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10-cube
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10-cube
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10-cube
Dekeract

Orthogonal projection
inside Petrie polygon
Orange vertices are doubled, and central yellow one has four
Type Regular 10-polytope e
Family hypercube
Schläfli symbol {4,38}
Coxeter-Dynkin diagram
9-faces 20 {4,37}
8-faces 180 {4,36}
7-faces 960 {4,35}
6-faces 3360 {4,34}
5-faces 8064 {4,33}
4-faces 13440 {4,3,3}
Cells 15360 {4,3}
Faces 11520 squares
Edges 5120 segments
Vertices 1024 points
Vertex figure 9-simplex
Petrie polygon icosagon
Coxeter group C10, [38,4]
Dual 10-orthoplex
Properties convex, Hanner polytope

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.

It can be named by its Schläfli symbol {4,38}, being composed of 3 9-cubes around each 8-face. It is sometimes called a dekeract, a portmanteau of tesseract (the 4-cube) and deka- for ten (dimensions) in Greek, It can also be called an icosaronnon or icosa-10-tope as a 10 dimensional polytope, constructed from 20 regular facets.

It is a part of an infinite family of polytopes, called hypercubes. The dual of a dekeract can be called a 10-orthoplex or decacross, and is a part of the infinite family of cross-polytopes.

Cartesian coordinates

[edit]

Cartesian coordinates for the vertices of a dekeract centered at the origin and edge length 2 are

(±1,±1,±1,±1,±1,±1,±1,±1,±1,±1)

while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9) with −1 < xi < 1.

Other images

[edit]

This 10-cube graph is an orthogonal projection. This orientation shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:10:45:120:210:252:210:120:45:10:1.
Orthographic projections
B10 B9 B8
[20] [18] [16]
B7 B6 B5
[14] [12] [10]
B4 B3 B2
[8] [6] [4]
A9 A5
[10] [6]
A7 A3
[8] [4]

Derived polytopes

[edit]

Applying an alternation operation, deleting alternating vertices of the dekeract, creates another uniform polytope, called a 10-demicube, (part of an infinite family called demihypercubes), which has 20 demienneractic and 512 enneazettonic facets.

References

[edit]
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10-cube

View on Grokipedia
from Grokipedia
A 10- is a ten-dimensional analog of a , known formally as a regular 10-polytope or in R10\mathbb{R}^{10}. Its vertices are the points with coordinates (±1,±1,…,±1)(\pm 1, \pm 1, \dots, \pm 1) in ten dimensions, and it is the convex hull of these points, forming a convex body bounded by 20 mutually perpendicular 9-dimensional as facets. The vertices of a 10- number 210=10242^{10} = 1024, each connected by edges to ten others, yielding 10×29=512010 \times 2^{9} = 5120 edges in total. Higher-dimensional faces include (102)×28=11520\binom{10}{2} \times 2^{8} = 11520 squares (2-faces), (103)×27=15360\binom{10}{3} \times 2^{7} = 15360 (3-faces), and so on, up to the 20 9-, following the general formula for the number of kk-faces in an nn- as (nk)2n−k\binom{n}{k} 2^{n-k}. Its Schläfli symbol is {4,38}\{4, 3^{8}\}, denoting a sequence of regular polytopes with right-angled cells built recursively from lower dimensions. The symmetry group of the 10-cube is the hyperoctahedral group of order 210×10!=3,715,891,2002^{10} \times 10! = 3,715,891,200, comprising all sign changes and permutations of the coordinates, half of which are proper rotations preserving orientation. As a measure polytope, it has uniform density and serves as a fundamental object in higher-dimensional geometry, with applications in combinatorial optimization and parallel computing architectures modeled after its graph structure. The dual polytope is the 10-dimensional cross-polytope (10-orthoplex), which has 20 vertices and 1024 9-simplex facets.

Definition and Terminology

Definition

The 10-cube is a 10-dimensional analog of the , generalizing the geometric structure of lower-dimensional s to of 10. It is defined as the of its vertices, which are the points in R10\mathbb{R}^{10} with coordinates (±1,±1,…,±1)(\pm 1, \pm 1, \dots, \pm 1), an alternative unit version uses coordinates 0 or 1 in each ; a centered version uses coordinates ±1/2\pm 1/2. This construction ensures the 10-cube is a bounded, embedded in 10-dimensional space. The notion of the n-cube extends naturally from the 2-cube, known as the square, which is the convex hull of four points in the plane, to the 3-cube or ordinary cube, the convex hull of eight points in . This pattern of dimensional progression was formalized in the as part of the development of higher-dimensional geometry, with Ludwig Schläfli providing a systematic treatment in his 1852 work on n-dimensional continua. The 10-cube represents the specific case where n=10, maintaining the same recursive structure of faces and symmetry as its lower-dimensional counterparts. As a , the 10-cube is convex and equilateral, with all facets being congruent 9-cubes and the figure invariant under the action of the hyperoctahedral group, establishing it as a foundational object for studying combinatorial and metric properties in high-dimensional geometry. This abstract definition sets the stage for explorations of its structural elements and without specifying numerical counts of subelements.

Naming Conventions

The 10-cube, also known as the 10-dimensional , is the standard terminology used in mathematical literature to denote the in ten-dimensional that generalizes the to higher dimensions. This naming convention extends the pattern from lower dimensions, where the 3-cube is simply called a and the 4-cube a , emphasizing the dimensional progression without implying a specific geometric embedding. Although occasionally described as a "10-dimensional tesseract" in informal contexts, this usage is imprecise since "tesseract" conventionally refers exclusively to the 4-cube. The Schläfli symbol for the 10-cube is {4,38}\{4,3^{8}\}, or expanded as {4,3,3,3,3,3,3,3,3}\{4,3,3,3,3,3,3,3,3\}, which compactly describes its structure as a regular polytope with square faces and successive vertex figures that are octahedra up to the ninth dimension. This notation highlights its regularity, where two 9-cubes meet at each 8-dimensional face, maintaining the hypercubic uniformity across dimensions. The concept of n-dimensional cubes, including the 10-cube, was first systematically introduced by Ludwig Schläfli in his 1852 treatise Theorie der vielfachen Kontinuität, where he developed the theory of regular polytopes in arbitrary dimensions by analogy to Platonic solids and regular tessellations. In specialized contexts, alternative notations appear: in , the 1-skeleton (edge graph) of the 10-cube is denoted Q10Q_{10}, representing the binary hypercube graph with 2102^{10} vertices connected by edges differing in exactly one bit. Additionally, its full is the of type B10B_{10}, known as the hyperoctahedral group, which encodes the reflections generating the polytope's isometries.

Combinatorial Structure

Vertices and Edges

The 10-cube possesses exactly 210=[1024](/page/1024)2^{10} = [1024](/page/1024) vertices, each uniquely identified by a binary string of length 10, where the coordinates are either 0 or 1. These vertices represent all possible combinations of 10 binary choices, forming the 0-skeleton of the . The edges of the 10-cube connect pairs of vertices that differ in exactly one coordinate, resulting in 10×29=512010 \times 2^{9} = 5120 edges. Each vertex has degree 10, as flipping any one of the 10 coordinates yields a neighboring vertex. In the standard geometric realization as the unit , all edges have equal length 1, measured in the Euclidean metric. The 1-skeleton of the 10-cube corresponds to the Q10Q_{10}, which is bipartite, with vertices partitioned into two sets based on the parity of the number of 1s in their binary representations. This graph is Hamiltonian, admitting a cycle that visits each vertex exactly once, and has 10, equal to the maximum between any two vertices. It is a distance-regular graph with intersection array {10,9,8,7,6,5,4,3,2,1;1,2,3,4,5,6,7,8,9,10}. Additionally, Q10Q_{10} satisfies strong isoperimetric inequalities, ensuring that subsets of vertices have boundary sizes that grow efficiently with their cardinality, a property central to its use in and .

Cells and Higher Faces

The faces of the 10-cube encompass all proper sub-polytopes from 2-dimensional squares up to 9-dimensional facets, each of which is itself a regular of the corresponding lower . In general, for an n-dimensional , the number of k-dimensional faces is given by the formula (nk)2n−k\binom{n}{k} 2^{n-k}, where (nk)\binom{n}{k} counts the ways to choose the k free coordinates that vary along the face, and 2n−k2^{n-k} accounts for the fixed values (0 or 1) in the remaining n-k coordinates. This enumeration highlights the combinatorial regularity of the , building recursively from lower-dimensional cubes embedded within it. For the 10-cube specifically (n=10), the counts of these higher faces are as follows:
Dimension (k)TypeNumber of k-faces
2Squares(102)28=11,520\binom{10}{2} 2^{8} = 11{,}520
3Cubes(103)27=15,360\binom{10}{3} 2^{7} = 15{,}360
4Tesseracts(104)26=13,440\binom{10}{4} 2^{6} = 13{,}440
55-cubes(105)25=8,064\binom{10}{5} 2^{5} = 8{,}064
66-cubes(106)24=3,360\binom{10}{6} 2^{4} = 3{,}360
77-cubes(107)23=960\binom{10}{7} 2^{3} = 960
88-cubes(108)22=180\binom{10}{8} 2^{2} = 180
99-cubes(109)21=20\binom{10}{9} 2^{1} = 20
These values are computed directly from the general formula and illustrate the symmetric distribution of faces, with the maximum occurring around the middle dimensions due to the peak of the binomial coefficients. The 9-dimensional facets of the 10-cube number and are regular 9-cubes, forming the bounding hypersurfaces that enclose the interior. Each such facet arises by fixing one of the 10 coordinates to either 0 or 1, yielding the 10 × 2 = possibilities. This boundary structure underscores the hypercube's recursive nature, where the facets are themselves hypercubes one lower, mirroring the construction of the full 10-cube from two parallel 9-cubes connected by edges in the 10th direction. Regarding connectivity among faces, each k-face of the 10-cube is incident to exactly (10 - k) higher-dimensional (k+1)-faces. This follows from the 's coordinate-based embedding: a k-face has (10 - k) coordinates fixed, and extending to a (k+1)-face requires freeing exactly one of those fixed coordinates to vary, providing (10 - k) choices without altering the original face. This incidence relation reinforces the recursive layering of the 10-cube, where lower-dimensional faces aggregate to form higher ones in a uniform, dimension-dependent manner.

Geometric Representation

Cartesian Coordinates

The 10-cube, as a in 10-dimensional , is commonly represented using Cartesian coordinates for its vertices. In the standard unit hypercube formulation, the vertices consist of all points (x1,x2,…,x10)(x_1, x_2, \dots, x_{10}) where each xi∈{0,1}x_i \in \{0, 1\}. This embedding positions the as the of ten unit intervals [0,1][0, 1], with edges connecting vertices that differ in exactly one coordinate by 1, resulting in an edge length of 1. A symmetric, centered variant has vertices at all combinations of (±1,±1,…,±1)(\pm 1, \pm 1, \dots, \pm 1) in 10 dimensions, forming the [−1,1]10[-1, 1]^{10} with edge length 2. An alternative unit-edge centered representation shifts the unit to align with the origin, with vertices at all combinations of (±1/2,±1/2,…,±1/2)(\pm 1/2, \pm 1/2, \dots, \pm 1/2) in 10 dimensions. This is obtained by translating the original coordinates by −1/2-1/2 in each , equivalently viewing the 10-cube as the product of ten intervals [−1/2,1/2][-1/2, 1/2]. Edges in this centered form connect vertices differing by 1 in a single coordinate (from −1/2-1/2 to +1/2+1/2), yielding an edge length of (1)2=1\sqrt{(1)^2} = 1
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