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Golden ratio — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The whirling squares. Squares with Fibonacci sides 1, 1, 2, 3, 5, 8, 13, each attached to the long side of what came before. They fill a 13 by 21 rectangle exactly.

    The rectangle that eats itself

    Cut a square off a golden rectangle and what is left is a golden rectangle. That single property is the whole of the golden ratio, and it explains both what the number really does and most of what is wrongly claimed for it.

    part 1 · geometry
  2. Rotating by φ − 1 of a turn, 21 times. Points on a circle produced by repeatedly turning through the same angle.

    Three gaps and no more

    Turn a circle by the same irrational angle over and over. The points never repeat and never settle, and yet at every single stage the gaps they leave take at most three different lengths — never four, at any number of steps, for any angle.

    part 2 · dynamics
  3. A pentagon, its pentagram, and the pentagon inside. A regular pentagon with its diagonals drawn as a pentagram, enclosing a smaller pentagon, repeated 3 times inward; every pentagon has diagonal-to-side ratio φ.

    The diagonal no unit measures

    Draw the five diagonals of a regular pentagon and they make a star with a smaller pentagon at its centre. Subtract the side from the diagonal and what is left is the smaller pentagon's diagonal; subtract that from the side and what is left is its side. The pentagon has handed back a smaller copy of itself, and it will do so for ever — which means no unit, however small, measures both the side and the diagonal an exact whole number of times.

    part 3 · geometry
  4. A patch of Penrose's rhombs. A disc of Penrose rhombus tiling after 5 subdivisions, thick and thin rhombs shaded differently: 550 thick and 340 thin half-rhombs.

    Tiles that never repeat

    Two rhombs with angles taken from the pentagon, cut each into halves, and cut each half into smaller copies of the two halves by a fixed rule. Repeat, and the pieces fill the plane with no gaps and no overlaps, in a pattern with five-fold stars everywhere and no period anywhere. The reason it cannot repeat is a single number: thick tiles outnumber thin ones by φ, and a repeating pattern would make that ratio a fraction.

    part 4 · geometry
  5. The tree of solutions of x² + y² + z² = 3xyz. Markov triples from (1, 2, 5) to depth 4; 231 Markov numbers below 10^15; first 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325.

    The numbers that come after the golden ratio

    The golden ratio is the number worst approximated by fractions: no constant larger than √5 works in Hurwitz's theorem. Set it aside and √2 is worst, with √8. Set that aside and the next is a number whose continued fraction repeats 2, 2, 1, 1, with √221/5. Markov proved in 1879 that the list continues for ever below 3, one entry for each solution of x² + y² + z² = 3xyz.

    part 5 · geometry
  6. A bracket that shrinks by the golden ratio. 2: [0.00000, 1.00000]; 3: [0.00000, 0.61803]; 4: [0.23607, 0.61803]; 5: [0.23607, 0.47214]; 6: [0.32624, 0.47214]; 7: [0.32624, 0.41641]; 8: [0.36068, 0.41641]; 9: [0.36068, 0.39512]; peak 0.374750.

    The search that reuses a point

    To find the peak of a function that rises and then falls, evaluate it at two points and throw away the part beyond the lower one. Place the points so that the survivor can be used again, and there is exactly one way to do it: at the golden ratio. Each new evaluation then shrinks the bracket to 0.618 of its width, and with a fixed number of evaluations nothing does better than Fibonacci's numbers.

    part 6 · geometry

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