Wiki

Wiki

The mathematics the tree stands on, one idea a page, written from scratch for a reader with school mathematics: parity, the Kronecker product, the Cantor set, the Mobius function, Ford circles and the rest. The pages hold known mathematics; the tree's own results are in the notes.

A page works small cases by hand and ends on In the tree, the notes and demos that use the idea. A page that needs another first names it as a prerequisite. Parity and the Kronecker product need none, and every design on the site is built from those two.

Every page needs only the pages above it.

Start here

The Basel problem

Add up the reciprocals of the squares and the total stops at pi squared over six, a circle appearing in a question that never mentioned one.

The difference coarray

Put a few sensors on a line and list every distance between two of them. Those distances are what the array can measure, and a well-placed handful of sensors covers far more distances than it has sensors.

The Euler-Mascheroni constant

Add 1 and a half and a third and on to one over n, take away the logarithm of n, and the difference settles on 0.5772157, a number nobody has yet placed.

Euler's number

Split a year's interest into more and more payments and the yearly growth climbs, slows and stops at 2.718281828.

Famous formulas

Eight elementary rules run out to infinity. Five close on a constant, one counts the primes, one refuses to reach zero and one refuses to settle down at all.

The Farey sequence

Every fraction between zero and one whose bottom number is at most Q, written in order; each new Q slips a few new fractions between the old ones and never moves one.

The greatest common divisor

The largest number dividing two others at once; Euclid finds it by taking remainders, and when it comes out 1 the two numbers share nothing.

The Kronecker product

Stamp a small picture into every filled cell of itself and it grows a level; that one move builds every design on this site.

The Leibniz series

Add one, take away a third, add a fifth, and the running total swings over and under a quarter of pi, closing on it very slowly.

Parity

Every whole number is even or odd, and that single bit, read on each coordinate at once, is the whole of what a design uses to choose its cells.

Prime numbers

A prime is a number whose stones make only one rectangle; every other number is built from primes, and the supply never runs out.

Rajchman measures

Spread a unit of mass round a circle and listen to it at higher and higher frequencies; if the sound dies away the measure is Rajchman, and the Cantor dust is the famous one that never goes quiet.

Visible lattice points

Stand at a corner of the grid and some points hide behind nearer ones; the ones you can see are the pairs sharing no divisor, and counting them hands back pi.

The Wallis product

Multiply 4/3 by 16/15 by 36/35 and keep going, and the running product climbs forever without ever reaching half of pi.

One step in

Burnside's lemma

Counting arrangements that turning and flipping should not tell apart, by averaging how many each symmetry leaves untouched.

After Parity

Cellular automata

A row of cells, a rule that reads each cell with its two neighbours, and a new row written underneath; run it and the rows draw a picture.

After Parity

The divisor function

d(n) counts the ways to split n into a product, 2 at every prime and 12 at 60; it never grows as fast as any power of n, and on average it is log n.

After Prime numbers

Euler's totient

Phi of n counts how many of the numbers 1 to n share nothing with n; it sinks for numbers with many small factors and hits its ceiling at every prime.

After The greatest common divisor

Ford circles

Above every fraction sits a circle resting on the number line, the simpler the fraction the bigger the circle, and no two of them ever overlap.

After The Farey sequence

The Gaussian integers

Numbers of the form a plus b times i, drawn as a square lattice, where each point has a size and some ordinary primes break in two while others stay whole.

After Prime numbers

Graphs

Dots joined by lines and nothing else; count the lines at a dot for its degree, write the joins in a table of ones, and a design's filled cells become a network.

After The Kronecker product

The Mobius function

Mu of n is 0 when a square divides n, and otherwise plus or minus one depending on whether n has an even or an odd number of prime factors.

After Prime numbers

Moire

Lay two rulings over each other and they beat; the points both of them mark are the ones their scales agree on, and stacking many scales lights the points most scales share.

After The Farey sequence, The greatest common divisor

Pascal's triangle

A triangle of numbers where every entry is the sum of the two directly above it, and colouring the odd ones draws a fractal.

After Parity

The prime counting function

How many primes are there below a number? The answer is a staircase with one riser per prime, and two smooth curves that chase it without ever quite catching it.

After Prime numbers

The Riemann zeta function

Add one over every whole number raised to the power s; the answer is zeta of s, it equals pi squared over six at s equal to 2, and it can be rewritten as a product over the primes.

After Prime numbers

The Sierpinski carpet

Cut a square into nine, throw the middle one away, then do the same to the eight that are left, and keep going for ever.

After The Kronecker product

The spirograph

A small wheel rolls inside a big ring carrying a pen, and the fraction made by the two radii decides how many petals the pen draws and when it comes home.

After The greatest common divisor

Stern's diatomic sequence

Write 1 1, then slip the sum of every two neighbours between them, again and again; the numbers that pile up count sums of powers of two, each power at most twice, and, read in pairs, list every positive fraction exactly once.

After The Farey sequence

The Thue-Morse sequence

A string of noughts and ones built by writing a block and then its opposite for ever, which is also the parity of the ones in each place number written in binary.

After Parity

Truchet tiles

A square with two quarter circles on it, laid on a grid in one of two turns, joins up into closed loops and long strands that never branch and never cross.

After Parity

The Ulam spiral

Wind the whole numbers outwards on a square spiral, light up the primes, and they fall on diagonal streaks instead of scattering evenly.

After Prime numbers

The Wallis sieve

Cut a square into nine and drop the middle, cut each survivor into twenty-five and drop the middle, and keep going; the area left is a quarter of pi.

After The Wallis product, The Kronecker product

Two steps in

Beurling generalised primes

Pick any climbing list of real numbers and call them primes; their products are the integers. Beurling's theorem says how straight the integer count must run for the prime number theorem to hold, and the exponent is sharp.

After Prime numbers, The prime counting function

The Eisenstein integers

Numbers built from a cube root of one, drawn as a hexagonal lattice, with six units and a size rule that turns the honeycomb into a number system.

After The Gaussian integers

The Euler characteristic

Count the corners, subtract the edges, add the faces; the answer is 2 for anything shaped like a ball, and bending the shape never changes it.

After The Sierpinski carpet

Goldbach's conjecture

Every even number past two is a sum of two primes. Machines have checked it further than anyone can list and nobody has proved it.

After Prime numbers, The prime counting function

The graph Laplacian

Degree minus adjacency, one table per network; its eigenvalues are the tones the network can ring at, and the staircase they make says how it is knit together.

After Graphs

The horocycle flow

Fold a horizontal line of the hyperbolic plane onto the modular surface and it closes into a loop of length 1/y; as the line sinks the loop spreads evenly over the whole surface, at a pace set by the zeros of zeta.

After Ford circles, The Riemann zeta function

The Menger sponge

The carpet's solid cousin: cut a cube into 27, drill out the middle and the middle of each face, then do the same to the 20 cubes that are left.

After The Sierpinski carpet

The Mertens function

Mark every whole number plus one, minus one or nothing, then keep a running total. How far that total is allowed to wander is the Riemann hypothesis.

After The Mobius function

The transfer matrix

A table that says which state may follow which; multiply it by itself and it counts the walks of any length, and those counts grow like a fixed power.

After Graphs

Three steps in

Domino tilings

Cover a region of the square grid with dominoes, each on two cells that share a side; the number of ways is one determinant, and a random way freezes near the corners and melts in the middle.

After Parity, Graphs, The transfer matrix

Fractal dimension

Cover a shape with boxes, shrink the boxes, and watch how fast the count grows; the growth rate is the dimension, and for a carpet it is not a whole number.

After The Sierpinski carpet, The Menger sponge

The Mahler measure

Average the logarithm of a polynomial's size once round the unit circle and you get one number; it sees only the leading coefficient and the roots outside the circle, and for whole-number polynomials it is zero exactly when the leading coefficient is plus or minus 1 and every nonzero root is a root of one.

After The transfer matrix

Nodal domains

A drum skin at one pure tone splits into patches that swing up while their neighbours swing down; the still lines between them are the nodal set, the patches are the nodal domains, and Courant's theorem says the k-th tone has at most k of them.

After The graph Laplacian

The spectral radius

Multiply an arrow by the same matrix again and again and it settles on one direction; the factor it stretches by there is the spectral radius.

After The transfer matrix

Four steps in

The Cantor set

Cut the middle third out of a line, then the middle third of every piece left, for ever. What survives has no length, as many points as the line, and added to itself fills an interval.

After Fractal dimension

The random walk

A walker on a graph steps to a neighbour picked at random; after n steps on a grid it is about the square root of n away, and on a fractal it is slower.

After Graphs, Fractal dimension

Space-filling curves

A line drawn without lifting the pen can pass through every point of a square, and points that are close along the line stay close in the plane.

After Fractal dimension

Subshifts of finite type

Write an endless string of letters under a rule that forbids a few short words; the strings that obey form a subshift of finite type, and the number of words it allows grows at a rate read off one small table.

After The transfer matrix, The spectral radius

Substitution tilings

Cut every tile into smaller copies of the tiles you started with, scale the result back up, and repeat; a handful of shapes then covers the plane with the same structure at every scale.

After The Kronecker product, The spectral radius

Five steps in

Minkowski content

Fatten a set by a small radius and weigh what it swallows. How fast that weight shrinks is the set's dimension; whether the rescaled weight settles down is decided by the logarithms of its ratios for a self-similar set on the line, while off the line the lattice half is known only under extra conditions on the holes.

After Fractal dimension, The Cantor set

Missing-digit numbers

Strike out every whole number that uses the digit 7. Almost all of them go, yet the survivors are a Cantor set in disguise and still hold infinitely many primes.

After Prime numbers, The Cantor set

Rep-tiles

A shape that cuts into a few smaller copies of itself, all of one size, can be cut again and again, and run backwards the same cut grows it to cover the whole plane.

After Substitution tilings, Fractal dimension

Wang tiles

Square tiles with a colour on every edge, laid so that touching edges agree; whether a given set can cover the whole plane is a question no computer program can always answer.

After Subshifts of finite type, Cellular automata