A large cube stands above the ground. As the animation runs it implodes: it shrinks uniformly toward its own center, every vertex traveling down its own radial line, while the starting cube is held in place and each vertex’s path is traced in gold. When the implosion rests, the figure is complete: an outer cube, an inner cube, and eight straight edges joining corner to corner — twelve, twelve, and eight, the thirty-two edges of the tesseract in its canonical perspective image. The four-dimensional cube is never imported; it is produced, as the time-graph of a three-dimensional event.
This is the claim the figure proves by drawing it. Perspective projection from a viewpoint on the fourth axis maps depth along that axis to uniform scaling about the line of sight: the near cubical cell projects large, the far cell projects small, concentric with it. A time-graph of a uniformly imploding cube maps time to uniform scaling about the center. The two constructions therefore produce the identical figure, edge for edge — and the correspondence is quantitative: with the inner cube drawn at 0.42 of the outer, the four-dimensional eye stands at distance (1+0.42)/(1−0.42) ≈ 2.45 along the fourth axis. On the dimensional account developed in The Projection of Gravity and The Spiral of the Year, this identity is no coincidence: what perspective does to the higher axis is exactly what a graph does to time, because time, seen spatially, is the higher axis — the perspective view of a tesseract simply is the graph of a cube imploding over time.
In the four-dimensional cube, every edge meets its neighbors at right angles: the eight golden trails are, in the higher register, perpendicular at once to all twelve edges of the outer cube and all twelve of the inner. But a direction perpendicular to the whole of space has no direction left within space, and so travel along it appears in our register as scaling — each vertex borne down its own radial line toward the common center. This is the perpendicular ontology drawn as geometry: the higher register stands perpendicular to the entirety of the lower, and motion along the perpendicular appears below as approach toward the center. The implosion is eight simultaneous free falls toward one point — the gravitational figure in miniature — and each trail is straight, as the higher paths are: the curvature of gathering belongs to the lower reading, while in the fourth dimension the tesseract’s edges run as straight and as equal as every other edge it has.
The cube’s anatomy is the twenty-six of the corpus: 8 vertices + 12 edges + 6 faces = 26, the count established in The Significance of 26 — which is 3³ − 1, since along each of the cube’s three axes an element stands in one of three states: at the low pole, at the high pole, or spanning between them, and the single case that spans all three axes at once is the solid body itself, which the twenty-six excludes. The fourth axis multiplies this anatomy by three. The tesseract has 3⁴ − 1 = 80 proper elements — 16 vertices, 32 edges, 24 faces, 8 cells — and the figure shows exactly how twenty-six becomes eighty: every one of the cube’s twenty-six appears twice, at the beginning of the journey and at its end (2 × 26 = 52); every one of the twenty-six sweeps one new element as its trail through the fourth dimension — each vertex an edge, each edge a face, each face a cell (+26); and the solid cube itself, which the twenty-six could not include, is included twice over by the higher figure, as its outer and inner cells (+2). Fifty-two, twenty-six, and two: eighty. The eight golden trails are the visible first rank of the sweep; the twelve swept faces and six swept cells stand in the wire figure unshaded. Each new perpendicular axis multiplies the anatomy of form by three — the two poles and the span between them — and the ladder runs 2, 8, 26, 80, 242: the same expansion, register upon register.