4D Regular Polytopes: The Six Shapes Beyond the Platonic Solids

One dimension further than intuition goes

A regular polytope is the pattern that continues past the polygon (2D) and the polyhedron (3D): a 4-polytope is bounded by 3D polyhedral cells the way a polyhedron is bounded by 2D faces. There are exactly six convex regular 4-polytopes — one more than the three-dimensional case's five Platonic solids — and this simulation shows three of them plus the best-known non-regular relative, the tesseract. None of this is metaphor or approximation: these are precisely-defined mathematical objects with exact coordinates, exact symmetry groups, and exact cell counts, worked out rigorously by Ludwig Schläfli in the 1850s, decades before anyone had a way to picture them.

The simplest: the 5-cell

The 5-cell (or pentachoron) is the 4D analogue of the tetrahedron — the simplest possible regular polytope in any dimension, built from the minimum number of vertices needed to be fully connected. It has 5 vertices, 10 edges, 10 triangular faces and 5 tetrahedral cells, and like the tetrahedron it is self-dual: swap its vertices and cells and you get the same shape back. Its vertices can be placed at 5 points in 4D space that are all mutually equidistant, the direct generalisation of an equilateral triangle's 3 points or a tetrahedron's 4.

The tesseract and its dual, the 16-cell

The tesseract (8-cell, or hypercube) is the 4D analogue of the cube: 16 vertices, 32 edges, 24 square faces, and 8 cubical cells, with coordinates simply (±1, ±1, ±1, ±1) — every combination of plus and minus one in four coordinates. Its dual, the 16-cell (hyperoctahedron), is the analogue of the octahedron: 8 vertices at (±1,0,0,0) and permutations, 24 edges, 32 triangular faces and 16 tetrahedral cells — exactly the tesseract's vertex/cell counts swapped, which is the defining signature of a dual pair.

tesseract vertices:  all 16 combinations of (+-1, +-1, +-1, +-1)
16-cell vertices:    8 points at (+-1,0,0,0), (0,+-1,0,0), (0,0,+-1,0), (0,0,0,+-1)

5-cell    (5V, 10E, 10F, 5C)   self-dual        -- 4D analogue of the tetrahedron
tesseract (16V, 32E, 24F, 8C)  dual: 16-cell    -- 4D analogue of the cube
16-cell   (8V, 24E, 32F, 16C)  dual: tesseract  -- 4D analogue of the octahedron
24-cell   (24V, 96E, 96F, 24C) self-dual        -- has NO 3D analogue at all

The 24-cell: a shape with no 3D relative

The 24-cell is the strangest of the six — it has no lower-dimensional analogue whatsoever, because it exploits a coincidence that only happens in exactly four dimensions. It has 24 vertices, 96 edges, 96 triangular faces and 24 octahedral cells, is self-dual like the 5-cell, and can be built by combining the tesseract's 16 vertices with the 16-cell's 8 vertices (suitably scaled) into one unified vertex set of 24 points. Its existence, along with two further regular 4-polytopes not shown here (the 120-cell and 600-cell, built from dodecahedral and tetrahedral cells respectively), is why four is the unique dimension with more regular polytopes than three — every dimension five and above has exactly three, the direct generalisations of the tetrahedron, hypercube and orthoplex, and nothing extra.

How to actually see it: double rotation and projection

4D objects do not project down to a 2D screen in one step — this simulator goes 4D → 3D → 2D, first applying a genuine 4D perspective projection (dividing by a 'w' coordinate exactly the way ordinary perspective divides by z) to collapse to 3D, then the ordinary 3D-to-2D projection from the perspective-projection article. The rotation itself uses a feature unique to four and higher dimensions: a rotation in 4D happens in a plane, not around an axis (there is no single axis left over once you fix a plane in 4D, unlike 3D where an axis is the natural leftover), and two independent, non-intersecting planes of rotation — for instance the xy-plane and the independent zw-plane — can rotate simultaneously at different rates, a genuinely 4-dimensional motion called a double rotation that has no 3D equivalent and is what gives a tumbling tesseract its famously disorienting, turning-inside-out appearance.

Frequently asked questions

Try it live

Everything above runs in your browser — open 4D Regular Polytopes and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.