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Concept

Unit circle

The circle of radius one centred at the origin, on which the trigonometric functions are read off directly. Every point on it satisfies the Pythagorean relation, and its rational points are exactly the Pythagorean triples in disguise.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

A circle unrolled into a sine wave. On the left a radius turns through an angle; on the right the height of its tip is plotted against the angle, tracing a sine curve.

A sine wave is a circle seen from the side

Sine is introduced as a ratio in a right triangle, which is true and explains nothing about why its graph is a wave. There is a better picture.

analysis · Circular functions Multiplying two complex numbers. In the complex plane, multiplying adds the two angles and multiplies the two lengths.

Multiplying is turning

Complex numbers are introduced as an algebraic dodge for square roots of negatives. They are better understood as the arithmetic of rotation, at which point every rule stops needing to be remembered.

algebra · Complex numbers 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 The unit ball at p = 2.00. The set of points one unit from the origin, when distance is measured by the p-th power sum. At p = 1 it is a diamond, at p = 2 a circle, and as p grows it fills out a square.

Circles that are diamonds and squares

The theorem hands over a formula for distance. Take the formula as a definition, change the exponent in it, and the set of points one unit from the origin stops being round — while remaining, in every sense that matters, a circle.

geometry · Pythagoras 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 Three polynomials of signs around the circle. The size of the polynomial around the unit circle divided by √64 for all-plus, random and Rudin–Shapiro sign polynomials of length 64; maxima 8.00, 2.50, 1.41.

The flattest polynomials of signs

A polynomial whose coefficients are all +1 or −1 has average size √n on the unit circle. Keeping it near √n everywhere is the problem Littlewood posed: the Rudin–Shapiro polynomials never exceed √2 times it, searching every sign pattern up to length 22 finds the best are the Barker sequences, and whether the maximum can come arbitrarily close to √n is still open.

algebra · Polynomial roots

Named alongside it

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

CosineParametrisationPiRadianSineAreaArgumentAutocorrelationBijectionComplex exponentialConicConvexity

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