In geometry, **demihypercubes** (also called *n-demicubes*, *n-hemicubes*, and *half measure polytopes*) are a class of *n*-polytopes constructed from alternation of an *n*-hypercube, labeled as *hγ _{n}* for being

They have been named with a *demi-* prefix to each hypercube name: demicube, demitesseract, etc. The demicube is identical to the regular tetrahedron, and the demitesseract is identical to the regular 16-cell. The demipenteract is considered *semiregular* for having only regular facets. Higher forms don't have all regular facets but are all uniform polytopes.

The vertices and edges of a demihypercube form two copies of the halved cube graph.

An *n*-demicube has inversion symmetry if *n* is even.

Thorold Gosset described the demipenteract in his 1900 publication listing all of the regular and semiregular figures in *n*-dimensions above 3. He called it a *5-ic semi-regular*. It also exists within the semiregular *k*_{21} polytope family.

The demihypercubes can be represented by extended Schläfli symbols of the form h{4,3,...,3} as half the vertices of {4,3,...,3}. The vertex figures of demihypercubes are rectified *n*-simplexes.

They are represented by Coxeter-Dynkin diagrams of three constructive forms:

- ... (As an alternated orthotope) s{2
^{1,1,...,1}} - ... (As an alternated hypercube) h{4,3
^{n−1}} - .... (As a demihypercube) {3
^{1,n−3,1}}

H.S.M. Coxeter also labeled the third bifurcating diagrams as **1 _{k1}** representing the lengths of the 3 branches and led by the ringed branch.

An *n-demicube*, *n* greater than 2, has *n*(*n*−1)/2 edges meeting at each vertex. The graphs below show less edges at each vertex due to overlapping edges in the symmetry projection.

n | 1_{k1} | Petrie polygon | Schläfli symbol | Coxeter diagrams A _{1}^{n}B _{n}D _{n} | Elements | Facets: Demihypercubes & Simplexes | Vertex figure | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | 7-faces | 8-faces | 9-faces | |||||||

2 | 1_{−1,1} | demisquare(digon) | s{2} h{4} {3 ^{1,−1,1}} | 2 | 2 | 2 edges | -- | |||||||||

3 | 1_{01} | demicube(tetrahedron) | s{2^{1,1}}h{4,3} {3 ^{1,0,1}} | 4 | 6 | 4 | (6 digons )4 triangles | Triangle (Rectified triangle) | ||||||||

4 | 1_{11} | demitesseract (16-cell) | s{2^{1,1,1}}h{4,3,3} {3 ^{1,1,1}} | 8 | 24 | 32 | 16 | 8 demicubes (tetrahedra) 8 tetrahedra | Octahedron (Rectified tetrahedron) | |||||||

5 | 1_{21} | demipenteract | s{2^{1,1,1,1}}h{4,3 ^{3}}{3^{1,2,1}} | 16 | 80 | 160 | 120 | 26 | 10 16-cells 16 5-cells | Rectified 5-cell | ||||||

6 | 1_{31} | demihexeract | s{2^{1,1,1,1,1}}h{4,3 ^{4}}{3^{1,3,1}} | 32 | 240 | 640 | 640 | 252 | 44 | 12 demipenteracts32 5-simplices | Rectified hexateron | |||||

7 | 1_{41} | demihepteract | s{2^{1,1,1,1,1,1}}h{4,3 ^{5}}{3^{1,4,1}} | 64 | 672 | 2240 | 2800 | 1624 | 532 | 78 | 14 demihexeracts64 6-simplices | Rectified 6-simplex | ||||

8 | 1_{51} | demiocteract | s{2^{1,1,1,1,1,1,1}}h{4,3 ^{6}}{3^{1,5,1}} | 128 | 1792 | 7168 | 10752 | 8288 | 4032 | 1136 | 144 | 16 demihepteracts128 7-simplices | Rectified 7-simplex | |||

9 | 1_{61} | demienneract | s{2^{1,1,1,1,1,1,1,1}}h{4,3 ^{7}}{3^{1,6,1}} | 256 | 4608 | 21504 | 37632 | 36288 | 23520 | 9888 | 2448 | 274 | 18 demiocteracts256 8-simplices | Rectified 8-simplex | ||

10 | 1_{71} | demidekeract | s{2^{1,1,1,1,1,1,1,1,1}}h{4,3 ^{8}}{3^{1,7,1}} | 512 | 11520 | 61440 | 122880 | 142464 | 115584 | 64800 | 24000 | 5300 | 532 | 20 demienneracts512 9-simplices | Rectified 9-simplex | |

... | ||||||||||||||||

n | 1_{n−3,1} | n-demicube | s{2^{1,1,...,1}}h{4,3 ^{n−2}}{3^{1,n−3,1}} | ... ... ... | 2^{n−1} | 2n (n−1)-demicubes2 ^{n−1} (n−1)-simplices | Rectified (n−1)-simplex |

In general, a demicube's elements can be determined from the original *n*-cube: (with C_{n,m} = *m ^{th}*-face count in

**Vertices:**D_{n,0}= 1/2 C_{n,0}= 2^{n−1}(Half the*n*-cube vertices remain)**Edges:**D_{n,1}= C_{n,2}= 1/2*n*(*n*−1) 2^{n−2}(All original edges lost, each square faces create a new edge)**Faces:**D_{n,2}= 4 * C_{n,3}= 2/3*n*(*n*−1)(*n*−2) 2^{n−3}(All original faces lost, each cube creates 4 new triangular faces)**Cells:**D_{n,3}= C_{n,3}+ 2^{3}C_{n,4}(tetrahedra from original cells plus new ones)**Hypercells:**D_{n,4}= C_{n,4}+ 2^{4}C_{n,5}(16-cells and 5-cells respectively)- ...
- [For
*m*= 3,...,*n*−1]: D_{n,m}= C_{n,m}+ 2^{m}C_{n,m+1}(*m*-demicubes and*m*-simplexes respectively) - ...
**Facets:**D_{n,n−1}= 2*n*+ 2^{n−1}((*n*−1)-demicubes and (*n*−1)-simplices respectively)

The stabilizer of the demihypercube in the hyperoctahedral group (the Coxeter group [4,3^{n−1}]) has index 2. It is the Coxeter group [3^{n−3,1,1}] of order , and is generated by permutations of the coordinate axes and reflections along *pairs* of coordinate axes.^{ [2] }

Constructions as alternated orthotopes have the same topology, but can be stretched with different lengths in *n*-axes of symmetry.

The rhombic disphenoid is the three-dimensional example as alternated cuboid. It has three sets of edge lengths, and scalene triangle faces.

In geometry, the **tesseract** is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

In geometry, a **hypercube** is an *n*-dimensional analogue of a square and a cube. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in *n* dimensions is equal to .

In geometry, an **alternation** or *partial truncation*, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

In five-dimensional geometry, a **five-dimensional polytope** or **5-polytope** is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets.

In five-dimensional geometry, a **demipenteract** or **5-demicube** is a semiregular 5-polytope, constructed from a *5-hypercube* (penteract) with alternated vertices removed.

In geometry, a **6-cube** is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

In geometry, a **6-demicube** or **demihexteract** is a uniform 6-polytope, constructed from a *6-cube* (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

In geometry, a **demihepteract** or **7-demicube** is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

In geometry, a **demiocteract** or **8-demicube** is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

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.

In geometry, a **uniform k_{21} polytope** is a polytope in

In geometry, a **hypercubic honeycomb** is a family of regular honeycombs (tessellations) in n-dimensions with the Schläfli symbols {4,3...3,4} and containing the symmetry of Coxeter group R_{n} for n>=3.

In geometry, a **10-cube** is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.

In five-dimensional geometry, a **rectified 5-simplex** is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

In 6-dimensional geometry, the **2 _{21}** polytope is a uniform 6-polytope, constructed within the symmetry of the E

In 8-dimensional geometry, the **4 _{21}** is a semiregular uniform 8-polytope, constructed within the symmetry of the E

In six-dimensional geometry, a **six-dimensional polytope** or **6-polytope** is a polytope, bounded by 5-polytope facets.

In geometry of five dimensions or higher, a **cantic 5-cube**, **cantihalf 5-cube**, **truncated 5-demicube** is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated 5-cube.

In six-dimensional geometry, a **runcic 5-cube** or is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

In seven-dimensional geometry, a **cantic 7-cube** or **truncated 7-demicube** as a uniform 7-polytope, being a truncation of the 7-demicube.

- T. Gosset:
*On the Regular and Semi-Regular Figures in Space of n Dimensions*, Messenger of Mathematics, Macmillan, 1900 - 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}) **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- (Paper 24) H.S.M. Coxeter,
*Regular and Semi-Regular Polytopes III*, [Math. Zeit. 200 (1988) 3-45]

- (Paper 24) H.S.M. Coxeter,

- Olshevsky, George. "Half measure polytope".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007.

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