geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, 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 ''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 hypercube name: demicube, demitesseract, etc. The demicube is identical to the regular

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...

, and the demitesseract is identical to the regular

16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the ...

. The

demipenteract In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' ( penteract) with alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only semiregular 5 ...

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 In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...

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 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 as half the vertices of . The

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...

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

Constructions

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

orthotope In geometry, an orthotopeCoxeter, 1973 (also called a hyperrectangle or a box) is the generalization of a rectangle to higher dimensions. A necessary and sufficient condition is that it is congruent to the Cartesian product of intervals. If all o ...

) s #... (As an alternated hypercube) h #.... (As a demihypercube) H.S.M. Coxeter also labeled the third bifurcating diagrams as 1_''k''1 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. 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 In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a paramete ...

(the

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...

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

s 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 The ''Messenger of Mathematics'' is a defunct British mathematics journal. The founding editor-in-chief was William Allen Whitworth with Charles Taylor and volumes 1–58 were published between 1872 and 1929. James Whitbread Lee Glaisher was th ...

, Macmillan, 1900 * John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''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