Matpic
mathematics, in pictures
Theme

The same thing twice — page 1

Two constructions that look unrelated and turn out to be the same object wearing different clothes.
The Pythagorean theorem by dissection. Two squares of the same size. Each holds four copies of one right triangle. The space left over is a single tilted square on the left and two upright squares on the right. Geometry

Two squares, four triangles, and no algebra

The Pythagorean theorem is usually met as a formula to be memorised. It is much better met as a rearrangement that can be checked by eye.

 Four conic sections from one cone. Circle, ellipse, parabola and hyperbola, produced by tilting a single cutting plane further and further. Geometry

One cone, four curves

The circle, the ellipse, the parabola and the hyperbola look like four separate objects with four separate equations. They are one object, cut at four angles.

 A circle unrolled into a sine wave. On the left a radius turns through an angle; on the right the height of its tip is plotted against the angle, tracing a sine curve. Analysis

A sine wave is a circle seen from the side

Sine is introduced as a ratio in a right triangle, which is true and explains nothing about why its graph is a wave. There is a better picture.

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

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

Pascal's triangle, in two colours

Shade the odd numbers in Pascal's triangle and a fractal appears. Nothing was designed to produce it, and the same shape arrives independently from a completely different construction.

 One point on the sphere for every point of the plane. Lines from the north pole of a sphere through each of its points land on a plane below, matching the sphere minus one point with the whole plane. Topology

A sphere is a plane plus one point

Remove a single point from a sphere and what is left can be flattened out to cover an infinite plane exactly. The construction is one straight line, repeated.

 The whirling squares. Squares with Fibonacci sides 1, 1, 2, 3, 5, 8, 13, each attached to the long side of what came before. They fill a 13 by 21 rectangle exactly. Geometry

The rectangle that eats itself

Cut a square off a golden rectangle and what is left is a golden rectangle. That single property is the whole of the golden ratio, and it explains both what the number really does and most of what is wrongly claimed for it.

 The dot product as a shadow. Two vectors and the shadow the first casts on the second. The shadow is 2.425 long and b is 4.123, so the dot product is 10.000. Algebra

The dot product is a shadow

Multiply the matching coordinates and add them up. That rule explains nothing, and it hides the fact that the answer is a length — how far one arrow reaches along another, times how long that other one is.

 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. Discrete

One sequence, counting everything

The number of ways to cut a polygon into triangles is 1, 2, 5, 14, 42. So is the number of ways to bracket a product, the number of binary trees, and the number of paths that never cross a diagonal. They are the same count, and the reason is one picture.

 V − E + F = 2, five times. Vertices, edges and faces of the five regular solids, with the alternating sum. The edges are counted from the faces rather than listed, and the sum is 2 in every row. Topology

Every corner pays for itself

Count the corners of any solid, subtract the edges, add the faces. The answer is two. It is two for a cube, for a pyramid, for a football, for anything squashed or stretched — and the number is measuring the shape it is wrapped around rather than the shape itself.

 The plane divided by nearest neighbour. 10 sites, and every point of the rectangle shaded by which site is closest to it. The boundaries are the places where two sites tie. Geometry

The plane, divided by whoever is nearest

Scatter some points and colour every other point of the plane by which one is closest. The result is a tiling nobody designed, and its dual triangulation has a property that no part of the construction mentions.

 Area is the undoing of slope. Above, a positive function with the area from 0 to 1.80 shaded. Below, that area plotted against where it stops. The lower curve's slope at 1.80 is 1.129, which is exactly the upper curve's height there. Analysis

Area is the undoing of slope

Two operations invented for unrelated reasons — measuring a region and measuring a rate — turn out to be inverse. The picture is two panels sharing one axis, and the claim is that the lower curve's steepness is the upper curve's height.

 The field that cannot be combed. A tangent field on the sphere, flowing along the meridians. Every arrow is tangent to the surface, and at the two poles there is no direction for an arrow to take — the field is zero there, and no rearrangement removes both zeros. Topology

Nothing on a sphere can be combed flat

Point an arrow along the surface at every place on a sphere, continuously, and somewhere an arrow has to vanish. On a doughnut it can be done. The difference between the two is a number that was already known from counting corners.

 φ as a continued fraction. The nested fraction, one quotient per step, descending to the right. Number

A fraction that never closes

Euclid's algorithm throws away everything except the number of squares it peeled at each step. Those counts are a second name for the number it started from — one that terminates exactly when the ratio is a ratio.

 The Stern–Brocot tree to depth 4. Every positive rational, each appearing exactly once, generated by taking mediants. Number

Every fraction, exactly once

Take two fractions, add the tops and add the bottoms. That is not how fractions are added, it is not an average, and repeating it produces every positive rational exactly once, already in lowest terms.

 Two factor trees of 360. The same number split two different ways, both ending in the same primes. Number

One way to factor, and no other

Every number breaks into primes in exactly one way. That is so familiar it is hard to see as a claim at all — until it is put beside an arithmetic where it is false, and where six has two different factorisations that cannot be reconciled.

 The circle of radius √25 on the integer lattice. A circle drawn on the whole-number grid, with the lattice points it passes through marked. Number

Two squares, and a lattice

Whether a prime is the sum of two squares is decided entirely by its remainder on division by four. A fact about circles is settled by a fact about remainders, and neither statement contains any hint of the other.

 Rational points on the unit circle. Lines of rational slope through the left-hand point of a circle, each meeting it again at a rational point. Number

Every triple, on one circle

Draw a line of rational slope through a single point of a circle. Wherever it comes out is a rational point, and clearing the denominators turns it into a Pythagorean triple — so every triple there is comes from one line through one point.

 the logistic map at 3.2, iterated from 0.2. A map drawn as a curve with the diagonal across it, and the staircase that iterating it produces. Dynamics

The staircase that shows the whole orbit

Take a number, feed it to a rule, feed the answer back in. There is a way of drawing that on the rule's own graph which turns the entire future of a starting point into a shape — and the shape is legible.

 A fixed point that attracts, and one that does not. The same map at two parameters, with the staircase walking towards the crossing in one and away in the other. Dynamics

A point that pulls, and a point that pushes

Every crossing of a curve with the diagonal is a value the rule leaves alone. Whether anything ever arrives there is decided by one number — the slope at the crossing — and the picture makes the reason obvious.

 Where the period doubles, and by how much the gaps shrink. The parameters at which the period doubles, with the ratio of consecutive gaps beside them. Dynamics

A constant that does not care which map

The gaps between successive period doublings shrink by a factor. Measure that factor for the logistic map and you get 4.669. Measure it for a completely different map and you get 4.669, and nobody expected that.

 The Lyapunov exponent, 2.8 to 4. The average rate at which nearby orbits separate, plotted against the parameter. Dynamics

How fast two orbits part

The word "sensitive" is an adjective. Averaging the logarithm of one derivative along an orbit turns it into a number — one that says how many steps of prediction the map allows, and whose sign says whether it allows any.

 Elementary cellular automaton, rule 90. A row of cells evolving downward, each cell decided by the three above it. Dynamics

Eight rules and a triangle

A row of cells, each one deciding its next state from the three above it. Eight cases, one bit of output each — a rule that fits in a byte, and 256 of them in total. One of those bytes draws Pascal's triangle.

 Rotating by φ − 1 of a turn, 21 times. Points on a circle produced by repeatedly turning through the same angle. Dynamics

Three gaps and no more

Turn a circle by the same irrational angle over and over. The points never repeat and never settle, and yet at every single stage the gaps they leave take at most three different lengths — never four, at any number of steps, for any angle.

All themes