Generator

The Stern–Brocot tree to depth 4

A generator in the number library, called 34 times across 8 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.

mediant 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

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

Ford circles up to denominator 7

The Farey sequence of order 7

Every positive rational, in one sequence

The tree as words in two matrices

The Farey tessellation, and a line down to √2 − 1

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.

?(1/11) is the binary fraction in the same position ×1023
0/1 and 1/13 have determinant one and nothing simpler between them ×359
3/2 is in lowest terms ×108
the circles on 0/1 and 1/7 touch ×56
s(1) and s(2) are coprime ×48
the 1th ratio's numerator agrees with the Calkin–Wilf walk ×48
the ratio 1/1 appears once ×48
0/1 and 1/7 are Farey neighbours ×24
1/1 arrives already in lowest terms ×15
and its mediant 1/1 is already in lowest terms ×13
s(2) counts the hyperbinary representations of 1 ×12
the 1th arc crossed is the descent's 1th interval ×10
run 1 of the turns is the continued fraction's quotient ×5
the sequence has one entry per coprime pair up to 7 ×2
? of the expansion of all ones is 2/3 ×1
? of the expansion of all twos is 2/5 ×1
?(1 − x) = 1 − ?(x) ×1
?(x/(1 + x)) = ?(x)/2 ×1
√2 − 1 has every quotient two ×1
and at most its right bound ×1
and is closer to the target ×1
and its denominator does too ×1
and the path closes on the target ×1
and their lengths add to one ×1
circles that are not neighbours stay clear of each other ×1
depth d cuts the interval into 2^d pieces ×1
each path is a word of at most eight L's and R's ×1
each record has a larger denominator than the last ×1
how many terms are drawn is a whole number between 8 and 64 ×1
how many turns are taken is a whole number between 6 and 34 ×1
neighbouring nodes have determinant one ×1
no fraction is produced twice ×1
no two words produce the same matrix, so the tree never rejoins ×1
reading the drawing left to right reads the fractions in increasing order ×1
the depth is a whole number between 2 and 5 ×1
the depth of the tree checked is a whole number between 3 and 14 ×1
the descent passes at least three fractions within 1/q² of the target ×1
the descent produces several record approximations ×1
the drawing scale is a whole number between 200 and 400 ×1
the endpoint each crossed arc keeps spells the descent's word ×1
the function never decreases ×1
the largest denominator drawn is a whole number between 5 and 30 ×1
the length carrying half the rise falls with depth ×1
the line crosses exactly as many arcs as the descent has intervals ×1
the matrix for "L" has determinant ±1 ×1
the matrix for "LR" has determinant ±1 ×1
the matrix for "LRL" has determinant ±1 ×1
the matrix for "LRLR" has determinant ±1 ×1
the matrix for "LRLRL" has determinant ±1 ×1
the matrix for "R" has determinant ±1 ×1
the matrix for "RL" has determinant ±1 ×1
the matrix for "RLR" has determinant ±1 ×1
the matrix for "RLRR" has determinant ±1 ×1
the matrix for "RR" has determinant ±1 ×1
the matrix for "RRL" has determinant ±1 ×1
the matrix for "RRLR" has determinant ±1 ×1
the matrix for "the root" has determinant ±1 ×1
the mediant is at least its left bound ×1
the mode of the mediant family is one of tree, farey, ford, matrices, diatomic, approx, tessellation, questionmark, singular ×1
the number is one of phi, root2, pi, e ×1
the order is a whole number between 2 and 12 ×1
the target is one of phi, root2, pi, e ×1
the tree has 2^D − 1 nodes to depth D ×1
the tree is full to its stated depth ×1
the window is a sub-interval of the unit interval ×1
two to six increasing depths, up to 18 ×1
up to which index the hyperbinary count is checked is a whole number between 4 and 24 ×1
φ − 1 has every quotient one ×1

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.

Number

A tree that holds every triple

Three fixed matrices, applied to 3-4-5 over and over, produce every primitive Pythagorean triple there is — each of them once, none of them twice, and with no test for common factors anywhere in the procedure.

Number

Approached too fast to be algebraic

An algebraic number of degree d cannot be approached by fractions faster than the denominator's dth power. So a number that is approached faster than that is the root of no polynomial at all — and one can be built by choosing where its decimal digits go.

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.

Number

Every rational in one sequence

The tree lists every positive fraction once and needs a tree to do it. One recursion on the whole numbers lists them in a single row — and each term of it counts something nobody was asking about, which is why the enumeration works.

Number

The arcs a line crosses on its way to a number

Draw a semicircle over every pair of neighbouring fractions and the half-plane above the number line is cut into curved triangles that never overlap. A straight line dropped towards any number crosses those arcs one after another, and the arcs it crosses, and the side it leaves each triangle by, are exactly the steps of the Stern–Brocot descent towards that number.

Number

The fractions that beat every smaller one

Walking down the tree towards a number produces a sequence of fractions closing in on it. Most of them are steps along the way; a few are the best approximations there are — closer than every fraction with a smaller denominator — and which few is decided by where the turns change direction.

Number

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

Two matrices that generate the tree

A node of the Stern–Brocot tree is not really a fraction — it is the pair of fractions it lies between. Written as the columns of a matrix, the two turns of the tree become two multiplications, and the determinant that kept everything in lowest terms becomes a property of a product.

The whole library · What the figures prove