The Essentials of Dimensional Geometry

Slide a point and its trail is a line. Slide the line sideways: a square. Lift the square: a cube. Slide the cube in a fourth direction — perpendicular to all three you know — and its trail is the tesseract. The corners double every time: 1, 2, 4, 8, 16 … In n dimensions a cube has 2ⁿ corners. You cannot point in the fourth direction, but the recipe never changes, so the figure is as real as the square.

A sphere passing through Flatland appears to the Flatlander as a point, a growing circle, the equator, a shrinking circle, a point — then nothing. A four-dimensional ball passing through our space would look the same: a marble that appears, swells, shrinks, and vanishes. And every picture of a tesseract is a shadow: the 4-cube casts a 3-D shape, and the page flattens it once more. A shadow always loses one dimension and bends what it keeps.

Corners 2ⁿ; edges n·2ⁿ⁻¹ — sixteen corners at the fourth, 1,024 at the tenth. The figures grow more crowded and more round: the outline of the n-cube is a polygon of 2n sides, forever approaching a circle it never becomes.

Two dimensions allow infinitely many regular polygons. Three allow exactly five — the Platonic solids — and no sixth is possible. Four allow exactly six, the richest dimension there is. From five dimensions on, only three remain, forever: the census of perfection is not a slope but a spike.

The simplex — the family of the triangle and the tetrahedron; the hypercube; and its dual, the orthoplex. Everything else perfection ever produced was a low-dimensional luxury. These three are the permanent civil service of space.

Volume flees to the corners and then to the skin; the ball of measure crests and dies; almost all of a high-dimensional orange is peel. The heights are simple and strange at once — which is exactly what the theology requires of them.