5-cell · 16-cell · 24-cell · double-plane 4D rotation · perspective projection
Just as the five Platonic solids are the only regular polyhedra in 3D, there are exactly six convex regular polytopes in 4-dimensional space. This simulator renders three of them — the 5-cell (the 4D analog of a tetrahedron, built from 5 tetrahedral cells), the 16-cell (built from 16 tetrahedral cells, the 4D analog of an octahedron) and the 24-cell (a shape with no direct 3D analog, built from 24 octahedral cells) — plus, for completeness, the 8-cell or tesseract, the famous hypercube already explored in more depth elsewhere on this site. Every vertex is a genuine 4D coordinate (x, y, z, w); edges are drawn only between vertices that are truly adjacent in the polytope's real 4-dimensional structure.
A 4D object rotates simultaneously in two independent planes rather than around a single "axis" — there is no single axis of rotation in 4D, only a fixed plane that is unaffected while everything else turns. The 24-cell is the only regular 4D polytope with no 3D analog at all: it is self-dual and, remarkably, it can be built entirely from unit hypercube edges, meaning it fits perfectly between the tesseract and its dual, the 16-cell. What you see on screen is a projection twice removed from reality — 4D vertices are perspective-divided down to 3D, and that 3D shadow is projected again to the 2D screen, exactly the way a 3D wireframe cube casts a flat 2D shadow.
This simulator computes real 4-dimensional vertex coordinates for the 5-cell, 16-cell, 24-cell and 8-cell (tesseract), rotates them simultaneously in two independent 4D rotation planes, then perspective-divides down through 3D to a flat 2D canvas — the same "shadow of a shadow" technique used to visualise any higher-dimensional object on a 2D screen.
Each polytope's vertices are generated from exact coordinate formulas (permutations of small integers, or an orthonormal projection of the 5-point simplex). Edges are found by measuring every pairwise distance and connecting only the pairs at the true minimum edge length, so the wireframe is combinatorially correct, not approximated.
Choose a polytope, then use the XY-plane and ZW-plane speed sliders to set two independent 4D rotation rates — notice they can even spin in opposite directions. The 4D camera distance slider controls how strongly the hidden w-coordinate bends the projection. Pause and Reset Angles let you study a single fixed orientation.
The 24-cell has no 3-dimensional counterpart — it is one of a kind among all regular polytopes in any dimension. It is also self-dual: its own vertex-cell relationship mirrors itself, a property shared in 3D only by the tetrahedron.
A regular polytope is the higher-dimensional generalisation of a regular polygon (2D) or Platonic solid (3D): a shape built from identical regular cells, meeting identically at every vertex, edge and face. There are exactly six convex regular polytopes in 4 dimensions, compared with five Platonic solids in 3D and infinitely many regular polygons in 2D.
The 16-cell uses all sign permutations of (1,0,0,0); the 24-cell uses all sign permutations of (1,1,0,0); the 8-cell (tesseract) uses all sign combinations of (1,1,1,1). The 5-cell is built by projecting five equally spaced points from a 5-dimensional simplex onto the 4-dimensional hyperplane they span, using Gram-Schmidt orthonormalisation, which produces five points that are genuinely equidistant in true 4D space.
Two steps. First, a perspective divide from 4D to 3D: each vertex is scaled by camera-distance / (camera-distance minus its w-coordinate), exactly as a 3D perspective camera divides by depth. Second, the resulting 3D point is projected to the 2D canvas using ordinary 3D-to-2D perspective, the same method used throughout this site's other 3D simulations.
In 3D, rotation happens around an axis — a line that stays fixed. In 4D, what stays fixed during a simple rotation is an entire plane, and there is a second, independent plane (fully orthogonal to the first) in which rotation can happen simultaneously and separately. That is why this simulation exposes two speed sliders rather than one.
The tesseract (8-cell) is the most famous 4D regular polytope and already has its own dedicated, more detailed simulation on this site. It is included here too, alongside its lesser-known relatives, so you can compare all four shapes' vertex counts, edge structures and rotation behaviour side by side using the same engine.