The collection

Every essay — page 12

Page 12 of 43, 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

Algebra

Structure — what stays true when the numbers change.

The twenty maps x ↦ ax + b modulo 5, as permutations of five roots. A four-by-five grid of pentagons; in each, arrows show where the map ax + b modulo 5 sends each of the five corners. Rows are the slopes 1, 4, 2, 3; columns the shifts 0 to 4.

The quintics that have a formula

Cardano's formula for x³ − 3x + 1, drawn: three real roots from conjugate cube roots. The complex plane with u³ = −0.50 + 0.866i and its conjugate, their three cube roots each, and the vertical pairings whose sums are the real roots −1.879, 0.347, 1.532.

Three real roots and no real radicals

Every pair of rearrangements of four letters, and what the two generate. A 24 by 24 grid of ordered pairs of permutations of four letters: 216 generate S4, 96 generate A4, 264 generate a smaller subgroup.

Two random shuffles reach every shuffle

The 3,600 pairs of icosahedral rotations, and what each pair generates. A 60 by 60 grid of ordered pairs of elements of A5 shaded by the order of the subgroup they generate; 2280 generate A5.

Nineteen copies and no more

Every cubic x³ + bx + c up to 30, sorted by its Galois group. A 61 by 61 grid of cubics x³ + bx + c: 309 reducible, lying on lines c = −r³ − rb; 20 cyclic with square discriminant; 3392 with group S3.

Twenty-seven million cubics, sorted by their symmetry

Even quartics never have the full group, and one step away almost all do. Even quartics x^4 + bx^2 + d to 20: reducible 199, S4 0, A4 0, D4 1338, C4 20, V4 124. With +x: reducible 157, S4 1524, A4 0, D4 0, C4 0, V4 0.

The quartics whose roots stay paired

6 figures $Every fraction has a bulb. The cardioid with 277 computed bulbs of period 2 to 30; the bulbs of period up to five labelled 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5.$

The fraction written on every bulb

Two rays land at the root of every bulb. Parameter rays 1/3 & 2/3, 1/7 & 2/7, 5/7 & 6/7, 1/15 & 2/15, 13/15 & 14/15, 9/31 & 10/31, 21/31 & 22/31 to depth 160; furthest ray end from its root 0.0197.

Two rays for every fraction

Three coordinates and a fourth power. Ball sizes at r = 10, 20, 44: Z² 221, 841, 3961; Z³ 1561, 11521, 117569; Heisenberg 4309, 68079, 1600703; local exponent at 30: Heisenberg 4.005, Z³ 2.946.

Three coordinates and a fourth power

6 figures · new

Every magic square of order three is a point in a space of three. Basis J, U = (1,-1,0,-1,0,1,0,1,-1), V = (0,-1,1,1,0,-1,-1,1,0); Lo Shu = 5J − 3U + V; dimension of the order-3 magic space 3.

The magic squares are a space

6 figures · new

Three ways to pick a rotation at random. Mean rotation angle: uniform 126.70° (exact 126.48°), uniform Euler angles 127.91°, uniform axis and angle 90.18°. Densities by 5° bin, uniform: 0.00, 0.00, 0.01, 0.01, 0.03, 0.04, 0.04, 0.06, 0.09, 0.09, 0.13, 0.15, 0.16, 0.20, 0.23, 0.25, 0.28, 0.29, 0.34, 0.36, 0.38, 0.41, 0.44, 0.46, 0.47, 0.52, 0.54, 0.55, 0.57, 0.61, 0.60, 0.64, 0.62, 0.61, 0.64, 0.64.

A rotation chosen at random

6 figures · new

Eigenvalues that will not crowd together. Unfolded spacings, eigenvalues: 0.36 1.52 1.15 0.88 1.23 1.13 0.34 0.75 2.05 0.65 1.66 0.64 1.07 1.01 0.86 1.23 0.69 1.30 0.77 0.94 0.99 0.71 0.92 2.49 0.32 1.07 0.62 1.34 0.07 0.51 0.98 2.25 0.64 0.78 0.35 0.95 1.36 1.89 0.21 0.79 0.46 0.84 1.12 1.52 0.96 0.65 1.09 0.29 1.30 1.06 1.83 1.50 0.28 0.43 1.19 1.01 0.52 1.54 1.51 1.56. Independent points: 0.34 0.01 0.31 3.72 0.25 2.29 0.98 2.14 1.99 0.54 0.15 0.58 0.96 0.22 0.26 1.11 0.04 0.38 0.55 0.74 0.89 0.11 1.47 0.52 0.06 2.11 1.29 3.04 0.44 0.28 2.49 3.97 0.87 1.04 0.83 0.77 0.79 0.69 1.76 1.82 0.14 0.07 1.03 0.88 1.05 0.37 0.06 0.82 0.70 1.72 0.90 1.24 0.32 0.25 1.07 0.55 1.59 3.23 0.36 0.40. Gaps under a quarter of the mean: 2 against 9.

Eigenvalues that keep their distance

7 figures · new

Discrete

Counting, graphs, and things that come in whole pieces.

Königsberg as a graph. The four landmasses as circles and the seven bridges as edges; every circle has an odd number of edges.

Seven bridges, and the invention of throwing things away

Pascal's triangle mod 2, 32 rows. Only the odd entries are drawn; the pattern that appears is the Sierpiński triangle.

Pascal's triangle, in two colours

Ulam's spiral to 900. The integers up to 900 laid out in a square spiral, with the primes marked; they crowd onto diagonal lines.

The primes on a spiral, and a pattern nobody ordered

13 into 12. 13 items spread as evenly as 12 boxes allow. Even at their most even, some box holds 2, because 13 is more than 12 × 1.

More things than boxes

A wheel of 5 rim regions needs 4 colours. A hub touching 5 rim regions arranged in a ring. The rim is odd, so the whole map needs 4 colours and no fewer.

Four colours, and a proof nobody can read

Every triangulation of a 6-gon. All 14 ways of cutting a convex 6-gon into triangles with non-crossing diagonals — the 4th Catalan number, counted by drawing them.

One sequence, counting everything

Arithmetic on a dial of 12. A dial with 12 positions. Starting at 8 and stepping forward 9 places lands on 5, because the walk passes the top 1 time on the way.

Numbers that wrap

Six people, and the trio that cannot be avoided. The fifteen pairs among six people, coloured at random. Whatever the colouring, three people are all mutual acquaintances or all mutual strangers — here 1, 2, 5.

Six people at a party

A closed walk on the 3-cube changing one place at a time. The corners of a 3-dimensional cube with a path through every one of them exactly once, each step moving along an edge, and the last corner one step from the first.

A walk that changes one thing at a time

A lattice polygon of area 22.5. A polygon with all its corners on the integer grid, with the 20 grid points strictly inside and the 7 on its boundary marked; its area is the first count plus half the second, less one.

Area by counting dots

Two graphs that will not lie flat

All 16 trees on 4 labelled points. Every tree on 4 labelled points, drawn one by one. There are 16 of them, which is 4 to the power 2.

Sixteen trees on four points