Generator

V − E + F = 2, five times

A generator in the topology library, called 29 times across 9 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.

euler-solid 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

V − E + F = 2, five times. Vertices, edges and faces of the five regular solids, with the alternating sum. The edges are counted from the faces rather than listed, and the sum is 2 in every row.

A solid flattened into a graph

The gap at a corner of the cube

Two trees, sharing every edge between them

A solid where V − E + F is 0

Twelve units of shortfall, on every solid with three faces at a corner

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.

in dimension 1 the sum is 1 − (−1)^d ×8
the alternating sum of the 5-cell is 0 ×5
cyclic, 6 corners: cells = edges − corners ×4
cyclic, 6 corners: faces = 2 × (edges − corners) ×4
cyclic, 6 corners: the alternating sum is 0 ×4
5-cell: cells = edges − corners ×3
5-cell: faces = 2 × (edges − corners) ×3
5-cell: the alternating sum is 0 ×3
V − E + F is 0 for this solid ×3
the whole boundary of the 3-cube has alternating sum 1 − (−1)^3 ×2
a honeycomb on the torus has V − E + F = 0 ×1
a spanning tree has one fewer edge than the graph has vertices ×1
and 720° is 360° times V − E + F ×1
and its twelve edges ×1
and six sides to every face ×1
between one and three holes are punched ×1
each unfolded face closes back on the corner ×1
every edge lies between exactly two faces ×1
every edge of the cube is shared by two faces ×1
every edge of the dodecahedron is shared by two faces ×1
every edge of the icosahedron is shared by two faces ×1
every edge of the octahedron is shared by two faces ×1
every edge of the surface is shared by exactly two faces ×1
every edge of the tetrahedron is shared by two faces ×1
every partial boundary before the last facet sums to 1 ×1
no leftover edge closes a loop among the faces ×1
the alternating sum is not 2, on a solid with flat faces and straight edges ×1
the alternating sum of the tesseract is 0 ×1
the counts obey Euler's formula ×1
the cube's faces fall short of hexagons by exactly 12 ×1
the defects on the cube come to 720° ×1
the defects on the dodecahedron come to 720° ×1
the defects on the icosahedron come to 720° ×1
the defects on the octahedron come to 720° ×1
the defects on the tetrahedron come to 720° ×1
the dodecahedron's faces fall short of hexagons by exactly 12 ×1
the faces at a corner of the cube do not close up flat ×1
the faces at a corner of the dodecahedron do not close up flat ×1
the faces at a corner of the icosahedron do not close up flat ×1
the faces at a corner of the octahedron do not close up flat ×1
the faces at a corner of the tetrahedron do not close up flat ×1
the faces short of six sides fall short by 12 in all ×1
the flattened cube keeps its eight corners ×1
the flattened cube still gives 2 ×1
the hexagonal prism's faces fall short of hexagons by exactly 12 ×1
the leftover edges are what the tree did not use ×1
the leftover edges join every face to every other ×1
the leftover edges number one fewer than there are faces ×1
the mode is one the family draws ×1
the solid is one of the five regular ones ×1
the solid is one the family builds ×1
the tesseract has 32 edges ×1
the tesseract's counts are the 16-cell's, reversed ×1
the tetrahedron's faces fall short of hexagons by exactly 12 ×1
the tree is grown from one of the graph's own vertices ×1
the tree reaches every vertex ×1
the triangular prism's faces fall short of hexagons by exactly 12 ×1
the truncated icosahedron's faces fall short of hexagons by exactly 12 ×1
the truncated octahedron's faces fall short of hexagons by exactly 12 ×1
the truncated tetrahedron's faces fall short of hexagons by exactly 12 ×1
the two trees together use every edge exactly once ×1
three edges at every corner ×1
three faces meet at every corner of the cube ×1
three faces meet at every corner of the dodecahedron ×1
three faces meet at every corner of the hexagonal prism ×1
three faces meet at every corner of the tetrahedron ×1
three faces meet at every corner of the triangular prism ×1
three faces meet at every corner of the truncated icosahedron ×1
three faces meet at every corner of the truncated octahedron ×1
three faces meet at every corner of the truncated tetrahedron ×1
V − E + F = 2 for the cube ×1
V − E + F = 2 for the dodecahedron ×1
V − E + F = 2 for the hexagonal prism ×1
V − E + F = 2 for the tetrahedron ×1
V − E + F = 2 for the triangular prism ×1
V − E + F = 2 for the truncated icosahedron ×1
V − E + F = 2 for the truncated octahedron ×1
V − E + F = 2 for the truncated tetrahedron ×1
V − E + F is -2 for this solid ×1
V − E + F is 2 for the cube ×1
V − E + F is 2 for the dodecahedron ×1
V − E + F is 2 for the icosahedron ×1
V − E + F is 2 for the octahedron ×1
V − E + F is 2 for the tetrahedron ×1
which is V − E + F = 2 ×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.

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.

Topology

Every surface is a sphere with handles

Take a square and say which edges are to be glued to which, and which way round. Four such rules give four different surfaces — and two numbers computed from the rule, without ever building the surface, say which one.

Topology

Nothing on a sphere can be combed flat

Point an arrow along the surface at every place on a sphere, continuously, and somewhere an arrow has to vanish. On a doughnut it can be done. The difference between the two is a number that was already known from counting corners.

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.

Discrete

Seven regions on a doughnut

A map on a torus can need seven colours, and the proof is a picture — seven regions, each sharing a border with all six others. The plane needed a computer and eighty-six years; the harder surface was settled in 1890 by drawing something.

Topology

The solid where the answer is not two

A slab with a hole through it has flat faces, straight edges and sixteen corners, and its alternating sum is zero. It is not a trick and not a degenerate case — it is the object that shows the theorem had a hypothesis nobody had written down.

Topology

Twelve pentagons, whatever the hexagons

A football has twelve pentagons and twenty hexagons. A molecule of sixty carbon atoms has the same pattern, a molecule of seventy has twelve pentagons and twenty-five hexagons, and a geodesic dome of any size has twelve places where the pattern of six breaks. None of this is a coincidence of design: Euler's formula, rearranged, says that faces meeting three at a corner must fall short of hexagons by exactly twelve in total, and the hexagons are free.

Topology

Two trees, and every edge in exactly one of them

Euler's formula is usually proved by deleting things until nothing is left. There is a better argument that deletes nothing — a tree through the corners and a tree through the faces, which between them use every edge once and can therefore be counted.

Topology

Zero in four dimensions

Corners minus edges plus faces is two for every solid. One dimension up, corners minus edges plus faces minus cells is zero for every one of the six regular four-dimensional solids, from the five-cell to the six-hundred-cell, and for every other convex solid in four dimensions. The alternating sum does not break when the dimension rises: it alternates, two in odd dimensions and zero in even ones, because it is measuring a sphere and not a solid.

The whole library · What the figures prove