Generator

Cube

A generator in the geometry library, called 69 times across 13 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

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

Cube. A cube drawn in projection with 6 faces.

A cube passes through a hole in a smaller cube

The shadows a cube can cast

All five Platonic solids pass through themselves

The hole has to be cut on the slant

Passages have to be searched for

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
2ᵏ times the A determinant is k + 1 ×1
2ᵏ times the E determinant is 9 − k ×1
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 cube can pass through a hole in a cube of the same size scaled down by 3√2/4 ×1
a label above six never extends past two mirrors ×1
a star polygon has between 5 and 13 points ×1
along this arc the factor peaks near 1.04, short of the 3√2/4 a tunnel off the arc reaches ×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
and the group has that many mirrors ×1
as a surface it has thirty-two corners ×1
at several numbers of charges most starts miss the lowest valley ×1
below sixteen charges every start finds the same arrangement; from sixteen on, several valleys appear ×1
compact hyperbolic diagrams stop after five mirrors ×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 diagram found has a name ×1
every edge of the solid has the same length ×1
every other valley is above the lowest by less than one per cent ×1
every Platonic solid can pass through a hole in a copy of itself ×1
every shadow's area lies between 1 and √3 ×1
every turn has an axis to be about ×1
exactly five vertex figures close up ×1
face on, the square shadow holds a square of the same size and no larger ×1
five charges settle as a triangular bipyramid, below the best square pyramid ×1
for these numbers of charges every one of the eight random starts settles into the same arrangement ×1
from random starts a local search sometimes finds a passage and sometimes gets stuck below factor 1 ×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
order of the group A₃ ×1
order of the group A₄ ×1
order of the group B₃ ×1
order of the group B₄ ×1
order of the group D₄ ×1
order of the group F₄ ×1
order of the group H₃ ×1
order of the group H₄ ×1
read as twelve pentagrams meeting five at a point, the number is −6 ×1
sixteen charges have two valleys, at 92.911655 and 92.920354 ×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 correction to N²/2, divided by N^(3/2), stays between −0.6 and −0.4 ×1
the cube and the dodecahedron are not ×1
the cube and the octahedron reach the same factor ×1
the diagrams on one to eight mirrors number 1, 4, 3, 5, 3, 4, 4, 4 ×1
the energy falls steadily as the twist grows from a cube to the antiprism ×1
the faces at a corner close into a ring ×1
the group is finite ×1
the hole is cut through a hexagonal shadow, seen along a direction near a long diagonal ×1
the kaleidoscope is tetrahedral, octahedral or icosahedral ×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 point is off at least one mirror ×1
the projected solid has a measurable extent ×1
the search reaches the published minimum energy ×1
the shadow of a unit cube has area 1 face on and √3 along a long diagonal ×1
the shortfalls from six neighbours add to exactly twelve, as Euler's formula for the sphere requires ×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 tetrahedron, octahedron and icosahedron are minima ×1
the three mirrors generate the icosahedral group of order 120 ×1
the three mirrors generate the octahedral group of order 48 ×1
the three mirrors generate the tetrahedral group of order 24 ×1
the traced solid has Euler characteristic two ×1
the turn in four dimensions is at most half a circle ×1
the view is one the family draws ×1
triangles where the determinant's sign matches the angle sum ×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
turning one square through 45° gives the minimum for eight charges ×1
two hundred starts for fifty-five charges find at least five valleys, the lowest at 1287.772721 ×1
untwisted, the best height gives exactly the cube ×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.

Geometry

A cube through a hole in a cube

Prince Rupert of the Rhine bet that a hole could be cut through a cube large enough for an equal cube to pass through it, and won. The trick is to look at the cube along a direction near its long diagonal, where its shadow is a hexagon wide enough to hold a square face with room to spare: a cube 6% larger can pass. All five Platonic solids turn out to have such a passage, found by searching over the directions from which a solid is seen — and in 2025 a solid was found that has none.

Algebra

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.

Topology

Every 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.

Geometry

One determinant for every kaleidoscope

Three mirrors close up into a kaleidoscope when the angles of their triangle add to more than half a turn. In more dimensions there is no triangle to add up, but there is a matrix — the inner products of the mirrors' directions — and the mirrors close up exactly when it is positive definite. Trying every arrangement of up to nine mirrors against that one test finds Coxeter's list: three families in every dimension, a fork that runs out at eight mirrors because its determinant reaches zero at nine, and five exceptions. The same search finds the hyperbolic kaleidoscopes, and finds that they stop at five mirrors.

Topology

Seven 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.

Geometry

Six 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.

Geometry

The 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.

Geometry

The 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.

Geometry

Thirteen 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.

Geometry

Three mirrors make every solid

Every symmetry of a regular solid, reflections included, is produced by just three mirrors meeting at its centre, reflected in one another over and over — a kaleidoscope. Put a single point between the three mirrors and its reflections are the corners of a solid: the regular solid itself if the point sits in a corner, and every one of its truncated and expanded relatives if it sits anywhere else.

Geometry

Valleys a search cannot count

Let equal charges on a sphere push each other apart from a random start, and they settle into an arrangement where nothing moves. Up to fifteen charges every start settles into the same one. Sixteen is the first number with two, a hundredth of a unit apart in energy; by fifty-two there are at least four, and for fifty-six charges only one start in ten finds the lowest. The other valleys sit a few parts in a million above it, and nothing in the arrangement a search returns says whether it is the bottom.

Geometry

Where charges settle on a sphere

Put equal electric charges on a sphere and let them push each other apart until nothing moves. Four settle at the corners of a tetrahedron, six of an octahedron, twelve of an icosahedron — but eight do not make a cube, and twenty do not make a dodecahedron. The cube loses to two squares turned half a corner apart, because charges on a sphere prefer triangles; and however many charges there are, exactly twelve more of them have five neighbours than have seven.

Geometry

Why 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.

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