The collection

Every essay — page 11

Page 11 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 three pairings of the roots of x⁴ + x + 1. Three panels each showing the same four roots of a quartic in the complex plane, joined in a different way into two pairs, with the value of the sum of the pair products beneath.

Three ways to pair four roots

Discs from the rows, and the eigenvalues inside them. The complex plane with one disc per row of a 3×3 matrix — centred on its diagonal entry, with radius the sum of the sizes of the rest of the row — and the 3 eigenvalues marked: 4.14, −5.09, −1.06.

Discs that fence in the eigenvalues

The largest value on each plane, over every direction of space. A longitude–latitude map of directions u in three dimensions, each shaded by the largest value of xᵀAx on the plane perpendicular to u. The smallest such value is the middle eigenvalue 1.829, reached at ±v₁; the largest is 3.622, reached along a great circle.

The highest point on the sphere is an eigenvalue

The free group on two generators with its middle removed: 4 pieces. The Cayley graph of the free group on two generators, with the elements within 0 steps of the identity greyed out and the remaining elements coloured by which connected piece they fall in.

What is left when the middle is taken out

One lattice, 3 generating sets, 3 shapes of ball. Lattice points reached within a fixed number of steps in the integers squared, for one step along either axis, axis steps and one diagonal, a king's moves, each drawn inside the polygon spanned by its steps and scaled by the radius.

The polygon a lattice becomes from far away

How a polynomial breaks modulo the primes

The units of a cubic field, as a lattice of logarithms. Points for 66 units of the field of a root of x³ − 3x + 1, plotted by the logarithms of two of their conjugates, lying on the lattice spanned by the logarithms of θ and θ − 1.

Units that form a lattice

Every reachable arrangement of the 3×3 sliding puzzle, by moves from solved. A bar chart of the 181440 reachable arrangements of the 3×3 sliding puzzle by the fewest moves that solve them, rising to a peak at 24 and falling to 2 at the maximum distance 31.

Thirty-one moves from solved

7 figures $Which arrangements sliding tokens reach, on nine graphs, against Wilson's theorem. A table of 9 small graphs drawn as icons, each with the searched count of reachable token arrangements and the fraction of all arrangements it is: a six-cycle 5/120; K₂,₃ 12/24; the 2×3 tray 60/120; a house 24/24; a six-cycle with one chord 120/120; a wheel of six 120/120; θ, inner paths 2, 2, 2 2520/5040; θ₀, inner paths 1, 2, 2 120/720; θ, inner paths 1, 3, 3 20160/40320.$

Which graphs let the tokens go anywhere

The same two lengths at 3 angles, and the area largest at the right angle. Parallelograms spanned by columns of lengths 1.6 and 1.25 at angles 38, 90, 142 degrees. Their areas are 1.23, 2.00, 1.23; the bound 2.00 is the product of the lengths and is reached only when the columns are perpendicular.

The biggest box built from signs

A map that folds the plane over itself, with every point still counted once. The map (u, v³ + uv): its domain shaded by the sign of the Jacobian determinant and its image with the grid carried across. At 5 marked target points the preimages number 3, 3, 1, 1, 1 and their signed counts are all 1; the determinant integrates to 4.447, equal to the integral of the signed count.

The count a fold cannot change

Two graphs the eigenvalues cannot tell apart

A loop that is a hole, and the same loop filled in. a hollow triangle: 3 points, 3 edges, 0 triangles; holes in each dimension 1, 1, 0; a filled triangle: 3 points, 3 edges, 1 triangles; holes in each dimension 1, 0, 0.

A hole is a cycle that bounds nothing

The 3 periods of the 13th roots of unity, and the equation they solve. Roots of unity modulo 13 grouped by the cosets of the index-3 subgroup; the periods 0.274, 1.377, −2.651 are the roots of x³ + x² − 4x + 1.

The sums that obey a smaller equation

The primitive 15th roots of unity, raised to a power and added. c₁₅(1) = 1; c₁₅(3) = −2; c₁₅(5) = −4; c₁₅(6) = −2; c₁₅(10) = −4; c₁₅(15) = 8.

The sums of primitive roots are always whole

The roots of random polynomials of degree 30 and 100: few are real. degree 30: 2 real roots at -2.335, -1.074; degree 100: 2 real roots at -0.327, 0.994.

Almost none of the roots are real

The roots of every polynomial of degree 11 with coefficients ±1. 22528 roots; radii from 0.500 to 2.000; 42% within 0.1 of the unit circle.

Random roots crowd onto the circle

A ball of the square grid with random travel times. A ragged roughly round region of grid cells shaded in bands by the time a signal from the centre first reaches them.

The shape a random ball grows into

Which pairs of the square's symmetries commute: 40 of 64. A 8-by-8 grid over the elements of the 8 symmetries of a 4-gon, rows and columns grouped into 5 conjugacy classes, with a filled square wherever the two elements commute: 40 filled squares, and each row's count of filled squares written at its end.

Five-eighths of the pairs, and no more

How often a walk comes home, on three lattices and a tree. Four curves of the logarithm of the return probability after 2n steps, for n up to 60: three lattices flattening, and the free group falling along a straight line.

How rarely a walk on a group comes home

Three polynomials of signs around the circle. The size of the polynomial around the unit circle divided by √64 for all-plus, random and Rudin–Shapiro sign polynomials of length 64; maxima 8.00, 2.50, 1.41.

The flattest polynomials of signs

The Mandelbrot set counted cell by cell. A 96-cell-wide grid over the Mandelbrot set: 2034 cells certainly inside (1.3794), 290 undecided after 60 steps (0.1967).

The area of the Mandelbrot set, from two sides

One form completed three ways, and the same straddle each time. The plane coloured by the sign of 2x² + 6xy + 3y². Three pairs of lines, one for completing with x first, one for y first and one along the eigenvectors; their coefficients are 2 and −3/2; 3 and −1; 5.541 and −0.541, and every pair has one line in a positive wedge and one in a negative wedge.

The signs no completion can change

Every slice of a tilted bell is a bell, centred off the axis. Contour ellipses of a two-variable bell with correlation 0.6, five vertical slices drawn as bells, their centres on the line y = 0.6x, and the ellipses' long axis on the diagonal y = x.

Regression is a square completed halfway