Matpic
mathematics, in pictures
The collection

Every essay — page 21

Page 21 of 47, 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.

The seven graphs of the Petersen family. Seven small graph drawings, K₆, G7, K₃,₃,₁, K₄,₄ minus an edge, G8, G9, Petersen graph, arranged by vertex count with 6 arrows for the triangle-to-star exchanges between them.

Seven graphs that must link

5 figures Four equal quarters of an L-shaped plate. An L-shaped plate cut by two perpendicular halving lines at 48.8 degrees into four pieces of equal area.

Four equal quarters with two lines

5 figures Seven points split into three groups whose hulls share a point. Seven random points in a square split into three groups of sizes 2, 2, 3 whose convex hulls share a common point; 4 of the 301 splits work.

Seven points split three ways

5 figures Counting targets from sensor readings with mistakes in them. Three 80 by 72 sensor fields over 8 targets: true counts (integral 8), counts with 415 mistaken readings (integral 160), and the mistaken counts smoothed (rounded persistence count 8).

Counting targets when the sensors make mistakes

6 figures Sperner's corridor to the same fixed point, on a gentle map and a twisted one. Two 32 by 32 labelled grids; the corridor crosses 44 triangles reading 100 corners on the gentle map and 148 triangles reading 204 corners on the twisted one.

What it costs to find the point that stays put

6 figures A random surface and the region above height 1. A smooth Gaussian random field on a 64 by 64 torus and its excursion set above 1: 14 pieces, 0 holes, Euler characteristic 14; expected 13.8.

Pieces minus holes on a random surface

6 figures A loop that cannot be undone, and the surface it bounds. The loop aba⁻¹b⁻¹ on the figure eight, which cannot be pulled tight, beside a square whose boundary reads the same word, showing that the loop bounds a surface.

What homology forgets about a loop

6 figures Pascal's triangle mod 2, and the room for the 2-dimensional projective space. Pascal's triangle with odd entries filled, rows 0 to 15, with row 3 marked as the tangent ledger 1 + a + a² and row 1 as the normal ledger 1 + a.

The room a projective space needs, read off Pascal's triangle

6 figures The Lorenz orbits LR and LLR, linked. Projection of the periodic orbits LR, LLR of the Lorenz system, linking number 1.

Every loop of the butterfly links every other

6 figures A trefoil, and how far its tangent turns. The (2, 3) torus knot with tube radius 0.45; total curvature 16.8556 = 5.3653π.

A knot must turn twice, and a little more

6 figures The three-sphere filled with trefoils. Seven fibres of the (2,3) Seifert fibration of S³, stereographically projected: five trefoils and two exceptional circles; pairwise linking numbers 6, 3, 2 and 1 verified.

Torus knots that fill the three-sphere

6 figures The pieces of a random vibration. Random plane wave, 240×240 grid, wavelength 24: 23 positive and 18 negative domains.

The pieces no formula counts

6 figures Cutting the square down to a fixed point. λ = 0.999 with turning; accuracy 0.001: cutting 4 readings, iterating 7215.

A cut that gets sharper as the map gets lazier

6 figures The two-handled surface's octagon, eight to a corner. Regular hyperbolic octagon with angles π/4 and 64 neighbouring copies in the Poincaré disc.

An octagon only the hyperbolic plane can hold

5 figures Four ways to say which pixels touch. curve 8, rest 4: curve 1 pieces, rest 2 pieces; curve 8, rest 8: curve 1 pieces, rest 1 pieces; curve 4, rest 4: curve 12 pieces, rest 2 pieces; curve 4, rest 8: curve 12 pieces, rest 1 pieces.

A curve of pixels needs two kinds of touching

5 figures An astrolabe is a stereographic projection in brass. Latitude 52°: 16 engraved curves, worst departure from a circle 8.88e-15; altitude-circle formulas within 2.2e-16; Cancer radius 0.6564, Capricorn 1.5235 of the equator's.

An instrument that computes with circles

5 figures A random walk winding round a puncture. Lattice walk of 4000 steps around (½, ½); final winding -4.861 turns; box -12..26 × -29..22.

How a random path winds round a point

6 figures Three random loops and the knots they tie. 0₁: 12 crossings; 3₁: 18 crossings; 4₁: 16 crossings.

Which knots a random loop ties

6 figures Five squares on one octagon. Octagon [[1.0756,0.5516],[0.6082,1.2964],[-0.466,1.2294],[-0.8758,0.3153],[-0.797,-0.3348],[-0.3706,-0.7683],[0.2718,-0.6558],[1.1693,-0.4453]]; squares with sides 1.4975, 1.4588, 1.4407, 1.3929, 1.3921.

An odd number of squares on every polygon

6 figures How many times the tangent turns. circle: 1; figure-eight: 0; limaçon with a loop: 2; two inner loops: 3; three inner loops: 4; three-lobed star: -2.

The turns a tangent makes

6 figures Every closed curve with three crossings. Six sphere curves with 3 crossings; plane counts 2, 3, 3, 3, 4, 5 (total 20); symmetries 12, 6, 4, 4, 2, 2.

How many ways a curve can cross itself

6 figures Every failed branch datum up to eight sheets. Exceptions with three branch points: 4: 31|22|22; 6: 51|222|222; 6: 42|411|222; 6: 42|222|222; 6: 33|33|3111; 6: 33|321|222; 8: 71|2222|2222; 8: 62|5111|2222; 8: 62|2222|2222; 8: 611|422|2222; 8: 53|3311|2222; 8: 53|2222|2222; 8: 521|422|2222; 8: 5111|44|2222; 8: 5111|422|422; 8: 44|44|311111; 8: 44|3221|2222; 8: 431|332|2222; 8: 422|332|2222.

The lists no covering can have

6 figures Thirteen subgroups of index three. a=012 b=120; a=021 b=102; a=021 b=120; a=102 b=021; a=102 b=120; a=102 b=210; a=102 b=201; a=120 b=012; a=120 b=021; a=120 b=102; a=120 b=120; a=120 b=210; a=120 b=201

Counting the subgroups of a free group

5 figures Ten points, thirteen halving lines. points 0.253,1.518 0.242,0.216 0.717,0.119 -0.427,0.222 0.554,0.409 0.299,0.346 0.806,0.006 -1.101,0.118 1.209,-0.414 0.445,0.714; halving pairs 0-9 1-2 1-3 1-5 2-4 2-6 3-5 3-7 4-5 4-9 5-6 5-9 6-8; degrees 1,3,3,3,3,5,3,1,1,3.

Lines that leave half on each side

5 figures