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Concept

Binary

Notation in which every number is written with two digits only, each place worth twice the one to its right. It is the notation in which a number's digits record which powers of two it is built from, one for each place.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

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 A closed walk on the 3-cube changing one place at a time. The corners of a 3-dimensional cube with a path through every one of them exactly once, each step moving along an edge, and the last corner one step from the first.

A walk that changes one thing at a time

Counting from nothing to fifteen in binary changes four digits at once somewhere in the middle. There is another order through the same sixteen words in which every step changes exactly one — and it is a closed walk on a four-dimensional cube.

discrete · Hamiltonian cycles The share that provably comes down. The proportion of starting values that fall below their own start within k steps, plotted against k up to 12. The proportion rises towards one; at the largest k drawn it is 0.94.

Almost every number comes down

The Collatz conjecture is open and a great deal about it is not. Whether a number falls below its own start in the first few steps is decided entirely by its remainder on division by a power of two, and the share of numbers for which it happens can be counted exactly.

dynamics · Collatz Every positive rational, in one sequence. The first 32 terms of Stern's diatomic sequence as bars, with the ratios of consecutive terms beneath. Every ratio is in lowest terms, no two agree, and each term counts the hyperbinary representations of its index.

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 · Stern brocot

Named alongside it

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

Counting two waysIterationParityRecursionBijectionCellular automatonCollatzComplete graphCountabilityCounting argumentDensityDeterminism

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