// 05 — Geometry

Shadows of higher forms

Nine interactive models: the six regular four-dimensional polytopes, the 600-cell and 120-cell merged as one dual pair, and two families of nested polyhedral shells, shown in three dimensions.

Rotate each form, flatten it along its symmetry axes into an exact two-dimensional pattern, switch its shells on and off, and read every rod length needed to build it.

// Atlas

Choose a form

drag to rotate · scroll or pinch to zoom · double-tap to reset

100%

Shadow

Look along

Show

Shells

Outer radius mm

Rod	Joins	Count

The build is the shadow: physical rods and hubs placed where the shadow's edges and vertices fall. Where the far half of a 4D form lands exactly on the near half, only one half is listed.

// Companion study

The Five-Fold Ladder

Why the pentagon family stops at four dimensions. Start from a circle divided into five, find the golden ratio it forces, and follow it up through the icosahedron to the 600-cell, where the ladder ends.

Open the study →

// Findings

What the geometry shows

Geometric results computed from exact coordinates. Each card opens the form it refers to. Physical measurements on built frames will be added as they are made.

Method and sources

Every model is generated from exact coordinates, not drawn. The 600-cell's 120 vertices come from the golden-ratio coordinates ½(±φ, ±1, ±1/φ, 0) and their companions; its edges are the pairs at distance 1/φ; its 600 tetrahedral cells are the groups of four mutually joined vertices. The 120-cell is built from these as the dual: one vertex at the centre of every tetrahedron. The thirty-fold shadows are projections onto the Coxeter plane, found from the product of the four H₄ mirror reflections. Every count and alignment stated on this page was checked numerically before publication.

Reference: H. S. M. Coxeter, Regular Polytopes, 3rd edition, Dover, 1973.