Concept

Exponential growth

Growth by a fixed factor in every equal step of time, so that the quantity doubles at regular intervals. It eventually overtakes every polynomial, and a count or cost that grows this way is usually the sign of a search that cannot be finished.

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

The slopes the equation demands, and the curves that obey them. A field of short segments whose slope at each point is 0.9 times the height there, with 3 solution curves integrated through it; each doubles over an interval of 0.770 wherever that interval is taken.

The equation with only one answer

A rate of change proportional to the current amount is the most common description in nature, and it pins down the function completely. There is exactly one curve through each starting point, and a half-life and a doubling time are the same measurement.

analysis · The exponential

Eight corners of a squashed cube, visited in order by the simplex method. Klee and Minty's program in 3 variables drawn as its own deformed cube and as a plain one. The simplex method with the largest-price rule visits all 8 corners, with objective values 0, 4, 6, 10, 15, 19, 21, 25; the optimum is one edge from the start.

The cube that takes every corner

The simplex method is fast on every program anybody meets in practice. In 1972 Victor Klee and George Minty squashed a cube so that the method, choosing the steepest edge each time, visits all of its corners — 2ⁿ − 1 moves in n variables, with the optimum one edge from the start.

applied · Duality

Named alongside it

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

Derivative Differential equation e, the number Gray code Half life Hypercube Initial condition Integral Limit Linear program Optimisation Polytope

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