Matpic
mathematics, in pictures
Concept

Topological invariant

A quantity left unchanged by bending and stretching, and therefore able to prove that two shapes are not the same shape. It can prove two shapes different and never that they are the same, which is the asymmetry every invariant argument carries.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

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 · Euler characteristic the trefoil. the trefoil, drawn as a closed curve with 3 crossings. At each crossing the strand passing underneath is broken, which is the only information the flat picture carries that the curve alone does not.

Three moves, and what they cannot undo

A knot is a closed loop of string, and two knots are the same if one can be wiggled into the other. Reidemeister reduced all possible wiggling to three local pictures — which is what makes it possible to prove that a knot is knotted.

topology · Knots The field that cannot be combed. A tangent field on the sphere, flowing along the meridians. Every arrow is tangent to the surface, and at the two poles there is no direction for an arrow to take — the field is zero there, and no rearrangement removes both zeros.

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 · Fixed points A point, a ray, and 9 crossings. A closed curve wound into a spiral corridor, with a marked point, a ray from it and every crossing marked; an odd count means the point is inside.

Which side of the line is inside

A closed curve with no self-crossings divides the plane into an inside and an outside. Nobody doubts it, almost nobody can prove it, and on a curve wound tightly enough nobody can see which side a given point is on either.

topology · Jordan curve The four ways to glue a square's edges in pairs. Squares with their edges arrowed to show which is glued to which and which way round, each with the vertices, edges and faces the gluing leaves, and the surface those numbers name.

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 · Surface classification 3 loops in one ring, and the number that separates them. Loops drawn in an annulus, each labelled with how many times it goes round the hole. Loops with different counts cannot be deformed into one another without leaving the ring.

A loop that cannot be pulled tight

A hole is a strange thing to point at, because it is precisely where the surface is not. What can be pointed at is a loop of string lying on the surface — and the hole announces itself by refusing to let that loop be pulled in to a point.

topology · Homotopy How many colourings each knot allows. Three knots, and the number of ways their arcs can be coloured with three, five and seven colours under the crossing rule, beside the determinant computed separately from the same crossings.

Colours that count more than three

Three colours prove the trefoil is knotted and say nothing at all about the figure-eight, which refuses them exactly as an unknotted loop does. The repair is to stop colouring and start counting — with five colours, or seven, and with the arithmetic done modulo the number of them.

topology · Knots Two graphs that will not lie flat, and one that will. K4, K5 and K3,3 in the best straight-line drawings a search could find. K4 has no crossings; the other two have one each, and Euler's formula shows that none can have none.

Two graphs that will not lie flat

Five points, every pair joined: no matter how the points are placed or how the lines are drawn, two of the lines cross. The proof is not about drawing at all — it counts edges against faces and finds one edge too many.

discrete · Planarity Three sets where a fixed point escapes, and one where it cannot. A ring turned about its centre, an open disc halved toward a point of its rim, the plane shifted sideways, and the closed disc turned and shrunk. Only the last has a point that its map leaves where it is.

Where the fixed point escapes

The theorem asks for a set that is closed, bounded and free of holes. Drop any one of the three and a map appears that moves every single point — and in each case the point that should have stayed still can be seen leaving.

topology · Fixed points the Hopf link, with every crossing signed. A diagram of the Hopf link with the under-strand broken at each crossing and each crossing between two components marked with its sign, which add to twice the linking number.

Two loops and one number

Give each crossing between two closed curves a sign, add them up, halve — and the answer does not depend on how the curves were drawn, how they are pushed about, or which way the picture was projected.

topology · Linking number The tent map and the logistic map, joined by a change of coordinate. Two cobweb diagrams side by side — the tent map at slope two and the logistic map at four — with the orbit of one carried to the orbit of the other by a curve drawn between them.

The same map in different coordinates

The tent map and the logistic map at four look nothing alike and are the same map, carried onto each other by a change of variable. Everything either one does the other does, and the change of variable is a sine squared.

dynamics · Iteration Reducing abcabc to a standard form. The gluing word abcabc rewritten step by step into one of the classification's standard forms, with the move used and the two invariants recomputed at each step.

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.

topology · Surface classification 4 surfaces with an edge, and the three numbers they need. Polygons whose gluing words leave some edges unpaired, each with its Euler characteristic, its sidedness and the number of boundary circles the unpaired edges form.

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.

topology · Surface classification Every way to pair the edges of a hexagon. Chord diagrams of all 15 pairings of a 6-gon's edges, shaded by the surface each gluing makes: 5 spheres, 10 tori.

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.

topology · Surface classification

Named alongside it

The objects these essays reach for when they reach for this one.

GenusOrientationEuler characteristicWinding numberGluing diagramNon-orientableBoundaryContinuityCrossing numberInvariantKnotReidemeister moves

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