geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, demihypercubes (also called ''n-demicubes'', ''n-hemicubes'', and ''half measure polytopes'') are a class of ''n''-

polytopes In elementary geometry, a polytope is a geometric object with Flat (geometry), flat sides (''Face (geometry), faces''). Polytopes are the generalization of three-dimensional polyhedron, polyhedra to any number of dimensions. Polytopes may exist ...

constructed from alternation of an ''n''-

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...

, labeled as ''hγ_n'' for being ''half'' of the hypercube family, ''γ_n''. Half of the vertices are deleted and new facets are formed. The 2''n'' facets become 2''n'' (''n''−1)-demicubes, and 2^''n'' (''n''−1)-simplex facets are formed in place of the deleted vertices. They have been named with a ''demi-'' prefix to each

name: demi

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

, demi

tesseract In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...

, etc. The demicube is identical to the regular

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

, and the demitesseract is identical to the regular 16-cell. The demipenteract is considered ''semiregular'' for having only regular facets. Higher forms do not 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.

Discovery

Thorold Gosset described the demipenteract in his 1900 publication listing all of the regular and semiregular figures in ''n''-dimensions above three. 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 symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

s of the form h as half the vertices of . The

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

s of demihypercubes are rectified ''n''- simplexes.

Constructions

They are represented by Coxeter-Dynkin diagrams of three constructive forms: #... (As an alternated orthotope) s #... (As an alternated

) h #.... (As a demihypercube) H.S.M. Coxeter also labeled the third bifurcating diagrams as 1_''k''1 representing the lengths of the three 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. In general, a demicube's elements can be determined from the original ''n''-cube: (with C_''n'',''m'' = ''m^th''-face count in ''n''-cube = 2^{''n''−''m''} ''n''!/(''m''!(''n''−''m'')!)) * 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 + 2³ C_''n'',4 (tetrahedra from original cells plus new ones) * Hypercells: D_''n'',4 = C_''n'',4 + 2⁴ C_''n'',5 (16-cells and 5-cells respectively) * ... * or ''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)

Symmetry group

The stabilizer of the demihypercube in the hyperoctahedral group (the Coxeter group

BC_n

,3^''n''−1 has index 2. It is the Coxeter group

D_n,

^{''n''−3,1,1}of order

2^n!

, and is generated by permutations of the coordinate axes and reflections along ''pairs'' of coordinate axes.

Orthotopic constructions

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.

References

* 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-Strauss, ''The Symmetries of Things'' 2008, (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,

** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45

External links