Concept

Square root

A number whose square is a given number, in whatever arithmetic is in use. Modulo a prime one may or may not exist, and when it does it can be lifted digit by digit to every power of the prime; among ordinary numbers taking one halves an exponent.

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

Breadth-first levels in a planar triangulation. A Delaunay triangulation of 160 random points with each point coloured by whether it is inside, on or outside the level 4 steps from the centre; that level of 23 points separates 31 from 106.

The few points that cut a flat graph

Any graph that can be drawn without crossings, however large, falls into pieces of at most two thirds once a few points are removed — about the square root of its size, never more than 2.83 times it. A grid shows the square root cannot be beaten, a ring of breadth-first neighbours comes close, and a cycle through a shallow tree finishes the job.

discrete · Planarity

Square roots of 2, lifted from one power of 7 to the next. A tree whose rows are the square roots of 2 modulo 7 to the powers 1 to 4, each joined to the root it reduces to; counts 2, 2, 2, 2.

A root lifted one digit at a time

On a dial of seven, 3 × 3 is 2. On a dial of forty-nine the square root of 2 must reduce to 3, so there are only seven candidates, and exactly one of them works: 10. On a dial of 343 exactly one lift of 10 works: 108. Each step adds one digit on the left, found by solving a linear equation, and the digits go on for ever — a number …21216213 whose square is 2, in a world where closeness means divisibility by seven.

discrete · Modular arithmetic

Named alongside it

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

Derivative Euler formula Graph Hensel lemma Lifting Local global principle Lower bound Modular arithmetic Newtons method P adic numbers Planar graph Planarity

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