Euler characteristic
Named by 53 essays across 5 fields — each of them below, with the objects they name alongside it.
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.
The surface with one side, and what happens when it is cut
A strip of paper, half a twist, and a join. The result has one side and one edge, and cutting it down the middle does not produce two of anything.
Four colours, and a proof nobody can read
Every map on a plane can be coloured with four colours so that no two neighbours match. The statement is understandable by a child, it resisted a century of attempts, and the proof that settled it cannot be checked by a human being.
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.
The plane, divided by whoever is nearest
Scatter some points and colour every other point of the plane by which one is closest. The result is a tiling nobody designed, and its dual triangulation has a property that no part of the construction mentions.
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.
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.
Area by counting dots
Draw a polygon with every corner on a grid of dots. Count the dots strictly inside, add half the dots on the edge, subtract one — and the answer is the area, exactly, with no measuring anywhere.
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.
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.
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.
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.
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.
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.
The bottle that needs a fourth dimension
Take the Möbius band's rectangle and glue the second pair of edges too. The result is closed, one-sided, and cannot be built in three dimensions without passing through itself — which is a fact about the room rather than about the surface.
A disc sewn to a Möbius band
The band has one boundary curve, and a disc has one boundary curve. Sew them together and the result is the smallest closed surface with one side — the one every other one-sided surface is built out of.
Two sheets over a one-sided surface
Above every one-sided surface sits a two-sided one, exactly twice as large, and the map between them forgets which of the two senses of turning a point was carrying. Building it turns a question about sides into a question about covers.
A table folded into a surface
Unfolding a square billiard gives a straight line on a torus. Unfolding any table whose angles are whole fractions of half a turn gives a straight line on some surface — and which surface it is decides how hard the dynamics will be.
Covering a surface multiplies its count
A covering of a closed surface is a permutation of the sheets for each edge of the surface's one face — with one condition that a covering of a graph never had to meet. When the condition holds, the cells of the cover can be counted directly, and the count is the base's count times the number of sheets. That multiplication decides which surfaces can cover which, before any cover is built.
What a branch point subtracts
Let the sheets of a covering meet at a few points and the count stops multiplying — but it fails by an amount that can be read off each point's permutation. Cut the sphere into a star, lift the cells, and the Riemann–Hurwitz formula falls out of a subtraction. The same count then turns out to be necessary and not sufficient.
Every word driven to a normal form
The classification is usually met as a statement: two numbers name the surface. The proof is a procedure — a short list of cut-and-reglue moves that drive any gluing word to one of the standard forms, with a measure that never rises to say why the procedure stops.
The third number a surface needs
Leave an edge unpaired in a gluing word and the surface acquires an edge of its own. Two numbers no longer name it — a count of boundary circles is needed as well — and with that third number the list is complete again, every triple occurring exactly once.
A count that can say zero
The branched count ends on a list of cycle shapes that passes every test and describes no covering. There is an exact formula for how many coverings a list has — a sum over the character table of a symmetric group — and it returns nought without giving any reason why.
The surface a knot bounds
Every knot is the edge of a surface with two sides, and Seifert found a way to build one from any diagram: smooth the crossings, fill the circles that result with discs, and join them with twisted bands. Counting the handles gives an upper bound on how complicated the knot is, the Alexander polynomial gives a lower one, and for the simplest knots the two meet.
Three ordinary lines from a count
Kelly's proof finds one line through exactly two of the points by minimising a distance. Melchior, seven years earlier, had found three — by turning every point into a line and counting the corners, edges and regions of the picture that results. Euler's formula for the projective plane does the rest, and it says exactly which configurations have no more than three.
Every way to pair a polygon's edges
A hexagon's six edges can be paired in fifteen ways. Glue each pair head to tail and five of the fifteen give a sphere and ten give a torus; an octagon's 105 pairings give 14 spheres, 70 tori and 21 surfaces with two handles. The spheres are exactly the pairings whose chords never cross, and the whole table obeys one recurrence found in 1986.
The surface a random gluing makes
Pair the edges of a large polygon at random and glue each pair head to tail. The surface almost always has nearly as many handles as the polygon allows: a thousand edges leave about seven and a half vertices, and the genus is within four of its ceiling of 250. The vertices behave like the cycles of a random permutation, and their average is a harmonic number.
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.
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 crossings an alternating knot cannot lose
Peter Guthrie Tait drew knots for years and believed, without proof, that a diagram whose crossings alternate over and under, and which has no twist that can be undone, is already drawn with the fewest crossings the knot allows. The proof took a century, and when it came it needed only the two most extreme ways of smoothing the diagram and Euler's count of the regions of a map.
The least wall for equal rooms
Divide the plane into rooms of equal area using as little wall as possible, and every wall does double duty. Bees settled on hexagons long ago, and the proof that nothing does better — not even rooms with curved walls — came in 1999. The straight-walled half of it is two facts: the rooms of any division average six sides, and more sides never cost more wall.
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.
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.
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.
Every surface is sewn from pants
A sphere with three holes — a pair of pants — is the smallest piece a surface with two or more handles can be cut into. Every such surface falls apart into them, and the Euler characteristic alone says how many pieces and how many cuts, however the cutting is done. What it does not say is the pattern, and the patterns are counted by drawing each one as a graph: two for two handles, five for three, seventeen for four.
The fewest corners a surface needs
Build a closed surface from triangles, any two meeting along a whole edge, at a single corner or not at all, and ask for the fewest corners. Two lines of counting give a floor for every surface, in terms of its Euler characteristic alone. The torus meets it with seven, the projective plane with six — and the Klein bottle, which the counting says could be built from seven, cannot, as a search of every possible arrangement shows.
A hole is a cycle that bounds nothing
A hollow triangle and a filled one have the same three edges and the same loop round them. In one the loop is the edge of something and in the other it is not, and that difference — a cycle that is not a boundary — is what a hole is. Counting holes is rank and nullity applied twice, and the Euler characteristic is what is left when the two applications cancel.
Two copies of the polygon, cross-matched
The two-sided surface that lies over a one-sided one can be built from the gluing word alone: take two copies of the polygon, one read each way round, glue a letter within the copies when it is used once each way, and across them when it is used twice the same way. The recipe works for every surface at once, doubles every count, and shows that a sphere with k cross-caps is covered by the surface with k − 1 handles — and that among all its connected double covers, exactly one is two-sided.
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.
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.
The signs no completion can change
A quadratic in several variables can be completed square by square in many orders, and each order hands back different coefficients. What no order changes is how many come out positive and how many negative — and that count, read off a completion of A − tI, says how many eigenvalues lie below t without finding a single one.
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.
An octagon only the hyperbolic plane can hold
Glue an octagon's sides in the pattern that makes a surface with two handles and its eight corners become one point, so their angles must add to a single turn. A flat octagon's add to three. The octagon that fits has corners of 45° and lives in the hyperbolic plane, where its area is forced to be exactly 4π and its four gluing moves are motions of the plane whose commutators multiply to the identity.
A curve of pixels needs two kinds of touching
On a screen a closed curve is a ring of pixels, and whether it has an inside depends on which pixels count as touching. Let diagonal pixels touch, and the inside leaks out through the corners; forbid it, and a curve can fence in a region while being twelve separate dots. Of the 1,048,575 sets of pixels in a small window, every simple closed curve obeys the Jordan curve theorem exactly when the curve and the rest of the plane are given different rules.
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.
How many ways a curve can cross itself
A closed curve that crosses itself three times can be drawn in exactly six essentially different ways on a sphere, and twenty in the plane. Four crossings give nineteen and eighty-two; six give 376 and 2,645. Every one of them is a word — the order in which the crossings are met — and a choice of direction at each crossing, and the whole census comes from asking which of those choices close up on a sphere.
The lists no covering can have
A covering of the sphere by itself with three branch points is three permutations of the sheets whose product is the identity. The Riemann–Hurwitz count says which lists of cycle shapes might work, and a search through every permutation says which do. Up to thirteen sheets, 150 lists pass the count and fail the search — and not one of them has a prime number of sheets.
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.
Where random discs change sign
Scatter discs at random over the plane and their union breaks into pieces with holes in them. Pieces minus holes has an exact average, η e^(−η)(1 − η) per disc area, where η is how many discs cover a point on average, and it changes sign at exactly η = 1, with 63.2% of the plane covered. The discs first join up across the whole plane later, at η ≈ 1.128. The sign change looks like the moment the union connects and is not; and for squares or long rectangles turned at random it comes earlier still, at 4πA/L², which only a disc reaches one.
A game decided before it starts
Draw some crosses on paper. Players take turns joining two free arms with a curve and adding a new cross on it. However they play, a game on n crosses lasts exactly 5n − 2 moves, so the winner is known before the first move: Euler's formula has already decided it. Its close relative Sprouts differs by one rule, leaving a real game whose winner for each size is known only by huge search, and which seems to depend on the size modulo six for no known reason.
The triangle a triple point buys
How many triangles can eight straight lines cut out? A count of line segments says at most 16, a parity argument takes off one, and the tables say 15. Yet every one of the 1,232,944 arrangements in which only two lines ever cross stops at 14. The fifteenth triangle exists only if three lines are made to meet at exactly one point, and moving a line by a hundredth loses it.
A number that counts what it cannot see
Whether a map has a fixed point seems to need the map. Lefschetz's number needs only what the map does to holes, and it stays the same however the map is bent. On a torus it is one determinant: |det(I − A)| fixed points, checked exactly on all 2,401 small matrices, 1, 5, 16, 45, 121 periodic points for Arnold's cat, and a formula that also counts solutions of equations over finite fields.
Peaks, pits and passes
Take any lumpy closed surface and any direction of height. Peaks plus pits minus passes is 2 on a sphere and 0 on a torus, whatever the lumps, in every direction. Average one vertex's score over all directions and it becomes the corner's angle defect, computed from nothing but comparisons. So Gauss–Bonnet follows from a count of which neighbours lie higher.
Named alongside it
The objects these essays reach for when they reach for this one.
GenusTopological invariantGluing diagramPolyhedronExhaustive searchDualityPlatonic solidsBoundaryNon-orientableOrientationPlanar graphTorus