Square Root of Three

√3 is the height of an equilateral triangle with side 2, the long diagonal of a unit cube's face pair, and the ratio between the diagonal and the side of a regular hexagon's opposing vertices. It is sometimes called the constant of Theodorus, after the Cyrenean who is said to have proved the irrationality of the square roots of the non-square integers up to 17. Its continued fraction, [1; 1, 2, 1, 2, ...], is periodic, as it is for every quadratic irrational.

Its most-used appearance is electrical. In a balanced three-phase system the line-to-line voltage is √3 times the line-to-neutral voltage, so a 400 V distribution board is 230 V to neutral, and three-phase power is P = √3·V_L·I_L·cos φ. The factor comes from the 120° phase separation and the vector subtraction of two phase voltages, and the number 1.732 is stamped on so many nameplates and printed in so many tables that electricians recognise it long before they meet the geometry.