Field

Algebra — page 4

Structure — what stays true when the numbers change.

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

The roots of a random polynomial are not scattered across the plane. They gather within about 1/n of the unit circle, and their directions spread evenly round it. Paul Erdős and Pál Turán proved in 1950 that this is not a fact about randomness at all: any polynomial whose coefficients are all of roughly one size has roots whose directions are nearly even, and the inequality says exactly how nearly, in terms of nothing but the coefficients.

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

Give every road of the square grid a random travel time and ask what can be reached from one point in time t. The region is ragged, and rescaled it converges to a fixed convex shape — but which shape is unknown for every natural law. Computing it shows a curve within a few per cent of a circle for continuous travel times, a flat side where fast roads percolate along a diagonal, a time per step that is still drifting at a hundred and twenty-eight steps, and fluctuations that grow like the distance to the power one third rather than one half.

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

Pick two symmetries of a square at random and do them in both orders: forty times in sixty-four the result is the same. No group that fails to commute does better. The reason is a count of pairs that turns into a count of conjugacy classes, and a two-line argument about the centre that caps the answer at five-eighths — reached by the square and the quaternions, approached from above by nothing, and approached from below by groups that commute a little more than half the time.

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

Walk at random on the picture of a group, one generator at a time, and ask for the chance of standing at the start after 2n steps. On the line, the plane and three-dimensional space it falls like a power of n. On the tree that pictures the free group it falls by the factor √3/2 every step, exponentially. Kesten proved in 1959 that this is no accident of two examples: the chance falls exponentially exactly when the group's balls are mostly boundary, so a probabilistic rate and a geometric ratio are the same measurement.

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

A polynomial whose coefficients are all +1 or −1 has average size √n on the unit circle. Keeping it near √n everywhere is the problem Littlewood posed: the Rudin–Shapiro polynomials never exceed √2 times it, searching every sign pattern up to length 22 finds the best are the Barker sequences, and whether the maximum can come arbitrarily close to √n is still open.

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

The Mandelbrot set has an area, and nobody knows it. Counting grid cells closes in on 1.5066 from both sides, but every count judges a cell by a single point, and points on the boundary never declare themselves. An exact formula exists — π times one minus a sum over the coefficients of the map onto the set's outside — and every partial sum is a guaranteed upper bound; after sixteen thousand coefficients the bound is still 1.786, and at the rate it falls it would need something like 10²⁵ more.

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

A quadratic in several variables can be completed square by square in many orders, and each order hands back different coefficients. What no order changes is how many come out positive and how many negative — and that count, read off a completion of A − tI, says how many eigenvalues lie below t without finding a single one.

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

A bell in two variables, sliced at a fixed x, is a bell in y — and completing the square in y alone says where it is centred: at ρx, on a line flatter than the ellipse's own axis. That gap is regression to the mean, and the same completion says it runs backwards in time just as well as forwards.

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

No formula solves every equation of degree five, and yet x⁵ − 2 is solved by a fifth root and x⁵ − 5x + 12 by a longer expression of the same kind. A quintic can be solved by radicals exactly when the symmetries of its roots fit inside one group of twenty — the maps x ↦ ax + b on the numbers modulo 5 — and two exact tests on its coefficients say whether they do.

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

The cubic x³ − 3x + 1 has three real roots, and Cardano's formula reaches every one of them by way of the cube roots of a complex number. That detour cannot be removed: Hölder proved in 1891 that no expression built from real radicals gives a root of an irreducible cubic whose roots are all real. The proof is three lines of the Galois correspondence, and its general form says that real radicals reach all-real roots only when square roots alone would.

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

Pick two ways of rearranging n letters at random and allow them to be repeated in any order. Almost always the two together reach every rearrangement there is, or every even one — and when they fail, the usual reason is a coincidence about a single letter that both happen to leave alone. Eugen Netto guessed it in 1882 and John Dixon proved it in 1969.

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

Two of the icosahedron's rotations, chosen at random, generate all sixty of them nineteen times in thirty. Philip Hall found the number in 1936 by inclusion and exclusion over every subgroup at once, weighted by the Möbius function of the subgroups' order — and the same count says that nineteen copies of the group, side by side, can still be generated by two elements, while twenty cannot.

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

Write down every cubic x³ + ax² + bx + c with coefficients up to 150 and sort each by its Galois group. 98.7% have all six symmetries of three roots. The rest are reducible at a rate of fifteen in every H², or cyclic at five in every H to the three-halves — and the cyclic ones turn up four times as often as chance would allow, because discriminants are not random numbers.

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

A quartic can have five different Galois groups, and in a box of nearly fourteen million quartics they do not come in order of size. The eight-element group D₄ outnumbers the twelve-element A₄ thirty-five to one. Every even quartic has a smaller group than S₄, and moving one step off that plane restores the full group almost everywhere.

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

The fraction written on every bulb

Round the edge of the Mandelbrot set's cardioid hangs one bulb for every fraction between nought and one. The fraction is the bulb's address, its period, and the turn its cycle makes round a fixed point, all at once, and it sets the bulb's size as well: sin(πp/q)/q², which 245 computed bulbs obey to within fourteen per cent.

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

Seen from far away, every point of the Mandelbrot set's outside has an angle, and the curves of constant angle come in to land on the set. At the root of the p/q bulb exactly two of them land, and their angles are written by the staircase under a line of slope p/q. The limbs those pairs fence off share out the whole circle of angles between them, by the identity Σ φ(q)/(2^q − 1) = 1.

Three coordinates and a fourth power

The Heisenberg group is made of triples of whole numbers, and two moves generate all of it. Count the triples within r moves of the start and the count grows like r⁴, not r³: at forty-four moves there are 1,600,703 of them, against 117,569 for the cubic lattice. The fourth power comes from one coordinate that the moves reach by enclosing area, so that height costs only the square root of itself — and seen from far away, the ball of reachable triples is a curved solid with a dimple at each pole.

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

A magic square is a grid whose rows, columns and diagonals share one sum. The conditions are linear, so the squares form a space: three-dimensional for order three, eight-dimensional for order four, n² − 2n in general. Inside the space of order three the eight classical squares are eight points with coordinates ±1 and ±3. Of the 880 squares of order four, 640 are singular matrices, and the search finds the reason: a symmetry of the rows and columns that carries every number to the one it adds to seventeen with.

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

Turn a solid body to an orientation chosen at random and it has usually been turned a long way: on average 126.5 degrees, and by less than ten degrees only three times in ten thousand. Unit quaternions make the right meaning of 'at random' a picture, and two recipes that look fair fail it in opposite directions.

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

Fill a large symmetric matrix with independent random numbers and its eigenvalues do something no list of independent numbers does: they refuse to crowd together. The reason is a count of conditions in a two-by-two matrix, and its consequences — a semicircle, gaps that vanish at nought, a spectrum almost as rigid as a lattice — reach from heavy nuclei to the zeros of the zeta function.

Every pair of landing angles to period eight. Lavaurs chords periods 2-8: 2:1, 3:3, 4:6, 5:15, 6:27, 7:63, 8:120; crossings 0.

Chords that never cross

Every component of the Mandelbrot set's interior has a root, and exactly two external rays land there, with angles that repeat under doubling. Which two can be found without drawing the set at all: list the angles of each period in order, and join each to the next one whose chord crosses no chord already drawn. The rule makes 63 pairs at period 7 and 2,010 at period 12, one for every component; traced into the plane, every pair's two rays end together, beside a centre of the right period and no other.

The correlations that reverse a coefficient. rho = 0.6: wedge shares 0.2051 and 0.2051 of the ellipse; arcsin(0.6)/pi = 0.2048.

Holding one still turns the other over

A quantity can rise with an outcome and still predict it downwards once a second quantity is held fixed. Completing the square in two predictors says exactly when: the pair of correlations has to fall in a wedge whose angle is arcsin of the predictors' own correlation. Averaged over every possible correlation matrix the chance is exactly 1/4 − 1/π², about one coefficient in seven; two coefficients never reverse together, and with two dozen predictors almost two in five do.

The class number is a degree

The integers of ℚ(√−23) fall into three classes of ideals, and that 3 can be found by counting forms. It is also the degree of a single number: the modular function j, evaluated at the three forms' points, gives three values that are not whole but are the roots of one cubic with whole-number coefficients. When there is only one class, the value is a whole number outright — and that is why e^(π√163) misses being whole by less than a trillionth.

How near nought a sum of roots can come

Five arrows of length one, each pointing at a 122nd root of unity, laid end to end, return to within fifteen millionths of where they started. No five of the 123rd roots come within a thousandth. Two, three and four arrows shrink their smallest miss in a tidy power of n; five scatter over four powers of ten from one n to the next, and nobody knows the law they follow.

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