Matpic
mathematics, in pictures
Generator

Multiplying two complex numbers

A generator in the algebra library, called 124 times across 24 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

complex-turn is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

Multiplying two complex numbers. In the complex plane, multiplying adds the two angles and multiplies the two lengths.

Three roots that one map rotates, for every t

Three roots that one map rotates, for every t. Shanks cubic roots for t in [-9,6]; at t=2: 3.507019, -0.221876, -1.285142; map permutation 1,2,0.

Every member is an equilateral triangle on a circle

Every member is an equilateral triangle on a circle. t=-1: roots 1.24698,-0.44504,-1.80194, angle 67.26; t=0: roots 1.87939,-0.34730,-1.53209, angle 80.00; t=1: roots 2.65109,-0.27389,-1.37720, angle 89.27; t=2: roots 3.50702,-0.22188,-1.28514, angle 95.61; t=5: roots 6.29590,-0.13706,-1.15883, angle 105.48; t=20: roots 21.09267,-0.04526,-1.04741, angle 115.41

A diagonal of cyclic cubics in a plane of full ones

A diagonal of cyclic cubics in a plane of full ones. c=−1, |a|,|b|≤40: cyclic on line 78, cyclic elsewhere 48 ((-36,-21) (-35,-12) (-31,-25) (-30,21) (-26,13) (-21,14) (-21,30) (-20,-9) (-17,10) (-15,-9) (-14,9) (-14,21)…), reducible 159, full 6276.

A parabola of square discriminants and a curve with almost none

A parabola of square discriminants and a curve with almost none. x^3 − t x − 1, |t| ≤ 1000000: square discriminant at t=3 (y=9).

The discriminant's root is a norm of an Eisenstein integer

The discriminant's root is a norm of an Eisenstein integer. t=-1: 7 = 7^1; t=0: 9 = 3^2; t=1: 13 = 13^1; t=2: 19 = 19^1; t=3: 27 = 3^3; t=4: 37 = 37^1; t=5: 49 = 7^2; t=6: 63 = 3^2·7^1; t=7: 79 = 79^1; t=8: 97 = 97^1; t=9: 117 = 3^2·13^1; t=10: 139 = 139^1; t=11: 163 = 163^1; t=12: 189 = 3^3·7^1; checked to t=2000 (2002 values).

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Algebra

A line of cubics that are all cyclic

In a box of cubics the full group of six symmetries is almost universal, and the three rotations are a rare exception. Daniel Shanks's x³ − tx² − (t + 3)x − 1 has the rotations at every whole t, and at every rational one. Its discriminant is the square of t² + 3t + 9, its roots are turned into one another by x ↦ −1/(1 + x), and on a suitable circle every member is an equilateral triangle.

 Algebra

A loop that cannot miss the middle

Feed a circle into a polynomial and a closed loop comes out. A small circle gives a loop that does not enclose the origin; a large one gives a loop that goes round it as many times as the degree. Something has to happen in between, and that something is a root.

 Algebra

A multiplication that remembers the order

Four characters — i² = j² = k² = ijk = −1 — define a multiplication in which ab and ba are different numbers. Everything follows from them, including the fact that a rotation of a solid body has two names, and that two full turns are needed to get one of them home.

 Algebra

A prime leaves no room for a comb

Take the table of powers of a root of unity, pick any rows and any equal number of columns, and take the determinant. At order 12 about one such determinant in six is exactly nought. At order 11, among 184,756 of them, none is, and in 1926 Chebotarev proved that none ever is at a prime. It means a signal on p places and its transform together occupy at least p + 1, while a comb on twelve places and its transform take 4 and 3.

 Algebra

Almost none of the roots are real

Pick the coefficients of a polynomial of degree a thousand at random, each one an independent draw from the bell curve, and ask how many of its thousand roots are real. The answer is about five. Mark Kac found in 1943 that the average grows only like (2/π) ln n, and the reason can be read off the real line itself: the real roots crowd towards +1 and −1 and spread evenly on a logarithmic scale of distance from them.

 Algebra

Every power sum, from the coefficients alone

Raise the roots of a polynomial to the k-th power and add them. However the roots turn, the total is a whole number when the coefficients are, and Newton's identities produce it from the coefficients one step at a time — no root is ever found. Run the rule on x³ − x − 1 and out comes Perrin's sequence, whose terms know which numbers are prime, nearly.

 Algebra

Every third coefficient

Add every third number in the twelfth row of Pascal's triangle and the answer is 1366 — a third of 4096, rounded up. Which way the rounding goes is decided by two arrows of length one in the complex plane, and the same average over the roots of unity counts dice totals, subsets and necklaces.

 Algebra

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.

 Algebra

Multiplying is turning

Complex numbers are introduced as an algebraic dodge for square roots of negatives. They are better understood as the arithmetic of rotation, at which point every rule stops needing to be remembered.

 Algebra

On the circle and never home

Every root of unity lies on the unit circle, and so does the point (3 + 4i)/5 — yet no power of it ever returns to 1. A root of a whole-number polynomial of degree ten does the same. What forces a point home is a condition on the numbers its polynomial ties it to, and how far one of them may stray is a question open since 1933.

 Algebra

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.

 Algebra

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.

 Algebra

The lattice that runs the other way

The symmetries of a polynomial's roots form a group, and the fields between the bottom and the top form a lattice. The two are the same picture, one of them turned over — a bigger group of symmetries fixes less, so it names a smaller field.

 Algebra

The polygon an equation forces

The n solutions of z to the n equals one are the corners of a regular polygon, and nearly everything about them — that they form a group, that they sum to zero, that the equation factors into pieces with whole-number coefficients — is that picture read carefully.

 Algebra

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.

 Algebra

The roots of the slope stay inside

Mark the roots of a polynomial in the complex plane and stretch a band around them. However the roots are arranged, the roots of the derivative land inside the band — never outside, never on a new frontier. The reason is a balance of pushes, the same reason makes the derivative of a cubic mark the foci of an ellipse nobody asked for, and a question about how far inside the roots must sit has been open since 1958.

 Algebra

The sums of primitive roots are always whole

Take the roots of unity of order exactly q, raise each to the power n and add them. The answer is always a whole number, it depends on n only through what n shares with q, and as n varies the sums behave like the sines and cosines of a Fourier series — so well that Ramanujan could rebuild the sum of the divisors of any number from them, and prove that their weighted total is nought, a fact that at n = 1 is the prime number theorem.

 Algebra

The sums of roots of unity that add to nothing

All n of the n-th roots of unity add to zero, and so does any regular polygon among them, turned. Those are not the only vanishing sums: six thirtieths of a turn close into a loop with no polygon in them. Which counts of roots can close at all is decided by the prime factors of n — no seven fifteenths ever add to zero — and the same question counts where the diagonals of a regular polygon cross.

 Algebra

The sums that obey a smaller equation

The twelve non-trivial thirteenth roots of unity satisfy an equation of degree twelve. Split them into three groups of four — the right three groups — and add each group: the three sums are the roots of x³ + x² − 4x + 1, an equation of degree three with whole-number coefficients. Gauss called such sums periods, found one for every divisor of p − 1, and used them to build the seventeen-gon from four quadratic equations.

 Algebra

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.

 Algebra

Three ways to pair four roots

Four roots can be split into two pairs in exactly three ways, and the three numbers r·r′ + r″·r‴ those pairings give are the roots of a cubic whose coefficients can be read straight off the quartic. That cubic is where Ferrari's formula gets its cube roots, and it is also a verdict: whether its roots are rational decides which of the twenty-four symmetries the quartic's roots actually have.

 Algebra

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.

 Algebra

What the coefficients already know

Finding the roots of a polynomial is hard and often impossible in closed form. Reading off their sum, their product and how many of them are real is none of those things — those numbers are sitting in the coefficients, and no root-finding is required to get at them.

 Analysis

When the period grows without bound

A repeating signal has a spectrum of separate lines. Stretch the gap between repeats and the lines crowd together while the curve they sit on stays exactly where it is — and at infinite period the lines are gone and the curve is the whole answer.

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