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Concept

Pythagorean triples

Triples of whole numbers that are the sides of a right triangle, so that the first two squares add to the third. They are parametrised by two coprime integers of opposite parity, and a tree of three matrices produces every primitive one exactly once.

Named by 4 essays across 2 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 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 tree of Pythagorean triples. A tree rooted at 3-4-5. Each triple has three children, obtained by three fixed integer matrices, and every primitive triple appears exactly once somewhere in it.

A tree that holds every triple

Three fixed matrices, applied to 3-4-5 over and over, produce every primitive Pythagorean triple there is — each of them once, none of them twice, and with no test for common factors anywhere in the procedure.

number · Pythagoras From a rational point on x² + y² = 17 to a whole one. The circle of radius √17 on the integer lattice, a rational point on it, and 4 reflections through nearest lattice points, the denominators 3757, 205, 25, 5, 1, ending at (1, 4).

A fraction on the circle forces a whole point

If a number is a sum of two squares of fractions, it is a sum of two squares of whole numbers. Draw the circle, mark the rational point, join it to the nearest lattice point and follow the line to where it meets the circle again: the new point is rational too, with a smaller denominator. Repeat, and the denominators fall until they reach one. The argument needs no primes at all — only the fact that every point of the plane is within distance one of the lattice.

number · Sums of two squares

Named alongside it

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

DescentBijectionPrimitive tripleRational pointsCongruenceConicConstructionConverseHypotenuseLatticeLaw of cosinesMatrix

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