The collection

Every essay — page 19

Page 19 of 46, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Topology

What survives bending, and what does not.

A circle stays a circle, unless it meets the pole. 3 circles on a sphere beside their stereographic images in the plane, which are circles, together with one circle through the projection point whose image is a straight line.

Angles survive and areas do not

A turn of the sphere, seen from the plane. A square grid in the plane and its image under the map obtained by lifting to the sphere, rotating by 62° about a tilted axis, and coming back down. The lines become arcs of circles and the crossings stay at right angles.

The sphere that complex numbers live on

Two charts on one sphere, meeting by the reciprocal. A sphere with its two polar caps marked, each the part missed by one of the two stereographic charts, and the band where both charts are defined shaded between them.

One chart is never enough

5 circles filling a three-sphere, every pair linked once. Several closed curves in space, nested on tori of different sizes, each pair passing through the other exactly once and none of them touching.

The circles that fill a three-sphere

3 symmetries over 3 sheets: a regular covering. A 3-sheeted covering of a wedge of 2 circles, with the permutations of its sheets that commute with every generator. There are 3, against 3 sheets.

The symmetries a cover has of its own

3 sheets, 8 of 26 words coming back. A table of reduced words in two generators with the sheet each sends the base sheet to. The words returning to it are the covering's subgroup, and the 3 sheets are its cosets.

A covering is a permutation

6 vertices folded to 4, and a graph that decides. The graph built from 3 generator words, folded until no vertex has two edges of one label leaving it. Reading a word from the base vertex decides membership, and 6 words are tested.

Folding a graph until it decides

3 sheets over a surface of genus 2: a surface of genus 4. A 3-sheeted covering of the closed surface of genus 2, drawn as 3 copies of its 8-sided face with each side coloured by its generator and numbered with the sheet it glues to. The Euler characteristic −6 is 3 times −2, and the cover has genus 4.

Covering a surface multiplies its count

2 sheets branched over 4 points of a sphere: a surface of genus 1. A 2-sheeted branched covering of the sphere with 4 branch points, drawn as 2 rows of sheets over a centre and the branch points, with the sheets joined where each point's permutation cycles them. Counting cells gives Euler characteristic 0, matching the Riemann–Hurwitz formula, and genus 1.

What a branch point subtracts

Two opposite points on a globe with the same temperature and the same pressure. A world map in longitude and latitude with one curve where each point's temperature matches its opposite point's and another where the pressures match, crossing at a pair of opposite points that are marked.

Two opposite points that agree twice

A necklace of 4 orange, 4 blue, 2 green beads, shared fairly with 3 cuts. A row of coloured beads cut at marked places into pieces, each piece labelled with the thief who receives it, so that both thieves get half of every colour.

As many cuts as colours

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

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

Six lists of cycle shapes, and how many coverings each has. A table of lists of cycle shapes over a sphere, each with the Euler characteristic the Riemann–Hurwitz count gives, the number of lists of permutations with that product, and the number of those that connect all the sheets.

A count that can say zero

A rule that meets the diagonal, and the same rule with a hole in it. Two panels, each the graph of a rule assigning a set of values to every point of the unit interval, drawn against the diagonal: the first meets the diagonal at a filled-in jump, the second has the jump left open and misses it.

A map that offers a choice

The Alexander matrix of the trefoil. The trefoil with its 3 arcs numbered and its 3 crossings lettered, beside the 3 by 3 matrix they give. A minor of the matrix is the Alexander polynomial t − 1 + t⁻¹, whose value at −1 is the determinant 3.

A polynomial behind the colourings

The Seifert circles of the trefoil. The trefoil with an orientation, cut at each of its 3 crossings and reconnected the way the orientation allows. The 6 segments form 2 circles, and the surface built from them has genus 1.

The surface a knot bounds

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 random pairing of 24 edges. Chord diagram of one uniformly random pairing of a 24-gon's edges, corners coloured by the vertex they become. a 24-gon with its edges paired at random and glued head to tail: the 24 corners fall into 3 vertices, so the surface has genus 5, against a most possible of 6.

The surface a random gluing makes

A labelled square and the edges that join opposite labels. A square grid of 121 vertices, each coloured by one of four labels, with opposite boundary vertices carrying opposite labels, and 3 edges drawn thick where a label meets its negative.

Opposite labels that have to meet

The pairs from five, joined when disjoint, need three colours. The Petersen graph drawn with its ten vertices labelled by pairs from one to five, edges joining disjoint pairs, and a proper colouring with three colours.

The colours a circle forces

Twelve units of shortfall, on every solid with three faces at a corner. A bar for each of 8 polyhedra with three faces at every vertex, divided into each face's shortfall from six sides; every bar has total length twelve, and the hexagons contribute nothing.

Twelve pentagons, whatever the hexagons

The six regular 4-polytopes, and an alternating sum of 0. A table of the six regular polytopes in four dimensions with their numbers of vertices, edges, faces and cells and the alternating sum, which is zero for each.

Zero in four dimensions

Loops on a torus that never cross themselves. Squares with opposite edges glued, each carrying one straight loop of a different slope, each labelled with its two crossing counts.

The loops on a torus that never cross themselves