An optional helper. The facts here are standard geometry, worked out from each shape itself; nothing on this page knows about any puzzle. It runs entirely in your browser, and your notes stay on this device.
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Which polychoron fits? Filter by facts
The density is how many times the cells wrap round the centre: 1 for the convex ones, more for the star ones.
Pick a polychoron
This browser can't draw the 4D view, but all the facts are below.
Drag to turn it in 3D, or click it and use the arrow keys. The sliders, or Shift with the arrow keys, turn it through the fourth dimension, which changes its shadow. Colour shows the fourth coordinate.
How to read the picture
A 4D shape can't be drawn directly, so this shows its shadow. A cube's shadow on paper is a square inside a square; in the same way, the tesseract's shadow in 3D is a cube inside a cube. The inner cube is just as big as the outer one, but it is further away in the fourth direction, so it looks smaller. That is the perspective view. The flat (orthographic) view drops the fourth coordinate instead, so nothing shrinks, but more edges land on top of each other.
Every cell is a solid, and cells of the same kind are all the same size, even when the shadow squashes them. Turning through a plane that includes W swaps a little of the fourth direction into the other three, so cells seem to turn inside out and swap places: the shape itself isn't changing, only its shadow.
“One cell” picks out one cell's edges; Next steps to the next one. “One vertex” picks out the edges that meet at one corner. For shapes with more than one kind of cell, the cell-type buttons pick out every cell of that kind.
Facts
Sizes are for edges of length 1. Tap + to copy a fact to your notepad.
Notepad
Saved in this browser only. Tap a note's value to put it in the last box you used below.
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Quick coordinate helpers
Digits go in order: degrees, then minutes with three decimals. 14 or 15 digits make a full coordinate; 7 or 8 make one half.
Takes decimal degrees, or degrees minutes seconds (N 45 12 20.7 W 93 54 19.3), and writes them the geocaching way.
Where the facts come from
The sixteen regular polychora here, convex and star, were built from scratch by the 4D kaleidoscope construction: Wythoff's construction with Coxeter's reflection groups A4, B4, F4 and H4. The counts, sizes and angles were worked out from the result, and the counts of vertices, edges, faces and cells were checked against group theory: each count times the symmetry of one of its pieces has to give the order of the whole group. The densities of the star ones were worked out two independent ways (counting how often random rays from the centre cross the cells, and adding up the 4D angle each cell takes up), and both give the published values.
The snub 24-cell and the grand antiprism aren't regular, but both can be cut from the 600-cell: removing the 24 vertices of a 24-cell inside it leaves the snub 24-cell, and removing two rings of 10 vertices at right angles leaves the grand antiprism. Their cells were found by a convex-hull search over those vertices.
Names come from standard references: H. S. M. Coxeter's Regular Polytopes for the Schläfli symbols and the regular figures; John Conway's names for the ten star polychora, from The Symmetries of Things (Conway, Burgiel and Goodman-Strauss); and Jonathan Bowers' short names, such as “hi” for the 120-cell and “gax” for the grand 600-cell. Other names that are in common use are listed under “Also called”.
The picture is drawn in your browser from each polychoron's 4D coordinates, rounded to four decimal places. The facts don't depend on the picture: they were worked out at full precision.