Matpic
mathematics, in pictures
Concept

Parametrisation

A map from an interval to a space, whose image is a curve and which carries more information than the image does. Length, speed and direction belong to the map rather than to the set of points, so one curve traced three times over is three times as long.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Rational points on the unit circle. Lines of rational slope through the left-hand point of a circle, each meeting it again at a rational point.

Every triple, on one circle

Draw a line of rational slope through a single point of a circle. Wherever it comes out is a rational point, and clearing the denominators turns it into a Pythagorean triple — so every triple there is comes from one line through one point.

number · Pythagoras One curve, 3 parametrisations, 3 lengths. Several maps from an interval with the same image drawn side by side, each with marks at equal parameter steps and its length beneath it — the same point set reported at different lengths.

The length belongs to the journey

Three maps from an interval with exactly the same image, and three different lengths. The picture of a curve is the set of points it passes through, and that set does not determine how far anything travelled along it.

analysis · Arc length A circular sector of area 1.10 and a hyperbolic sector of area 0.80. On the left, the unit circle with the sector from (1, 0) to (cos 2.2, sin 2.2), of area 1.100. On the right, the hyperbola x² − y² = 1 with the sector from (1, 0) to (cosh 1.6, sinh 1.6), of area 0.800. In both, the parameter is twice the shaded area.

The angle that is really an area

On a unit circle the angle t is the length of arc the point has walked, and it is also twice the area of the slice it has swept. The two readings agree on the circle and part company on the hyperbola — where arc length leads nowhere and area leads straight to cosh, sinh, and the exponential.

analysis · Circular functions A box with whole diagonals on every face. 44,117,240: face diagonals 267, 244, 125; space diagonal squared 73225.

A box with every face diagonal whole

Pythagoras gives a rectangle with whole sides and a whole diagonal — 3, 4, 5. A box with whole edges and whole diagonals on all three faces is far rarer: the smallest is 44 × 117 × 240, found in 1719, and there are only 113 essentially different ones with every edge below 300,000. Ask for the long diagonal through the box to be whole as well and the answer is unknown. No such perfect box has ever been found, and nobody has proved there is none.

geometry · Pythagoras

Named alongside it

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

Pythagorean triplesUnit circleApproximationArc lengthAreaBijectionConicConjectureContinuityCosineCounterexampleDerivative

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