Hypercube
A hypercube, also known as an n-cube, is an n-dimensional analogue of the square (2-cube) and cube (3-cube), defined geometrically as the set of all points (x_1, x_2, \dots, x_n) in n-dimensional Euclidean space \mathbb{R}^n such that $0 \leq x_i \leq 1 for each coordinate i = 1, 2, \dots, n.[1] This construction represents the unit hypercube, which can also be viewed as the Cartesian product of n unit line segments [0,1], making it a convex polytope with regular properties that extend those of lower-dimensional cubes.[1][2] The vertices of an n-dimensional hypercube are the $2^n points where each coordinate is either 0 or 1, corresponding to the corners of the figure; for example, the 1-cube is a line segment with 2 vertices, the 2-cube is a square with 4 vertices, and the 3-cube is a cube with 8 vertices.[1][2] The combinatorial structure of hypercubes is highly symmetric: an n-cube has n \cdot 2^{n-1} edges, as each of the $2^n vertices connects to exactly n others differing in one coordinate.[3] More generally, the number of k-dimensional faces (or k-cubes) in an n-cube is given by the formula \binom{n}{k} 2^{n-k}, which counts the ways to choose k dimensions to vary while fixing the others at 0 or 1.[3] The boundary of an n-cube consists of $2n facets, each an (n-1)-dimensional hypercube.[1] A notable example is the 4-dimensional hypercube, called a tesseract, which has 16 vertices, 32 edges, 24 square faces, and 8 cubic cells.[4] Tesseracts and higher-dimensional hypercubes are challenging to visualize directly but can be projected into lower dimensions, often appearing as nested or intersecting cubes.[3] These polytopes are fundamental in convex geometry and topology, serving as models for studying higher-dimensional spaces and their symmetries.[1]Fundamentals
Definition
A hypercube, also known as an n-cube, is the n-dimensional analog of a square in two dimensions and a cube in three dimensions. It is a convex polytope embedded in n-dimensional Euclidean space \mathbb{R}^n, formed as the convex hull of its vertices, which are all points with coordinates in the set \{0,1\}^n.[2] This structure generalizes the familiar lower-dimensional cases, such as the square as the 2-cube.[2] The edges of a hypercube connect pairs of vertices that differ in exactly one coordinate, corresponding to a Hamming distance of 1 between their binary representations.[2] In graph theory, the hypercube is often denoted as Q_n, where the vertices represent binary strings of length n, and edges represent single-bit flips.[5] To understand the hypercube, prerequisite concepts include polytopes, dimensions, and convexity. A polytope is a bounded geometric figure in n-dimensional space defined by the intersection of half-spaces, generalizing polygons and polyhedra to higher dimensions.[6] The dimension n refers to the number of independent coordinates needed to specify points in the ambient space \mathbb{R}^n. Convexity ensures that the line segment joining any two points within the hypercube lies entirely within it, making it a convex set.Low-Dimensional Examples
The concept of a hypercube begins with its lowest-dimensional manifestations, providing intuition for higher dimensions. The 0-cube, or zeroth-dimensional hypercube, is simply a single point with no extent in any direction, possessing 1 vertex and no edges or faces.[2] Progressing to the 1-cube, this takes the form of a line segment connecting two vertices at its endpoints, featuring 2 vertices and 1 edge, with no faces.[2] In two dimensions, the 2-cube appears as a square, which has 4 vertices, 4 edges, and 1 square face consisting of the square itself.[2] The familiar 3-cube, or cube, extends this to three dimensions with 8 vertices, 12 edges, 6 square faces, and 1 cubic cell that encloses the volume.[2] Each of these builds upon the previous by introducing a new perpendicular direction, effectively duplicating the lower-dimensional figure and connecting corresponding elements with edges or higher facets.[2] The 4-cube, known as the tesseract, further generalizes this pattern in four dimensions, comprising 16 vertices, 32 edges, 24 square faces, 8 cubic cells, and 1 tesseractic cell.[4] To visualize the tesseract in three-dimensional space, projections such as the Schlegel diagram are employed, where one cubic cell is represented as an outer cube, and the remaining seven cells are projected inward as smaller cubes connected by edges, preserving the topological structure.[4] This progression illustrates how each additional dimension duplicates the (n-1)-cube and links the copies along the new axis, fostering an intuitive grasp of hypercubic geometry.[2]Construction
Coordinate Representation
The vertices of an n-dimensional hypercube are represented as the set of all $2^n points in \mathbb{R}^n with coordinates (x_1, x_2, \dots, x_n), where each x_i \in \{0, 1\}.[2] This binary coordinate system embeds the hypercube directly in Euclidean space, with adjacent vertices connected by edges when their coordinate vectors differ in exactly one position.[2] The standard edge length in this representation is 1, as the Euclidean distance between adjacent vertices v and w satisfies \|v - w\|_2 = \sqrt{(1-0)^2} = [1](/page/1).[2] For low dimensions, the vertex coordinates are as follows:- For n=1: ([0](/page/0)), ([1](/page/1)).
- For n=2: ([0](/page/0),[0](/page/0)), ([0](/page/0),[1](/page/1)), ([1](/page/1),[0](/page/0)), ([1](/page/1),[1](/page/1)).
- For n=3: ([0](/page/0),[0](/page/0),[0](/page/0)), ([0](/page/0),[0](/page/0),[1](/page/1)), ([0](/page/0),[1](/page/1),[0](/page/0)), ([0](/page/0),[1](/page/1),[1](/page/1)), ([1](/page/1),[0](/page/0),[0](/page/0)), ([1](/page/1),[0](/page/0),[1](/page/1)), ([1](/page/1),[1](/page/1),[0](/page/0)), ([1](/page/1),[1](/page/1),[1](/page/1)).[2]
Recursive Construction
The recursive construction of an n-dimensional hypercube, denoted Q_n, builds upon lower-dimensional hypercubes by starting with two disjoint copies of the (n-1)-dimensional hypercube Q_{n-1} and connecting corresponding vertices between them with edges along the new dimension.[8] This method defines Q_n recursively as the Cartesian product Q_n = Q_{n-1} \square K_2, where K_2 is the complete graph on two vertices (a line segment), for n \geq 1, with the base case Q_0 being a single vertex.[9] Geometrically, this corresponds to extruding the Q_{n-1} along a direction perpendicular to its embedding in (n-1)-dimensional space, creating parallel "slices" connected by the new edges.[8] The vertex set of Q_n is formed as the union V_n = (V_{n-1} \times \{0\}) \cup (V_{n-1} \times \{1\}), where the two factors represent the two copies of Q_{n-1} distinguished by the new coordinate.[9] The edges of Q_n consist of the edges within each slice (identical to those of Q_{n-1}) and the new edges connecting vertices that differ only in the nth coordinate (i.e., pairs (v, 0) and (v, 1) for each v \in V_{n-1}).[8] This construction admits an inductive proof that Q_n is indeed an n-dimensional hypercube with the expected combinatorial structure. The base case for n=1 is the line segment Q_1 = K_2, which has 2 vertices and 1 edge.[9] Assuming Q_{n-1} has $2^{n-1} vertices and (n-1) \cdot 2^{n-1} edges, the inductive step adds $2^{n-1} new vertices (from the second copy) and $2^{n-1} new edges (the connections between copies), yielding |V_n| = 2^n and |E_n| = n \cdot 2^{n-1} total edges, matching the defining properties of the n-hypercube.[8] For visualization, applying this recursion to n=4 produces the tesseract Q_4 from two disjoint 3-cubes (ordinary cubes), where each vertex of one cube connects to its counterpart in the other via edges in the fourth dimension, forming the 8 cubic cells of the tesseract.[8]Structural Elements
Vertices, Edges, and Faces
The faces of an n-dimensional hypercube, also known as an n-cube, consist of all its k-dimensional sub-hypercubes for $0 \leq k \leq n, where a k-face is itself a k-cube embedded in the n-cube. The total number of k-faces is given by the formula \binom{n}{k} 2^{n-k}, which arises combinatorially from selecting the k dimensions that vary freely between 0 and 1, while fixing each of the remaining n-k dimensions to either 0 or 1.[2][3] For k=0, this yields $2^n vertices; for k=1, n 2^{n-1} edges; and for k=n, a single n-dimensional cell comprising the entire hypercube itself.[2] Each k-face can be explicitly constructed by choosing a subset of k coordinates from the n-dimensional space to vary over [0,1], with the orthogonal n-k coordinates held constant at specific binary values (0 or 1), thereby determining its position within the hypercube.[3] This coordinate-based representation highlights the hypercube's regularity, as reflected in its Schläfli symbol \{4, 3^{n-2}\} for n \geq 2, which encodes the fact that it is a regular polytope with square 2-faces and successive vertex figures that are (n-2)-simplices for n ≥ 3.[2] In terms of incidence relations within the face lattice, each k-face is contained in exactly (n - k) distinct (k+1)-faces, obtained by selecting one of the n-k fixed coordinates and allowing it to vary while keeping the others fixed.[3] Conversely, each (k+1)-face contains $2(k+1) k-faces as its boundary elements, following the general face-counting formula applied to a standalone (k+1)-cube; for instance, the 3-dimensional cube (a single cell of the 3-cube) is bounded by 6 square (2-dimensional) faces.[2]| Dimension n | Vertices (k=0) | Edges (k=1) | 2-Faces | ... | (n-1)-Faces | n-Cell (k=n) |
|---|---|---|---|---|---|---|
| 2 (square) | 4 | 4 | 1 | 4 | 1 | |
| 3 (cube) | 8 | 12 | 6 | 6 | 1 | |
| 4 (tesseract) | 16 | 32 | 24 | 8 | 8 | 1 |