Dual	?
Properties	convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a demienneract or 9-demicube is a uniform 9-polytope

9-polytope

In nine-dimensional geometry, a polyyotton is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets....

, constructed from the 9-cube, with alternated vertices deleted. It is part of a dimensionally infinite family of uniform polytope

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

s called demihypercubes.

Coxeter named this polytope as 1₆₁ from its Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

, with a ring on
one of the 1-length Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

 branches.

Cartesian coordinates

Cartesian coordinates for the vertices of a demienneract centered at the origin are alternate halves of the enneract

Enneract

In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces....

:

(±1,±1,±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

External links

• Multi-dimensional Glossary