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Concept

Pythagorean theorem

The statement that the squares on the two shorter sides of a right triangle together hold the area of the square on the longest. Read as a definition rather than a fact it says how distance is computed, and changing the exponent in it gives a whole family of distances.

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

A knotted rope pulled into a 3-4-5 triangle. A closed loop of rope carrying 12 equally spaced knots, held at three of them so the sides are 3, 4 and 5 knots long; the angle between the two shorter sides is 90.0 degrees.

The rope that squares a corner

The theorem turns two sides into a third. Run it backwards and it turns three lengths into a right angle — which is a different statement, needs its own proof, and is the only one of the two that has ever been used to build anything.

geometry · Pythagoras The diagonal of a box, by using the theorem twice. A box 12 by 4 by 3 with the diagonal of its floor drawn, and the diagonal of the box standing on it; the two right triangles share a side and give the sum of three squares.

Two right angles and the diagonal of a box

The theorem applied once gives the diagonal of a floor. Applied again, standing on the first result, it gives the diagonal of the room — and the pattern does not stop at three, which is where a fact about triangles quietly becomes the definition of distance.

geometry · Pythagoras A right triangle on a sphere. A spherical triangle with a right angle where the equator meets a meridian and legs of 50 and 60 degrees; its hypotenuse is shorter than the flat theorem predicts.

The triangle that a globe gets wrong

On a sphere, a right triangle with legs of fifty and sixty degrees has a hypotenuse of seventy-two, not seventy-eight. The theorem is not approximately true there — it is false, and what replaces it says exactly how much room the surface has.

geometry · 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

Named alongside it

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

Right angleCongruenceConstructionConverseConvexityCounting argumentCurvatureDe gua theoremDimensionDistanceGeodesicHamming distance

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