In five-dimensional

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 c ...

, a 5-cube is a name for a five-dimensional

hypercube 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, ...

with 32 vertices, 80

edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

s, 80 square

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

, 40 cubic cells, and 10

tesseract In geometry, a 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 eig ...

4-face In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a ''polyhedron''. In more technical treatments of the geometry of polyhedra ...

s. It is represented by

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

or , constructed as 3 tesseracts, , around each cubic

ridge A ridge or a mountain ridge is a geographical feature consisting of a chain of mountains or hills that form a continuous elevated crest for an extended distance. The sides of the ridge slope away from the narrow top on either side. The line ...

. It can be called a penteract, a

portmanteau A portmanteau word, or portmanteau (, ) is a blend of wordstesseract In geometry, a 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 eig ...

'' (the 4-cube). It can also be called a regular deca-5-tope or decateron, being a 5-dimensional polytope constructed from 10 regular

facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...

Related polytopes

It is a part of an infinite

family. The dual of a 5-cube is the

5-orthoplex In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces. It has two constructed forms, the first being regular with ...

, of the infinite family of

orthoplex In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...

es. Applying an '' alternation'' operation, deleting alternating vertices of the 5-cube, creates another

uniform 5-polytope In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope Facet (geometry), facets. The complete set of convex uniform 5-polytopes ...

, called a 5-demicube, which is also part of an infinite family called the

demihypercube 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 ''hγn'' for being ''half'' of the hy ...

s. The 5-cube can be seen as an ''order-3 tesseractic honeycomb'' on a 4-sphere. It is related to the Euclidean 4-space (order-4)

tesseractic honeycomb In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol , and constructed by a 4-dimensional packing of tesseract facets. Its ver ...

and paracompact hyperbolic honeycomb

order-5 tesseractic honeycomb In the geometry of Hyperbolic space, hyperbolic 4-space, the order-5 tesseractic honeycomb is one of five compact regular polytope, regular space-filling tessellations (or honeycomb (geometry), honeycombs). With Schläfli symbol , it has five 8-cel ...

As a configuration

This configuration matrix represents the 5-cube. 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-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.Coxeter, Complex Regular Polytopes, p.117

\begin\begin
32 & 5 & 10 & 10 & 5 \\ 
2 & 80 & 4 & 6 & 4 \\ 
4 & 4 & 80 & 3 & 3 \\ 
8 & 12 & 6 & 40 & 2 \\ 
16 & 32 & 24 & 8 & 10 
\end\end

Cartesian coordinates

The

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

of the vertices of a 5-cube centered at the origin and having edge length 2 are : (±1,±1,±1,±1,±1), while this 5-cube's interior consists of all points (''x''₀, ''x''₁, ''x''₂, ''x''₃, ''x''₄) with -1 < ''x''_''i'' < 1 for all ''i''.

Images

''n''-cube

Coxeter plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...

projections in the B_k

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 refl ...

s project into k-cube graphs, with power of two vertices overlapping in the projective graphs.

Projection

The 5-cube can be projected down to 3 dimensions with a

rhombic icosahedron The rhombic icosahedron is a polyhedron shaped like an oblate sphere. Its 20 faces are congruent golden rhombi; 3, 4, or 5 faces meet at each vertex. It has 5 faces (green on top figure) meeting at each of its 2 poles; these 2 vertices lie on it ...

envelope. There are 22 exterior vertices, and 10 interior vertices. The 10 interior vertices have the convex hull of a

pentagonal antiprism In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for ...

. The 80 edges project into 40 external edges and 40 internal ones. The 40 cubes project into golden rhombohedra which can be used to dissect the rhombic icosahedron. The projection vectors are u = , v = , w = , where φ is the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

\frac

Symmetry

The ''5-cube'' has

symmetry B₅, abstract structure

C_\wr S_

, order 3840, containing 25 hyperplanes of reflection. The

for the 5-cube, , matches the

Coxeter notation In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram ...

symmetry ,3,3,3

Prisms

All

have lower symmetry forms constructed as prisms. The 5-cube has 7 prismatic forms from the lowest 5-

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 Congruence (geometry), congruent to the Cartesian product ...

, ⁵, and upwards as orthogonal edges are constrained to be of equal length. The vertices in a prism are equal to the product of the vertices in the elements. The edges of a prism can be partitioned into the number of edges in an element times the number of vertices in all the other elements.

Related polytopes

The ''5-cube'' is 5th in a series of

: The

regular skew polyhedron In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra ...

can be realized within the 5-cube, with its 32 vertices, 80 edges, and 40 square faces, and the other 40 square faces of the 5-cube become square ''holes''. This polytope is one of 31

s generated from the regular 5-cube or

References

* H.S.M. Coxeter: ** Coxeter, ''

Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

'', (3rd edition, 1973), Dover edition, , 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,

*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* Norman Johnson ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966) *

External links

* *
Multi-dimensional Glossary: hypercube
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