Concept

Bisection

Halving an interval repeatedly and keeping the half in which a sign change, a target value or some other property persists. Each step costs one evaluation and halves the uncertainty, so precision grows by one binary digit a step.

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

One line, and both shapes halved. Two shapes and the single straight cut that divides each of them into two equal areas. The direction was found by sweeping every angle and watching the imbalance change sign.

One line that halves them both

Two shapes lying anywhere on a page, of any sizes and any shapes at all. There is always a single straight line that cuts both of them into two equal halves at once — and finding it needs no cleverness, only the observation that a quantity which reverses sign has to pass through zero.

topology · Borsuk ulam

One form completed three ways, and the same straddle each time. The plane coloured by the sign of 2x² + 6xy + 3y². Three pairs of lines, one for completing with x first, one for y first and one along the eigenvectors; their coefficients are 2 and −3/2; 3 and −1; 5.541 and −0.541, and every pair has one line in a positive wedge and one in a negative wedge.

The signs no completion can change

A quadratic in several variables can be completed square by square in many orders, and each order hands back different coefficients. What no order changes is how many come out positive and how many negative — and that count, read off a completion of A − tI, says how many eigenvalues lie below t without finding a single one.

algebra · Completing the square

Named alongside it

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

Antipodal pair Brouwer Completing the square Continuity Degree Determinant Eigenvalue Euler characteristic Exact arithmetic Exhaustive search Existence proof Fair division

All concepts