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8-cube

The 8-cube is a eight-dimensional and the generalization of the three-dimensional to eight dimensions, also known as an n-cube for n=8, octeract, or a measure polytope with {4,36}. It is defined as the of the 256 points in eight-dimensional where each coordinate is either -1 or +1, or equivalently, the set of all points with coordinates in {0,1}8 scaled appropriately. This possesses a rich combinatorial structure, featuring 256 vertices, 1,024 edges, 1,792 square 2-faces, 1,792 cubic 3-faces, 1,120 4-faces, 448 penteract 5-faces, 112 hexaract 6-faces, and 16 heptaract 7-faces. These counts follow the general formula for the number of k-dimensional faces in an n-cube: $$ \binom{n}{k} 2^{n-k} The [8-cube](/page/8-cube) can be constructed recursively by taking the [Cartesian product](/page/Cartesian_product) of a [7-cube](/page/7-cube) with a [line segment](/page/Line_segment).[](https://mathworld.wolfram.com/Hypercube.html) Its [symmetry group](/page/Symmetry_group) is the hyperoctahedral group of [order](/page/Order) $2^{8} \times 8! = 10,[321](/page/321),920$, consisting of all signed permutations of the coordinates, which acts transitively on the vertices and edges.[](https://polytope.miraheze.org/wiki/8-cube) In [graph theory](/page/Graph_theory), the 1-skeleton of the [8-cube](/page/8-cube) forms the [hypercube graph](/page/Hypercube_graph) $Q_{8}$, a [bipartite graph](/page/Bipartite_graph) with diameter 8 and [Hamiltonian](/page/Hamiltonian) laceability, useful in modeling [parallel computing](/page/Parallel_computing) architectures and interconnection networks.[](https://www.researchgate.net/figure/The-octeract-graph-of-an-8-cube-displaying-the-relationship-among-the-256-vertices-of_fig1_370612668) ## Definition and Properties ### Definition The 8-cube, also known as the octeract, is the eight-dimensional hypercube, serving as the generalization of the three-dimensional cube to higher dimensions and functioning as a convex regular polytope in eight-dimensional Euclidean space. As a regular polytope, it is bounded by 16 facets, each of which is a seven-dimensional hypercube.[](https://mathworld.wolfram.com/Hypercube.html) The Schläfli symbol of the 8-cube is \{4,3,3,3,3,3,3\}, which recursively defines its structure starting from square faces (\{4\}) and building iteratively through cubes and higher-dimensional analogs. This symbol encapsulates the uniformity where three such elements meet at each subelement across dimensions.[](https://mathworld.wolfram.com/SchlaefliSymbol.html)[](https://mathworld.wolfram.com/Hypercube.html) The 8-cube possesses key general properties including convexity, as the [convex hull](/page/Convex_hull) of its vertices; regularity, with all faces being congruent [regular](/page/Regular) polytopes and all edges of equal length; and central symmetry, remaining invariant under [point reflection](/page/Point_reflection) through its geometric center.[](https://mathworld.wolfram.com/Hypercube.html)[](https://polytope.miraheze.org/wiki/Hypercube) Higher-dimensional [geometry](/page/Geometry), including the 8-cube, was first rigorously explored by Ludwig Schläfli in his work completed in 1852, *Theorie der vielfachen Kontinuität* (published 1901), where he introduced [polytopes](/page/Polytope) as analogs of polygons and polyhedra in n dimensions greater than three; specific analyses of the 8-cube emerged in 20th-century polytope theory, particularly through H. S. M. Coxeter's systematic classification of [regular](/page/Regular) polytopes.[](https://mathshistory.st-andrews.ac.uk/Biographies/Schlafli/)[](https://books.google.com/books/about/Regular_Polytopes.html?id=iWvXsVInpgMC) ### Dimensional Elements The 8-cube, as a [regular](/page/Regular) 8-dimensional polytope, is composed of various lower-dimensional elements that form its [boundary](/page/Boundary) and structure. These elements include vertices, edges, faces, and higher-dimensional facets, each corresponding to regular polytopes of dimensions from 0 to 7. The counts of these elements follow a combinatorial pattern inherent to [hypercube](/page/Hypercube)s.[](https://www.math.brown.edu/tbanchof/Beyond3d/chapter4/section07.html) The number of $k$-dimensional faces ($f_k$) in an $n$-dimensional [hypercube](/page/Hypercube) is given by the formula f_k(n) = \binom{n}{k} 2^{n-k}, where $\binom{n}{k}$ is the [binomial coefficient](/page/Binomial_coefficient) representing the number of ways to choose $k$ dimensions out of $n$ to allow variation, and $2^{n-k}$ accounts for the two possible fixed positions (positive or negative) in each of the remaining $n-k$ dimensions.[](https://www.gregegan.net/APPLETS/29/HypercubeNotes.html) This derivation arises from the standard coordinate representation of the [hypercube](/page/Hypercube), where vertices are all points with coordinates $(\pm 1/2, \pm 1/2, \dots, \pm 1/2)$ scaled by the edge length; a $k$-face is then obtained by fixing $n-k$ coordinates to one of their two [boundary](/page/Boundary) values and allowing the remaining $k$ to vary fully along the edge length.[](https://www.gregegan.net/APPLETS/29/HypercubeNotes.html) For the specific case of the 8-cube ($n=8$), this [formula](/page/Formula) yields the following counts of elements: | Dimension ($k$) | Type | Number of elements | |-----------------|-----------------------|--------------------| | 0 | Vertices | 256 | | 1 | Edges | 1024 | | 2 | Squares | 1792 | | 3 | Cubes | 1792 | | 4 | Tesseracts | 1120 | | 5 | 5-cubes | 448 | | 6 | 6-cubes | 112 | | 7 | 7-cubes (facets) | 16 | These values are computed directly from $f_k(8) = \binom{8}{k} 2^{8-k}$.[](https://polytope.miraheze.org/wiki/8-cube) The 1-skeleton consists of the vertices and edges, forming the underlying [graph](/page/Graph) where each [vertex](/page/Vertex) connects to 8 others. Higher elements build upon this: 2-faces are the square boundaries between edges, 3-faces are cubic volumes enclosing those squares, and so on, up to the 7-faces, which are the 7-dimensional facets that bound the entire 8-cube.[](https://www.math.brown.edu/tbanchof/Beyond3d/chapter4/section07.html) The $n$-dimensional content (hypervolume) of a [hypercube](/page/Hypercube) with [edge](/page/Edge) length $a$ is $V_n = a^n$. For the 8-cube, this specializes to $V_8 = a^8$, which equals 1 when $a=[1](/page/1)$.[](https://math.stackexchange.com/questions/894378/volume-of-a-cube-and-a-ball-in-n-dimensions) ## Geometric Coordinates ### Cartesian Coordinates The vertices of the 8-cube can be embedded in 8-dimensional [Euclidean space](/page/Euclidean_space) $\mathbb{R}^8$ using Cartesian coordinates where each vertex has coordinates $\pm \frac{1}{2}$ in every dimension, resulting in all possible combinations of these values across the eight coordinates. This positioning centers the 8-cube at the origin and ensures a unit edge length, as the [Euclidean distance](/page/Euclidean_distance) between two adjacent vertices—differing in exactly one coordinate by 1—calculates to $\sqrt{1^2 + 0^2 + \cdots + 0^2} = 1$. An alternative standard embedding places the vertices at all points in $\{0, 1\}^8$, forming the unit 8-cube aligned with the positive [orthant](/page/Orthant).[](https://www.math.toronto.edu/mathnet/plain/questionCorner/eucgeom.html) In this representation, adjacent vertices differ by 1 in precisely one coordinate, yielding the same unit edge length of $\sqrt{1} = 1$. Both coordinate systems describe the same combinatorial structure with $2^8 = 256$ vertices, though the $\{0, 1\}^8$ version shifts the center to $(\frac{1}{2}, \frac{1}{2}, \dots, \frac{1}{2})$.[](https://www.math.toronto.edu/mathnet/plain/questionCorner/eucgeom.html) The set of all vertices in the $\pm \frac{1}{2}$ representation can be generated as the orbit of the point $(\frac{1}{2}, \frac{1}{2}, \dots, \frac{1}{2})$ under the action of the hyperoctahedral group $B_8$, which consists of all signed permutations of the coordinates—permutations of the eight axes combined with independent sign flips on any subset of coordinates.[](https://corescholar.libraries.wright.edu/cgi/viewcontent.cgi?article=2608&context=etd_all) This group action transitively maps any vertex to any other, reflecting the full symmetry of the 8-cube while preserving distances and the embedding.[](https://corescholar.libraries.wright.edu/cgi/viewcontent.cgi?article=2608&context=etd_all) ### Vertex Figure The vertex figure of the 8-cube is a regular 7-simplex, a uniform 7-polytope with [Schläfli symbol](/page/Schläfli_symbol) \{3,3,3,3,3,3\}. This figure arises as the intersection of the 8-cube with a [hyperplane](/page/Hyperplane) positioned near a vertex but in [general position](/page/General_position) so as not to pass through it or any other vertex, yielding a scaled and translated regular 7-simplex that describes the combinatorial and geometric arrangement of elements incident to that vertex.[](https://polytope.miraheze.org/wiki/Hypercube) Each vertex of the 8-cube connects to 8 other vertices via edges, and these 8 neighboring vertices form the vertex set of the vertex figure. The 1-skeleton of the vertex figure is the complete graph $K_8$ on these 8 vertices, where each edge of the graph corresponds to a 2-face of the 8-cube that contains exactly two of the emanating edges from the original vertex. The dihedral angle of the vertex figure—the angle between two adjacent 6-dimensional facets of the 7-simplex—is $\arccos\left(\frac{1}{7}\right) \approx 81.787^\circ$.[](https://arxiv.org/abs/1304.0967) This value generalizes from lower dimensions, such as the dihedral angle $\arccos\left(\frac{1}{3}\right) \approx 70.529^\circ$ for the tetrahedral vertex figure of the 4-cube (tesseract). ## Combinatorial Aspects ### As a Configuration In combinatorial geometry, the 8-cube can be modeled as a point-line [incidence structure](/page/Incidence_structure), where the 256 vertices serve as points and the 1024 edges as lines, forming a [configuration](/page/Configuration) in which each point is incident to 8 lines and each line is incident to 2 points.[](https://mathworld.wolfram.com/Hypercube.html) This structure captures the relational incidences between vertices and edges without reference to their embedding in [Euclidean space](/page/Euclidean_space).[](https://mathworld.wolfram.com/HypercubeGraph.html) More generally, the n-cube realizes a regular (2^n_n, n \cdot 2^{n-1}_2)-configuration, consisting of 2^n points and n \cdot 2^{n-1} lines, where two points are incident to a common line if and only if their representing binary coordinate vectors differ in exactly one position.[](https://mathworld.wolfram.com/HypercubeGraph.html) In this model, the incidences reflect the Hamming distance of 1 between adjacent vertices.[](https://mathworld.wolfram.com/HypercubeGraph.html) From a graph-theoretic perspective, this incidence structure corresponds to the 8-cube graph, which is the Cartesian product of eight complete graphs K_2 and exhibits a diameter of 8—the maximum shortest-path distance between any two vertices—and a girth of 4, indicating the smallest cycle length.[](https://mathworld.wolfram.com/HypercubeGraph.html)[](https://mathworld.wolfram.com/HypercubeGraph.html) Unlike the regular 8-simplex and 8-orthoplex, which have triangular 2-faces, the 8-cube features square 2-faces.[](https://mathworld.wolfram.com/Hypercube.html)[](https://mathworld.wolfram.com/Simplex.html)[](https://mathworld.wolfram.com/CrossPolytope.html) ### Face Lattice The face lattice of the 8-cube is the [partially ordered set](/page/Partially_ordered_set) (poset) consisting of all its faces, including the empty face and the 8-cube itself, ordered by [inclusion](/page/Inclusion). Each face corresponds to a choice, for each of the eight coordinates, of whether it is fixed at 0, fixed at 1, or free to vary between 0 and 1; the dimension of such a face is the number of free coordinates. This structure forms a [distributive lattice](/page/Lattice) with $3^8 = 6561$ elements.[](https://epubs.siam.org/doi/10.1137/0135057) In this poset, the covering relations occur when a $k$-dimensional face is covered by a $(k+1)$-dimensional face obtained by changing exactly one fixed coordinate to free in the lower-dimensional face.[](https://link.springer.com/article/10.1007/BF02483910) The number of $k$-faces is $\binom{8}{k} 2^{8-k}$, reflecting the choice of $k$ free dimensions and the $2^{8-k}$ ways to fix the remaining coordinates to 0 or 1. The [Euler characteristic](/page/Euler_characteristic) of the face lattice, computed as the alternating [sum](/page/Sum) of the face numbers $\sum_{k=0}^{8} (-1)^k f_k$ (where $f_k$ is the number of $k$-faces and $f_8 = 1$), equals [1](/page/1), corresponding to the [topology](/page/Topology) of the closed 8-ball; this follows from the binomial expansion $(2 - 1)^8 = 1^n = [1](/page/1)$. For the boundary complex excluding the interior 8-face, the [sum](/page/Sum) up to $k=7$ yields 0, consistent with the [Euler characteristic](/page/Euler_characteristic) of the 7-sphere. The Möbius function $\mu$ of the lattice satisfies $\mu(\emptyset, F) = (-1)^{\dim F}$ for any face $F$, a property holding for the face lattices of all convex polytopes.[](https://web.math.ucsb.edu/~mccammon/slides/durham4.pdf) This function facilitates inclusion-exclusion principles in computing polytope volumes, such as expressing the volume of the 8-cube as an inversion over its face lattice relative to the standard basis parallelotope. ## Symmetry Group ### Full Symmetry The full symmetry group of the 8-cube is the hyperoctahedral group $B_8$, a finite Coxeter group of type $B_8$ whose order is $2^8 \cdot 8! = 10{,}321{,}920$. This group is realized as the wreath product $\mathbb{Z}_2 \wr S_8$ and is generated by all permutations of the eight coordinate axes together with independent sign changes (reflections through the origin in each coordinate hyperplane).[](https://alco.centre-mersenne.org/item/10.5802/alco.316.pdf) As the Coxeter group $[4,3,3,3,3,3,3]$, $B_8$ admits a presentation via generators $r_1, \dots, r_8$ satisfying $r_i^2 = 1$ and [braid](/page/Braid) relations determined by the Coxeter-Dynkin diagram: a linear chain of eight nodes where all adjacent bonds represent angle $\pi/3$ except the final bond between $r_7$ and $r_8$, which represents $\pi/4$ (indicated by a double edge). The group $B_8$ acts transitively on the $2^8 = 256$ vertices of the 8-cube, which may be taken as the points $(\pm 1, \pm 1, \dots, \pm 1) \in \mathbb{R}^8$. To determine the stabilizer of a vertex, apply the orbit-stabilizer theorem: the orbit size is $256$ and $|B_8| = 10{,}321{,}920$, so the stabilizer order is $10{,}321{,}920 / 256 = 40{,}320 = 8!$. This stabilizer consists precisely of the $S_8$-action via coordinate permutations (with no sign changes, as any sign change would map the vertex to a distinct one), yielding an isomorphism to the symmetric group $S_8$. ### Rotation Group The rotation group of the 8-cube is the index-2 [subgroup](/page/Subgroup) of the full symmetry group, is the Weyl group of type $D_8$, consisting of all orientation-preserving isometries that map the 8-cube to itself. This [subgroup](/page/Subgroup) has order $2^7 \cdot 8! = 5{,}160{,}960$.[](https://projecteuclid.org/journals/experimental-mathematics/volume-5/issue-4/Attractors-with-the-Symmetry-of-the-n-Cube/em/1047565450.pdf) It is isomorphic to the Weyl group of type $D_8$, which can be realized as the group of signed permutations of the 8 coordinates where the number of negative signs (sign flips) is even. The generators of this group include even [permutations](/page/Permutation) of the coordinates paired with an even number of sign flips, ensuring the overall [determinant](/page/Determinant) is +1. Additionally, it is generated by [rotations](/page/Rotation) such as 90-degree turns around the coordinate axes; for instance, a 90-degree [rotation](/page/Rotation) in the [plane](/page/Plane) spanned by the first two [standard basis](/page/Standard_basis) vectors cycles the coordinates in that [plane](/page/Plane) while preserving the others, corresponding to a signed [permutation](/page/Permutation) with det 1. These generators produce all proper [rotations](/page/Rotation) that preserve the hypercubic [lattice](/page/Lattice). Conjugacy classes within the rotation group are classified according to the dimensions of the fixed-point subspaces (axes) and the rotation angles in the orthogonal complementary subspaces. For example, one class consists of 90-degree (or 270-degree) rotations in 2-dimensional planes, leaving the remaining 6 dimensions fixed; another includes 120-degree and 240-degree rotations in 3-dimensional subspaces, with the other 5 dimensions fixed. Further classes involve 180-degree rotations in 2D or higher even-dimensional subspaces, or combinations thereof that partition the 8-dimensional space, all with angles compatible with the hypercubic lattice (multiples of 90 degrees in even dimensions or 120/180 in odd). This classification arises from the action on the coordinate basis and the even-sign condition. ## Projections and Visualizations ### Orthogonal Projections The orthogonal [projection](/page/Projection) of the 8-cube onto a $k$-dimensional [subspace](/page/Subspace) is performed via a [linear map](/page/Linear_map) defined by an [orthonormal basis](/page/Orthonormal_basis) $\{\mathbf{u}_1, \dots, \mathbf{u}_k\}$ of the [subspace](/page/Subspace), where the projected coordinates of a [vertex](/page/Vertex) $\mathbf{v} \in \mathbb{R}^8$ are given by $(\langle \mathbf{v}, \mathbf{u}_1 \rangle, \dots, \langle \mathbf{v}, \mathbf{u}_k \rangle)$. The vertices of the 8-cube are the $2^8 = 256$ points with all possible combinations of coordinates $\pm 1/2$, ensuring the [hypercube](/page/Hypercube) is centered at the [origin](/page/Origin) with [edge](/page/Edge) [length](/page/Length) 1.[](https://www.gregegan.net/APPLETS/29/HypercubeNotes.html)[](https://arxiv.org/abs/2501.10257) This [projection](/page/Projection) preserves parallelism of edges and results in a zonotope as the [convex hull](/page/Convex_hull), specifically a $2 \times 8 = 16$-gon in [2D](/page/2D) or a more complex zonotope in [3D](/page/3D), with interior points corresponding to the projected vertices.[](https://link.aps.org/doi/10.1103/v291-9hxy) A basic 2D orthogonal projection onto the subspace spanned by the first two standard basis vectors $\mathbf{e}_1$ and $\mathbf{e}_2$ simply retains the first two coordinates of each vertex, yielding a square grid of 4 points at $(\pm 1/2, \pm 1/2)$, with all 256 vertices collapsing onto these locations due to the degeneracy in the higher dimensions. However, a non-degenerate full 8D projection onto a generic 2D subspace produces a dense set of 256 distinct points within the 16-gon convex hull, often visualized using principal component analysis (PCA) to maximize variance and reveal structural patterns like clustered vertices or fractal-like distributions in spin systems modeled by the hypercube.[](https://arxiv.org/abs/2501.10257) For instance, an isometric 2D projection aligns the contribution vectors of the edge directions to form symmetric patterns, such as a regular 16-gon boundary enclosing the projected vertex cloud.[](https://link.aps.org/doi/10.1103/v291-9hxy) In 3D, an orthogonal [projection](/page/Projection) of the 8-cube is obtained by selecting three mutually orthogonal directions in $\mathbb{R}^8$, such as via a $3 \times 8$ [projection matrix](/page/Projection_matrix) with orthonormal rows, mapping the 256 vertices to points in $\mathbb{R}^3$. The resulting figure is a complex arrangement of projected cubic cells forming a lattice-like structure with interconnected tunnels, where parallel edges remain parallel and the overall shape is a 56-faced zonohedron bounded by rhombic faces. This [visualization](/page/Visualization) highlights the hypercube's combinatorial connectivity, with inner "tunnels" representing projections of higher-dimensional paths between cubic facets, as seen in PCA-based projections that preserve key geometric relations.[](https://arxiv.org/abs/2501.10257) The [Schlegel diagram](/page/Schlegel_diagram) provides a generalization for projecting the [8-cube](/page/8-cube) radially from one of its [7-cube](/page/7-cube) facets onto a parallel 7D [hyperplane](/page/Hyperplane), embedding the interior structure without crossing edges. This perspective projection, using rays from a viewpoint near the reference facet, applies to lower-dimensional analogs (e.g., [3D](/page/3D) from a cubic [cell](/page/Cell)) to maintain combinatorial fidelity while minimizing overlaps. This method, analogous to Coxeter's projections for regular polytopes, facilitates studying the 8-cube's face lattice in reduced dimensions.[](https://www.math.brown.edu/tbanchof/Beyond3d/chapter6/section03.html) ### Perspective Projections Perspective projections of the 8-cube employ central [projection](/page/Projection) methods, where rays emanate from a designated viewpoint in 8-dimensional space and intersect the hypercube's elements before projecting onto a 7-dimensional [hyperplane](/page/Hyperplane), introducing radial [distortion](/page/Distortion) to convey depth. This technique scales elements based on their distance from the viewpoint, with closer features appearing larger and more prominent, analogous to how a 3D cube distorts in a vanishing-point [drawing](/page/Drawing). The mathematical framework for such n-dimensional [perspective](/page/Perspective) projections, applicable to the 8-cube, involves transforming coordinates via the [formula](/page/Formula) $ x_i' = \frac{(R - F) x_i}{R - x_n} $ for $ i = 1 $ to $ n-1 $, where $ (R, 0, \dots, 0) $ is the viewpoint, $ F $ defines the projection hyperplane (e.g., $ x_n = F $), and distortion arises from the denominator's variation with depth.[](http://reprints.gravitywaves.com/VIP/ViewExtraSpaceDims/Noll-1967_AComputerTechniqueForDisplayingNDimensionalHyperObjects.pdf) Typical viewpoints for 8-cube projections include a [vertex](/page/Vertex) or the center of a cubic [cell](/page/Cell), yielding renderings that depict the structure as a series of nested lower-dimensional hypercubes, with the nearest [vertex](/page/Vertex) expanding outward and remote sections contracting inward. These projections are computed iteratively—reducing from 8D to [7D](/page/The_7D), then 6D, and continuing to [3D](/page/3D)—to produce accessible [3D](/page/3D) models, where the 8-cube's 256 vertices and extensive edge network create intricate, self-intersecting wireframes that reveal hidden symmetries and topological features not visible in orthogonal views. In such [3D](/page/3D) renderings, "unfolded" perspectives often portray nested tesseracts or cubes with radial expansion from the viewpoint, emphasizing the hypercube's recursive layering.[](http://reprints.gravitywaves.com/VIP/ViewExtraSpaceDims/Noll-1967_AComputerTechniqueForDisplayingNDimensionalHyperObjects.pdf) Computational rendering of these projections utilizes software like Mathematica, which supports high-dimensional rotations and iterative projections for hypercubes up to and beyond 6D, enabling dynamic visualizations of the 8-cube through wireframe or shaded models. Advanced techniques extend ray tracing to 8D-to-[3D](/page/3D) mappings, tracing rays through the hypercube's [volume](/page/Volume) to simulate solid appearances, though self-intersections in the projected [3D](/page/3D) [space](/page/Space)—arising from the 8-cube's 1792 cubic cells overlapping in lower dimensions—necessitate careful handling to avoid visual artifacts. These methods highlight the unique distortions of high-dimensional geometry, such as exaggerated foreshortening of distant facets, providing intuitive depth cues via varying opacity or hue.[](https://demonstrations.wolfram.com/RotatingSquaresCubesAndHigherDimensionalHypercubes/) Early 20th-century visualizations by [Alicia Boole Stott](/page/Alicia_Boole_Stott), who pioneered intuitive unfoldings of 4D [polytopes](/page/Polytope) without formal mathematical training, laid foundational techniques for higher-dimensional geometry. Stott's net-based unfoldings inspired extensions in computational visualizations of [polytopes](/page/Polytope).[](https://www.sciencedirect.com/science/article/pii/S0315086007000973) ## Related Polytopes ### Derived Polytopes The derived polytopes of the 8-cube are convex uniform 8-[polytopes](/page/Polytope) generated via Archimedean-style operations on its regular form, utilizing the reflections of the B_8 [Coxeter group](/page/Coxeter_group) to ensure vertex-transitivity and uniform facets. These modifications, including [rectification](/page/Rectification) and [truncation](/page/Truncation), alter the edge structure while preserving the underlying [symmetry](/page/Symmetry), producing variants with mixed cell types derived from the original facets and vertex figures. The B_8 group's Wythoff constructions systematically yield these forms by selecting active reflection mirrors in the Coxeter diagram, enabling higher-order derivations like bitruncation and cantellation.[](https://polytope.miraheze.org/wiki/Uniform_8-polytope) The rectified 8-cube, denoted $ r\{4,3^6\} $, positions its vertices at the midpoints of the original 8-cube's edges, effectively truncating vertices and edges to their midsections. Its cells consist of [rectified 7-cubes](/page/Rectification), arising from the rectification of the original [7-cube](/page/7-cube) facets, and [regular](/page/Regular) 7-simplices, corresponding to the unchanged vertex figures of the 8-cube. This [structure](/page/Structure) maintains uniformity, with the rectified 7-cubes exhibiting cuboctahedral cross-sections in lower dimensions as a representative example of the operation's [effect](/page/Effect).[](https://polytope.miraheze.org/wiki/Rectified_8-cube) The truncated 8-cube, denoted $ t\{4,3^6\} $, shortens the original edges by fully truncating vertices, replacing them with new 7-simplex cells while transforming the 7-cube facets into truncated 7-cubes. These truncated 7-cubes feature regular 6-orthoplex vertex figures, illustrating the operation's insertion of new facets from the original edges. The resulting polytope remains uniform under the B_8 group, with the 7-simplices providing the scale for the vertex-derived elements.[](https://polytope.miraheze.org/wiki/Truncated_8-cube) Bitruncation and related Wythoff constructions further modify the 8-cube by applying successive truncations, often equivalent to truncating the [dual](/page/Dual) (itself an 8-orthoplex) and then the result. The bitruncated 8-cube, denoted $ 2t\{4,3^6\} $, has cells comprising bitruncated 7-cubes from the original facets and rectified 7-simplices from the vertices.[](https://polytope.miraheze.org/wiki/Bitruncated_8-cube) Other variants, such as the cantitruncated form $ t_{0,1,2}\{4,3^6\} $, include additional cell types like prism products between lower-dimensional elements, generated by marking multiple nodes in the B_8 diagram to expand edges and faces simultaneously. Cantellation and runcination represent higher-order derivations that insert prisms between the original cells and their boundaries, expanding the structure without truncating vertices directly. The cantellated 8-cube, denoted $ rr\{4,3^6\} $, features cells including 7-dimensional prisms over cuboctahedra (from edge expansions) and lower rectified forms, emphasizing the separation of original elements by cylindrical insertions.[](https://polytope.miraheze.org/wiki/Cantellated_8-cube) Runcination, as in $ t_{0,2}\{4,3^6\} $, extends this by runcitruncating facets, yielding cells such as runcinated 7-cubes and [simplex](/page/Simplex) prisms, which highlight the B_8 symmetry's capacity for complex uniform variants in eight dimensions.[](https://polytope.miraheze.org/wiki/Runcinated_8-cube) ### Analogous Polytopes The 8-cube belongs to the broader family of [hypercubes](/page/Hypercube), which are n-dimensional [convex polytopes](/page/Convex_hull) generalizing the square (2-cube) and [cube](/page/Cube) (3-cube) to higher dimensions. Each [hypercube](/page/Hypercube) is constructed as the [convex hull](/page/Convex_hull) of points with coordinates ±1 in all dimensions, scaled appropriately, and exhibits self-duality in certain combinatorial senses across dimensions.[](https://www.cs.cmu.edu/~ggordon/10725-F12/slides/17-dual-corresp.pdf) For instance, the 4-cube, or [tesseract](/page/Tesseract), consists of 8 cubic cells and serves as a direct analog in four dimensions, while the 16-cube extends the structure to 16 dimensions with 2^{16} vertices.[](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Markov.pdf) The dual polytope of the 8-cube is the 8-orthoplex, also called the 8-cross-polytope, which is the [convex hull](/page/Convex_hull) of the [standard basis](/page/Standard_basis) vectors and their negatives in 8-dimensional space. This duality maps vertices of the 8-cube to facets of the 8-orthoplex and vice versa, preserving the combinatorial structure through an inclusion-reversing [bijection](/page/Bijection) between their face lattices.[](https://www.dam.brown.edu/people/cklivans/kequal.pdf) The 8-orthoplex has 16 vertices and 256 7-simplex facets, contrasting with the 8-cube's 256 vertices and 16 [7-cube](/page/7-cube) facets, and represents the l1-ball in 8 dimensions dual to the l∞-ball of the [hypercube](/page/Hypercube).[](https://www.cs.cmu.edu/~ggordon/10725-F12/slides/17-dual-corresp.pdf)[](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Markov.pdf) In the sequence of cross-polytopes, the 8-orthoplex is analogous to the [octahedron](/page/Octahedron) (3-orthoplex) and the [16-cell](/page/16-cell) (4-orthoplex), each serving as the [dual](/page/Dual) to their respective hypercubes and sharing the property of being [regular](/page/Regular) polytopes with simplicial facets. This duality extends uniformly across dimensions, where the n-cross-polytope and n-hypercube are polar reciprocals, enabling applications in optimization and geometry where one bounds the other in normed spaces.[](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Markov.pdf)

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