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mathematics, in pictures
Concept

Stern brocot tree

The binary tree of fractions built by repeatedly taking the mediant of two neighbours. Every positive rational appears in it exactly once and already in lowest terms, because the construction preserves primitivity rather than restoring it by cancelling.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

11122113233231142535344353524115 fractions, all in lowest terms, none of them twiceand left to right they are already in order

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 · stern brocot 3-4-55-12-137-24-2548-55-7328-45-5320-21-2939-80-89119-120-16936-77-858-15-1733-56-6565-72-9712-35-37the rootfirst matrixsecondthird13 triples in 3 generations, each child got from its parent by one of three fixed matrices; the root is the 3-4-5 trianglegrown until the hypotenuse passes 200, the tree produces exactly the 32 primitive triples an exhaustive search over Euclid's m and n finds— the same set, by two procedures with nothing in common

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 · pythagoras

Named alongside it

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

BijectionContinued fractionsCounting two waysDescentFarey sequenceFord circlesLowest termsMatrixMediantPrimitive triplePythagorean triplesRecurrence

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