Dimensional Fusion: The Cartesian Product of Cubes

Factor A

Factor B

Fusion

Fusion progress t 0%

Resulting Object

Dimension

Vertices

Edges

dim = 2 + 1 = 3 V = 2² · 2¹ = 16 E = E_A·V_B + V_A·E_B = 4·2+4·1 = 12

How it works

A k-cube (hypercube) is the set of points in ℝᵏ with every coordinate in {−1, +1}, joined by edges wherever two vertices differ in exactly one coordinate. The Cartesian product of a k_A-cube and a k_B-cube "fuses" the two spaces into a single (k_A+k_B)-cube: every vertex is a pair (a, b), built by literally concatenating A's coordinates with B's coordinates.

Fusion rule (graph Cartesian product □):
  V(A □ B) = V(A) × V(B)            → |V| = |A|·|B| = 2^(kA+kB)
  E(A □ B) = E_A·V_B + V_A·E_B      → each A-edge repeated once per B-vertex,
                                       plus each B-edge repeated once per A-vertex

That single rule explains every classic construction: point □ line = line, line □ line = square, square □ line = cube, and square □ square = the 4D tesseract — a genuine 4-dimensional object that this simulator rotates in the X–W plane and projects into 3D with perspective foreshortening (points "further" in W render smaller), exactly like a 3D shadow of a 4D rotation.

Factor A / Factor B — pick the dimension of each starting cube (0–2D); the fused result is their sum.
Fusion progress t — at t=0 the |V_B| copies of A sit apart, scattered along the extrusion direction; at t=1 they lock into the true product-cube coordinates. B-type edges (the "fusing" connections) fade in as t grows.
4D rotation — only active when the result is 4-dimensional; rotates the tesseract through its extra axis before the 4D→3D projection.

This is the same construction used to build the k-dimensional lattice graphs behind hypercube networks in parallel computing, and it is the standard way higher polytopes (cube → tesseract → penteract…) are generated from lower ones.

Factor-A edges Fusing (Factor-B) edges

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