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mathematics, in pictures
Concept

Scaling

Enlarging or shrinking everything by one factor, so that shapes are preserved and sizes are not.

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

perioddoubles at r =gap ratio22.998304—43.4488464.743183.5438344.6385163.5643124.6464323.568719—each doubling is found by bisection, and each ratio is measured from the two gaps beside itthe last one is 4.646; Feigenbaum's constant is 4.6692, and it is the same for any map with a smooth hump

A constant that does not care which map

The gaps between successive period doublings shrink by a factor. Measure that factor for the logistic map and you get 4.669. Measure it for a completely different map and you get 4.669, and nobody expected that.

dynamics · period doubling 3112ad − bc = 6 − 1 = 5area of the drawn shape = 5positive: the corners keep their order1

The number that says how much room is left

A linear map takes the unit square to a parallelogram. The area of that parallelogram is one number, it is computable from the four entries of the matrix, and almost everything the determinant is used for is a restatement of that sentence.

algebra · determinant 3 sidesarea 4,3304πA/L² = 0.6054 sidesarea 5,6254πA/L² = 0.7856 sidesarea 6,4954πA/L² = 0.90712 sidesarea 6,9984πA/L² = 0.977the circlearea 7,1624πA/L² = 1every shape here has a perimeter of 300; only the area changes4πA/L² rises from 0.605 to 0.977 and reaches 1 only at the circle

The most area a fence can hold

One length of boundary, and the question of what shape to bend it into. The answer is a circle, everybody knows it, and the argument that convinced the nineteenth century turned out to prove something slightly different.

geometry · isoperimetric

Named alongside it

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

AreaBasisBifurcationCircleConstant widthConvergenceConvexityDeterminantEigenvalueExistence proofFeigenbaum constantInvariant direction

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