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mathematics, in pictures
Concept

Self-similarity

The property of a shape that contains smaller copies of the whole of itself. It can be exact, as in a construction, or statistical, as in a random walk seen at three magnifications.

Named by 26 essays across 6 fields — each of them below, with the objects they name alongside it.

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

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.

discrete · Pascals triangle 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.

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.

geometry · Golden ratio 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.

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.

dynamics · Iteration The logistic map's bifurcation diagram, 2.4 to 4. For each parameter, the values the orbit settles into, plotted as a column of points.

The road paved with doublings

Turn one dial slowly and watch what a map settles into. It settles on a point, then on two points, then four, then eight — faster and faster, and the doublings run out at a parameter that is finite.

dynamics · Period-doubling 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.

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.

dynamics · Period-doubling The Mandelbrot set. Points of the complex plane shaded by how long the iteration takes to escape, with the set itself the innermost region.

The shape in every picture of itself

One line of arithmetic, repeated, with a single complex number as its only input. Sort the numbers by whether the result stays bounded and the boundary between the two answers is the most complicated object anyone draws from a rule this short.

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

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.

dynamics · Cellular automata 6 cosines, and a curve with no tangent anywhere. Partial sums of a sum of cosines whose amplitudes shrink geometrically and whose frequencies grow faster. Each term adds finer detail; the curve converges and its slopes do not.

A curve with a corner at every point

Continuity means a curve can be drawn without lifting the pen. Differentiability means it has a tangent. The first was assumed to nearly imply the second until 1872, when Weierstrass exhibited a curve that is continuous everywhere and has a tangent nowhere — and it is a sum of cosines.

analysis · The derivative Middle thirds removed 6 times over. The interval with its middle third removed, then the middle third of each survivor, and so on. The lengths removed are a geometric series adding to the whole interval.

Almost none of it left, and still uncountably many

Remove the middle third of an interval, then the middle third of each piece left, and keep going. The lengths removed add to exactly the whole interval, so nothing measurable survives — and what survives can be paired off one for one with every point of the interval that was started with.

analysis · Measure One walk at three magnifications, and the shape it is heading for. The same random walk over three windows, each ten times longer than the last and scaled vertically by the square root of ten, so all three look alike. Beside them, the exact distribution of the position after a few step counts, standardised, closing on the bell curve.

The walk that becomes a curve

Shrink the steps of a random walk and it disappears. Shrink them while stretching the time in the right proportion — space by the square root of whatever time is divided by — and something is left behind, which is a curve nobody could draw.

probability · Random walk The rotation number against the parameter at K = 1. The measured rotation number of a circle map plotted against its parameter, forming a staircase that is flat over an interval at each simple rational.

The staircase that is flat almost everywhere

A map of the circle advances by an average amount each step. Plot that average against the parameter driving it and the graph is flat over an interval at every rational, rises only on a set of measure zero, and still climbs from nothing to one.

dynamics · Mode locking the Koch curve, after 5 steps. the Koch curve drawn from its own rule: replace the middle third of every segment with two sides of a triangle.

A dimension that is not a whole number

Cover a set with boxes of side ε and count how many are needed. For a line the count grows like 1/ε, for a region like 1/ε². For the Koch curve it grows like 1/ε to the power 1.26, and that exponent is as good a definition of dimension as the other two.

dynamics · Fractal dimension A set with no interval in it and half its length left, after 6 stages. Stages of removing a shrinking middle from every surviving interval, with the total length left printed at each stage, and the middle-thirds construction of the same depth drawn beneath for comparison.

No interval in it, and length to spare

The middle-thirds set has no length because the removed pieces add to one. Remove shrinking middles instead and they add to a half — leaving a set that still contains no interval anywhere, and still has half the length it started with.

analysis · Measure Stretch, fold, and what is left. 5 stages of the horseshoe map's surviving set: one square, then two strips, then four, up to 16, each narrower than the last by a factor of 3.

Stretch, fold, and what is left

A system that pushes every pair of nearby points apart and keeps them all inside a bounded region has only one option, and it is the one a baker uses. Stretching and folding is the mechanism, and what survives infinitely many folds is a Cantor set.

dynamics · Strange attractor The same banding at every magnification. The Hénon attractor drawn from 26000 points, followed by 2 magnifications of one part of it. Each magnification resolves what looked like a single curve into several parallel ones.

Neither a surface nor a solid

The attractor has no volume, because the flow shrinks volumes at a rate that can be read off the equations. It is also not a surface, because a surface cannot carry chaotic dynamics. What is left is an object of dimension a little over two, and that number is measurable.

dynamics · Strange attractor A staircase with no steps. The Cantor function drawn to several stages: a continuous non-decreasing curve from nought to one which is constant on every interval of the complement of the middle-thirds set, so its whole rise happens on a set of measure zero.

A staircase with no steps

A function that rises from nought to one, is continuous everywhere, and has derivative zero at almost every point. All of its climbing happens on a set of no length at all, which is possible because that set has uncountably many points.

analysis · Measure 3 carries in base 2, and 2 divides it 3 times. The addition of 5 and 7 written in base 2, column by column, with the carries marked. There are 3, and 2 divides the binomial coefficient 792 exactly 3 times.

The carries decide the divisibility

How many times a prime divides a binomial coefficient is not a fact about the coefficient at all. It is a count of the carries that happen when two numbers are added in that prime's base, which is a question about column addition and has nothing to do with choosing anything.

discrete · Pascals triangle Infinite below 0.6309, nought above it. The total of the s-th powers of the diameters in the natural cover of the middle-thirds Cantor set, plotted against s for 4 depths. Every curve passes through one at s = 0.6309 and they separate either side of it.

Infinite on one side and nought on the other

Box counting returns a growth rate. Hausdorff's definition returns a measure — a quantity that is infinite for every exponent below the dimension and zero for every exponent above it, and the dimension is the one place where it is neither.

dynamics · Fractal dimension A carpet whose two dimensions differ by 0.076. A self-affine carpet built by keeping 5 cells of a 4 by 2 grid and repeating 4 times. Its box dimension is 1.6610 and its Hausdorff dimension 1.5850.

A carpet with two dimensions

For every self-similar set met so far the two definitions of dimension agree, and the agreement is a theorem about sets built from copies of themselves scaled equally. Stretch one direction more than the other and the two numbers come apart, by an amount that can be computed exactly.

dynamics · Fractal dimension A mass split 10 times, 0.3 to the left and 0.7 to the right. A self-similar measure on the unit interval: the mass is split 10 times, 0.3 of each piece's share going left and 0.7 right, and each of the 1024 pieces is drawn as a bar as tall as its mass. The heaviest carries 0.0282 and the lightest 5.9 × 10⁻⁶.

A dimension for every rate of crowding

Spread a unit of mass over an interval by splitting it unevenly, again and again, and the result covers the whole interval while crowding almost all of its weight onto a set of smaller dimension. Every rate of crowding picks out its own set of points with its own dimension, and the whole family is read off one curve.

dynamics · Fractal dimension A graph of dimension 1.585, built by raising midpoints. The Takagi–Landsberg graph with w = 0.75 on the unit interval, drawn with 16 columns shaded by how far the graph rises and falls across each. Counting boxes at seven widths gives a slope of 1.582.

The room a jagged graph takes up

The graph of a continuous function is a curve, and a curve ought to have dimension one. Build the function by raising midpoints — by an amount that shrinks more slowly than the intervals do — and its graph needs more boxes than any line, at a rate fixed by one ratio. A random path built the same way needs exactly as many.

dynamics · Fractal dimension 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

The Stern–Brocot tree and the tree of binary fractions have exactly the same shape, so there is a function that sends each fraction to the binary fraction in the same position. It is continuous and increasing, it turns every quadratic irrational into an ordinary fraction, and it does all of its rising on a set of numbers so thin that at almost every point its slope is nought.

number · Stern brocot Two periodic orbits meeting at the edge of the 1/2 plateau. The circle at 4 parameter values, with the period-2 points marked — two orbits inside the plateau, drawing closer together towards its edge, and none outside it.

How a lock comes apart

Inside a plateau the map has two periodic orbits, one attracting and one pushing away. Track them to the plateau's edge and they run into each other and vanish together — so the edge of a tongue is a collision, and the width of the plateau is how far the two can be pulled apart.

dynamics · Mode locking The coordinate change that makes an unlocked map a rotation. The conjugating map built from one orbit, plotted as a staircase from the circle to itself, with the rigid rotation it turns the circle map into.

A rotation in different coordinates

At an unlocked parameter the map is not merely like a rigid rotation; it is one, after a change of coordinates built out of a single orbit. The theorem needs the map to be smooth enough, and the map that shows why is one whose orbit leaves a hole.

dynamics · Mode locking A pentagon, its pentagram, and the pentagon inside. A regular pentagon with its diagonals drawn as a pentagram, enclosing a smaller pentagon, repeated 3 times inward; every pentagon has diagonal-to-side ratio φ.

The diagonal no unit measures

Draw the five diagonals of a regular pentagon and they make a star with a smaller pentagon at its centre. Subtract the side from the diagonal and what is left is the smaller pentagon's diagonal; subtract that from the side and what is left is its side. The pentagon has handed back a smaller copy of itself, and it will do so for ever — which means no unit, however small, measures both the side and the diagonal an exact whole number of times.

geometry · Golden ratio A patch of Penrose's rhombs. A disc of Penrose rhombus tiling after 5 subdivisions, thick and thin rhombs shaded differently: 550 thick and 340 thin half-rhombs.

Tiles that never repeat

Two rhombs with angles taken from the pentagon, cut each into halves, and cut each half into smaller copies of the two halves by a fixed rule. Repeat, and the pieces fill the plane with no gaps and no overlaps, in a pattern with five-fold stars everywhere and no period anywhere. The reason it cannot repeat is a single number: thick tiles outnumber thin ones by φ, and a repeating pattern would make that ratio a fraction.

geometry · Golden ratio

Named alongside it

The objects these essays reach for when they reach for this one.

MeasureCantor setScalingBox dimensionOrbitBifurcationHausdorff dimensionIterationPeriodic orbitAttractorBinomial coefficientChaos

All concepts