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Pythagoras — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The Pythagorean theorem by dissection. Two squares of the same size. Each holds four copies of one right triangle. The space left over is a single tilted square on the left and two upright squares on the right.

    Two squares, four triangles, and no algebra

    The Pythagorean theorem is usually met as a formula to be memorised. It is much better met as a rearrangement that can be checked by eye.

    part 1 · geometry
  2. Euclid's proof, without moving anything. The square on a leg and its share of the square on the hypotenuse are each exactly twice the same triangle, so they are equal. Nothing in the figure is cut or rearranged; the triangle is only looked at from the other side.

    Euclid proves it without moving anything

    The rearrangement proof cuts and slides. Euclid's does neither — it shows that a square and a rectangle are each exactly twice the same triangle, seen from opposite sides, and that is harder to hold in the head for a reason worth understanding.

    part 2 · geometry
  3. 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.

    part 3 · number
  4. 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.

    part 4 · geometry
  5. 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.

    part 5 · geometry
  6. 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.

    part 6 · geometry
  7. 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.

    part 7 · number
  8. 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.

    part 8 · geometry

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