Cube
polyhedron is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
With nothing chosen
The five Platonic solids
Why the solids run out
The cube and its dual, the octahedron
The regular solids of four dimensions
Every turn that leaves a cube where it was
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- the order-4 axes carry 9 turns between them ×7
- a turn of order 3 sits on an axis of order 3 ×5
- and each such axis carries a turn of order 4 ×4
- 3 triangles at a vertex leave a gap ×3
- 5-cell: and the edge count agrees with that degree ×3
- 5-cell: every vertex is the same distance from the centre ×3
- 5-cell: every vertex meets the same number of edges ×3
- 3 hexagons at a vertex cannot leave a gap ×1
- 3 pentagons at a vertex leave a gap ×1
- 3 squares at a vertex leave a gap ×1
- 4 pentagons at a vertex cannot leave a gap ×1
- 4 squares at a vertex cannot leave a gap ×1
- 6 triangles at a vertex cannot leave a gap ×1
- a star polygon has between 5 and 13 points ×1
- and 32 − 90 + 60 = 2, the Euler number of any sphere ×1
- and at the depth found every edge is the same length ×1
- and every one of them is the same triangle ×1
- and sends every corner to a corner ×1
- and steps by at least 2, less than half of p ×1
- and steps by between 2 and 6 ×1
- and the angle at each point is π − 2πq/p ×1
- as a surface it has thirty-two corners ×1
- cuboctahedron: and lies on a circle, so it is regular ×1
- cuboctahedron: and the walk round a corner visits every face at it once ×1
- cuboctahedron: each face has equal sides ×1
- cuboctahedron: Euler's formula still holds ×1
- cuboctahedron: every edge is the same length ×1
- cuboctahedron: every vertex is surrounded by the same faces in the same order ×1
- each candidate is a turn rather than a reflection ×1
- each polytope is one this family builds: 5-cell, tesseract, 16-cell, 24-cell ×1
- each solid is one this family cuts: truncated-tetrahedron, truncated-cube, truncated-octahedron, cuboctahedron, icosidodecahedron, truncated-icosahedron ×1
- every turn has an axis to be about ×1
- exactly five vertex figures close up ×1
- icosidodecahedron: and lies on a circle, so it is regular ×1
- icosidodecahedron: and the walk round a corner visits every face at it once ×1
- icosidodecahedron: each face has equal sides ×1
- icosidodecahedron: Euler's formula still holds ×1
- icosidodecahedron: every edge is the same length ×1
- icosidodecahedron: every vertex is surrounded by the same faces in the same order ×1
- no neighbour lies straight along the corner's own axis ×1
- read as twelve pentagrams meeting five at a point, the number is −6 ×1
- tesseract: and the edge count agrees with that degree ×1
- tesseract: every vertex is the same distance from the centre ×1
- tesseract: every vertex meets the same number of edges ×1
- the axes account for every turn, the one that does nothing included ×1
- the corner polygon's edge starts shorter than the remaining edge and ends longer ×1
- the faces at a corner close into a ring ×1
- the number of distinct turns is twice the number of edges ×1
- the path visits every point exactly once ×1
- the path winds about the centre q times ×1
- the projected solid has a measurable extent ×1
- the solid is one of the five ×1
- the spiky surface is sixty triangles ×1
- the star has between 5 and 13 points ×1
- the step and the number of points share no factor, so the path closes once ×1
- the turn in four dimensions is at most half a circle ×1
- the view is one the family draws ×1
- truncated-cube: and lies on a circle, so it is regular ×1
- truncated-cube: and the walk round a corner visits every face at it once ×1
- truncated-cube: each face has equal sides ×1
- truncated-cube: Euler's formula still holds ×1
- truncated-cube: every edge is the same length ×1
- truncated-cube: every vertex is surrounded by the same faces in the same order ×1
- truncated-icosahedron: and lies on a circle, so it is regular ×1
- truncated-icosahedron: and the walk round a corner visits every face at it once ×1
- truncated-icosahedron: each face has equal sides ×1
- truncated-icosahedron: Euler's formula still holds ×1
- truncated-icosahedron: every edge is the same length ×1
- truncated-icosahedron: every vertex is surrounded by the same faces in the same order ×1
- truncated-octahedron: and lies on a circle, so it is regular ×1
- truncated-octahedron: and the walk round a corner visits every face at it once ×1
- truncated-octahedron: each face has equal sides ×1
- truncated-octahedron: Euler's formula still holds ×1
- truncated-octahedron: every edge is the same length ×1
- truncated-octahedron: every vertex is surrounded by the same faces in the same order ×1
- truncated-tetrahedron: and lies on a circle, so it is regular ×1
- truncated-tetrahedron: and the walk round a corner visits every face at it once ×1
- truncated-tetrahedron: each face has equal sides ×1
- truncated-tetrahedron: Euler's formula still holds ×1
- truncated-tetrahedron: every edge is the same length ×1
- truncated-tetrahedron: every vertex is surrounded by the same faces in the same order ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
A rotation of four-space takes two of them
One unit quaternion, conjugating, turns three-space about an axis. Two of them, multiplying from the left and the right, turn four-space — and a rotation of four-space has no axis at all, but two independent angles and two planes it spins in.
TopologyEvery corner pays for itself
Count the corners of any solid, subtract the edges, add the faces. The answer is two. It is two for a cube, for a pyramid, for a football, for anything squashed or stretched — and the number is measuring the shape it is wrapped around rather than the shape itself.
TopologySeven hundred and twenty degrees of gap
Unfold the faces around any corner of a solid and they do not close up. The gap left over is different at every corner and on every solid, and the gaps always add to two full turns.
GeometrySix in four dimensions, and three forever after
The count of regular solids goes five in three dimensions, six in four, and then three in every dimension above — for good. Four dimensions is the last place anything unusual happens, and it happens twice.
GeometryThe five solids as three groups
There are five regular solids and only three groups of rotations between them, because a solid and its dual share their symmetries exactly. The largest of the three is the smallest group with no way of coming apart, which is why the general equation of the fifth degree has no formula.
GeometryThe four that are allowed to cross themselves
Drop convexity from the definition of a regular solid and four more appear. Their faces are pentagrams, they pass through one another, and the alternating sum that gives two for every ordinary solid gives minus six for two of them.
GeometryThirteen more when one word is dropped
The list of regular solids stops at five because the definition asks for two things at once. Ask for only the second — every corner alike — and thirteen more appear, each of them cut off a Platonic solid at a depth found rather than chosen.
GeometryWhy the list of perfect solids stops at five
There are infinitely many regular polygons and exactly five regular solids. The reason is not deep, but it is very sharp, and it can be checked on a single row of corners.