Matpic
mathematics, in pictures
The collection

Every essay — page 29

Page 29 of 49, 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

Number

The whole numbers, and how much structure they turn out to have.

The tree as words in two matrices. 6 nodes of the Stern–Brocot tree, each as the word of turns reaching it, the matrix that word multiplies out to, its two columns as fractions, and the mediant of those columns.

Two matrices that generate the tree

5 figures Every positive rational, in one sequence. The first 32 terms of Stern's diatomic sequence as bars, with the ratios of consecutive terms beneath. Every ratio is in lowest terms, no two agree, and each term counts the hyperbinary representations of its index.

Every rational in one sequence

5 figures 13 record approximations in 26 turns. The distance from π to each fraction the descent passes, against its denominator, on logarithmic axes. 13 of them beat every fraction with a smaller denominator.

The fractions that beat every smaller one

5 figures The Farey tessellation, and a line down to √2 − 1. Semicircles over the unit interval joining every pair of Farey neighbours with denominators up to 13, and a vertical line at √2 − 1. The 6 arcs it crosses are the intervals of the Stern–Brocot descent to √2 − 1, and their turns spell LRRLL.

The arcs a line crosses on its way to a number

5 figures Minkowski's question-mark function. The graph of Minkowski's function ?(x) on the unit interval: continuous and increasing, sending each Stern–Brocot fraction to the binary fraction in the same position. It sends √2 − 1 to 2/5 and φ − 1 to 2/3.

The function that sends fractions to binary

5 figures Bhāskara's cyclic method on x² − 61y² = 1, step by step. A table of the cyclic method's rows: the helper m chosen at each step and the near miss a² − Db² = k it produces, ending at k = 1 with the fundamental solution.

A method that is allowed to miss

5 figures The states the continued fraction of √61 can be in. A grid of whole-number pairs with the band of reduced states shaded, the pairs that qualify marked, and the cycle of states visited by the expansion of the square root numbered in order.

Why the expansion has to repeat

5 figures The powers of 2 modulo 13, as a ring of 12. The non-zero residues modulo 13 placed on a circle, with the successive powers of 2 joined by straight lines into a closed walk of 12 steps.

One residue whose powers are all of them

7 figures The orders of the 8 units modulo 15. A strip of the units modulo 15 with each one's multiplicative order beneath it, the largest order marked at 4 against φ(15) = 8.

The exponent that is smaller than Euler's

8 figures The two squares of 97, produced by division. A table of the division chain on 97 and a square root of minus one modulo it, with each row's quotient and remainder, the point at which the remainder falls below the square root marked, and the two squares that add to 97.

The two squares actually produced

5 figures How rare a sum of two squares is. Two curves against the logarithm of the bound: the fraction of numbers below it that are sums of two squares, falling; and that count times the square root of the logarithm, divided by the bound, which is nearly constant.

Almost no number is one

5 figures Where the aliquot sequence of every number up to 1000 goes. A grid of the starting values 2 to 1000 coloured by the fate of each aliquot sequence: 964 reach 1, 19 reach a perfect number, 3 enter a cycle, and 13 pass 10²² undecided.

The sum of the parts, taken again

6 figures How the abundancy σ(n)/n is distributed, up to 1,000,000. The share of numbers up to 1000000 whose ratio σ(n)/n is at least t, for t from 1 to 4. It is 0.2475 at two, 0.0202 at three and 0.00023 at four, with a steep fall just above one, where the numbers with no small prime factor sit.

Why a quarter of numbers overshoot

6 figures The tangent built from a continued fraction. The curve tan x on (−1.55, 1.55) with 4 of Lambert's convergents: a straight line, then rational curves that bend ever closer to the tangent and follow it towards its poles.

The fraction Lambert built for the tangent

5 figures The continued fraction of e. Bars for the first 30 continued-fraction terms of e: mostly ones, with every third bar rising in a straight staircase, 2, 1, 2, 1, 1, 4, 1, 1, 6, ….

The pattern in e's continued fraction

5 figures 441 factored two ways among the numbers of the form 4k + 1. Two factor trees for 441 among the Hilbert numbers, those one more than a multiple of four: one splits it as 9 times 49, the other as 21 times 21, and every factor is irreducible there.

A factorisation that hides its primes

6 figures Where discs of radius one cover the lattice, and where they leave holes. Six lattices of algebraic integers drawn as points in the plane with a unit disc around each; for five of them the discs cover the whole plane and for the sixth, the integers of the field of the square root of minus nineteen, uncovered holes remain.

Factoring uniquely with no way to divide

7 figures A witness that 97 is prime: 5 has order 96. A table of the checks Lucas's test makes on 97 with base 5: the power (n − 1)/q for each prime q dividing n − 1, none of which is 1.

An order that proves a prime

5 figures The Fermat quotient of 2 at every prime up to 4000, and where it is 0. A scatter of the Fermat quotient of 2 modulo p, divided by p, against p for the primes up to 4000; the points spread evenly between zero and one and reach zero at the Wieferich primes.

Two primes where Fermat holds twice

5 figures The solutions of x² − 2y² = N, class by class. Rows for several right-hand sides N, each marking the solutions of x² − 2y² = N at the logarithm of x + y√2, coloured by class, over alternately shaded windows one unit-step wide.

Two families of solutions, and a box that holds both

5 figures How many digits Pell's first solution has, for D up to 1000. A scatter of the digit count of the smallest solution of Pell's equation against D, with the record-setting values ringed and labelled.

Sixty needs two digits and sixty-one needs ten

6 figures Random partitions of 1,000, scaled, against their limit shape. The outlines of 4 uniformly random partitions of 1000, scaled by the square root of 1000, lying close to the curve e^(−cx) + e^(−cy) = 1 with c = π/√6.

The shape a random partition takes

5 figures Two ways of counting that agree at every number. For n up to 40, the counts of partitions with gaps of at least two against partitions into parts congruent to 1 or 4 mod 5, on a logarithmic scale, equal at every n, with the second identity's counts beside them.

Two counts that agree for no visible reason

5 figures Numbers whose divisors add to two, three, four, five and six times themselves. A table of multiperfect numbers with the multiple their divisor sum makes of them, the number itself and its factorisation into prime powers.

Divisors that add to three times the number

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