Matpic
mathematics, in pictures
Concept

Optimality

The property of being best under a stated objective, together with a proof that nothing else does better.

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

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 13 rays leave the left focus in 13 directions and every one of them arrives at the rightfocuseach path is the same total length — 2.90 drawing units — which is the property thatdefines the ellipse and the reason the reflection works

Every ray comes back to the other focus

An ellipse has two foci and one property everybody remembers: the distances to them add to a constant. What that property forces is stranger and more useful — a mirror shaped like an ellipse sends every ray leaving one focus, in every direction, through the other.

geometry · conic sections

Named alongside it

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

AreaCircleConicConstant widthConvexityDandelin spheresDirectrixEccentricityEllipseExistence proofFocusParabola

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