Generator

V − E + F = 2, five times

A generator in the topology library, called 59 times across 14 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.

Counting overlapping targets from sensor counts alone, by integrating against χ

The Euler characteristic of shapes made of squares: pieces minus holes

A solid where V − E + F is 0

Inclusion and exclusion for the Euler characteristic

The level sets of the count, and their Euler characteristics

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
with exact readings the 40 by 36 grid counts the targets ×3
the whole boundary of the 3-cube has alternating sum 1 − (−1)^3 ×2
∫ χ(h ≥ s) ds is the pieces' lifetimes less the holes' ×1
a bipyramid lies on the line F = 2V − 4 ×1
a deeper overlap tolerates more smoothing ×1
a honeycomb on the torus has V − E + F = 0 ×1
a prism lies on the line V = 2F − 4 ×1
a pyramid has as many corners as faces ×1
a spanning tree has one fewer edge than the graph has vertices ×1
a wide gap separates the targets' features from the mistakes' ×1
about one sensor's width of smoothing is needed to swallow the mistakes ×1
and 720° is 360° times V − E + F ×1
and its twelve edges ×1
and nothing he forbids is reached ×1
and six sides to every face ×1
below the squared field's floor the region is the whole torus, with χ = 0 ×1
between 20 and 400 surfaces ×1
between one and three holes are punched ×1
corner cut off: V − E + F = 2 ×1
counting and discarding the edge pieces extrapolate to the same constant ×1
cube: each face has three sides and each corner three edges ×1
cube: Euler ×1
cutting a three-edged corner adds two corners and one face ×1
dodecahedron: each face has three sides and each corner three edges ×1
dodecahedron: Euler ×1
each ring has χ = 0 ×1
each unfolded face closes back on the corner ×1
every count Steinitz allows is reached by a pyramid and the two moves ×1
every detection region is a disc, χ = 1 ×1
every detection region is a ring, χ = 0 ×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
four levels ×1
gluing on a tetrahedron adds one corner and two faces ×1
icosahedron: each face has three sides and each corner three edges ×1
icosahedron: Euler ×1
no leftover edge closes a loop among the faces ×1
no mistake lives longer than twice the largest change smoothing left in the readings ×1
octahedron: each face has three sides and each corner three edges ×1
octahedron: Euler ×1
on a surface with no edges, maxima minus saddles plus minima is zero ×1
once divided by the formula's prefactor, every scale gives the same curve ×1
pentagonal bipyramid: each face has three sides and each corner three edges ×1
pentagonal bipyramid: Euler ×1
pentagonal prism: each face has three sides and each corner three edges ×1
pentagonal prism: Euler ×1
pentagonal pyramid: each face has three sides and each corner three edges ×1
pentagonal pyramid: Euler ×1
pieces minus holes matches the exact Euler characteristic ×1
rounding the lifetimes counts the targets ×1
sixteen times the sensors, about sixteen times the error ×1
smoothed, with each lifetime rounded, it counts them again ×1
tetrahedron glued on: V − E + F = 2 ×1
tetrahedron: each face has three sides and each corner three edges ×1
tetrahedron: Euler ×1
tetrahedron: V − E + F = 2 ×1
the 5-sided bipyramid has V − E + F = 2 ×1
the 5-sided prism has V − E + F = 2 ×1
the 5-sided pyramid has V − E + F = 2 ×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 average Euler characteristic follows the formula ×1
the blind rules tilt the balance of signs and the exact rule keeps it level ×1
the constant is below the percolation prediction by more than two standard errors ×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 error follows p/2 times the sensors that see nothing ×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 indices add up to the Euler characteristic counted directly ×1
the large pieces follow a power law near percolation's exponent ×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 level is between −3 and 3 ×1
the median gives way first; the rounded lifetimes hold to three in ten ×1
the mistaken readings alone throw the integer count far off ×1
the mode is one the family draws ×1
the rings overlap in two separate pieces ×1
the rounded count holds until the overlaps' lifetimes fall below a half ×1
the solid is one of the five regular ones ×1
the solid is one the family builds ×1
the squared field breaks the symmetry the Gaussian one keeps ×1
the sum over the level sets equals the sum of the regions' own χ ×1
the targets' regions are discs or rings ×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
the unrounded integral only falls as the smoothing widens ×1
the window closes as the overlap gets shallow ×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
up to one reading in ten, both repairs count exactly ×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
with every region a disc, the integral counts the targets ×1
with every region a ring, the integral sees nothing ×1
with exact readings the integral counts the targets ×1
χ above u is maxima minus saddles plus minima above u ×1
χ of a disc is 1 ×1
χ of a disc with two holes is -1 ×1
χ of a ring is 0 ×1
χ of two discs is 2 ×1
χ(A ∪ B) = χ(A) + χ(B) − χ(A ∩ B) ×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

Counting targets by their holes

The Euler characteristic adds up like an area: glue two shapes together and it is the sum of theirs minus that of their overlap. So it can be used to measure, and measuring with it does something no area can. A field of sensors that each report only how many targets they detect — not which, not where — can have its readings added up, weighted by the Euler characteristics of the regions where the count is high, and the answer is exactly the number of targets.

Topology

Counting targets when the sensors make mistakes

Add up the Euler characteristics of the regions where sensor counts are high and the answer is exactly the number of targets — until one sensor misreads. Each wrong reading is a new piece or a new hole, so the error grows with the number of sensors instead of averaging away. Smooth the readings and count each piece and hole by how long it survives as the threshold is lowered, and the count comes back exactly, with half the readings wrong, at the price of having to know how small a real feature can be.

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 count a solid can have

Euler's formula ties the corners, edges and faces of a convex solid together, but it does not say which numbers are possible. Ernst Steinitz answered in 1906: V corners and F faces occur together exactly when neither is more than twice the other less four. The proof in one direction is two lines of counting; in the other it is two moves — cut off a corner, glue on a tetrahedron — that walk from the pyramids to every allowed pair.

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

Pieces minus holes on a random surface

Cut a random landscape at some height and the region above it breaks into pieces with holes in them. How many pieces, and how many holes, has no formula. Their difference does: the average Euler characteristic of the region above height u is a constant times u multiplied by the bell curve at u, exactly, for every smooth Gaussian surface — because the Euler characteristic is a sum of peaks, pits and saddles, and the average number of those can be computed point by point.

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 pieces no formula counts

A random vibration of a large membrane splits the surface into regions where it bulges up and regions where it dips down. Pieces minus holes has an exact average; the number of pieces has none. Nazarov and Sodin proved it is proportional to the area, Bogomolny and Schmit predicted the constant from percolation, 0.0624, and counting the pieces of 51 random waves gives 0.0599 — below the prediction by several standard errors. The sizes of the pieces follow percolation's law; their number does not.

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