The golden ratio

The teaching

As the lore of the cut from Euclid to Pacioli and Kepler teaches it.

The cut in extreme and mean ratio

Euclid defines it in the sixth book of the Elements: a straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less. Call the shorter part one; then the longer is the number phi, about 1.618, and the whole is phi plus one. Its exact value is one plus the square root of five, all divided by two.

The number has properties no other has. Its square is itself plus one, about 2.618, and its reciprocal is itself less one, about 0.618, so phi, its square and its reciprocal share the same endless decimal tail. Each power of phi is the sum of the two powers before it, just as each number in the sequence of Fibonacci is the sum of the two before it.

Cutting a line by compass

To cut a given line in this ratio, call its ends A and B. At B raise a line at right angles, half as long as AB, and call its top C. Join A to C. With the compass point at C and the opening CB, cut the line AC at D. Then with the point at A and the opening AD, cut AB at E. The point E divides AB in the golden ratio: AB is to AE as AE is to EB.

Euclid reaches the same point by squares in his second book: the rectangle made from the whole line and the shorter part equals the square made on the longer part. The two statements are one, the first told in proportions and the second in areas.

The pentagon and its star

In a regular pentagon each diagonal is phi times the side, and where two diagonals cross, each cuts the other in the golden ratio. Draw all five diagonals and the pentagram appears, every line of it divided in extreme and mean ratio, with a smaller pentagon at its heart. This is why Euclid needs the cut before he can draw the pentagon at all.

Two triangles carry the ratio. The golden triangle, with a point of 36 degrees and two base angles of 72, has its long sides phi times its base; Euclid builds it in his fourth book as a triangle whose base angles are each double the angle at the top, and from it he sets the pentagon in a circle. Its companion, with angles of 108, 36 and 36, is the triangle cut from a pentagon by one diagonal.

To draw the pentagon in a circle, draw two diameters at right angles and mark the top point A. Halve the right-hand radius at M. With centre M and opening MA, swing an arc to cut the left-hand radius at N. The distance AN is the side of the pentagon: step it five times round the circle from A and join the points. The short length from the centre to N is the side of the ten-sided figure, and the radius is phi times it.

The golden rectangle and its spiral

A rectangle whose sides are in the ratio is a golden rectangle. Cut a square from it and what remains is again a golden rectangle, turned on its side; cut a square from that, and so on without end, the squares winding inward toward one point. Join opposite corners of each square with a quarter circle and a spiral appears, close to the true golden spiral, which grows by phi with every quarter turn.

Fibonacci's rabbits

Leonardo of Pisa put it as a puzzle. A pair of rabbits is shut in a walled place; each month every pair old enough bears a new pair, which can breed from its second month. How many pairs will there be at the end of a year? Month by month the count runs 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, each number the sum of the two before it, and the answer is 377.

Divide each number by the one before it and the answers swing above and below phi and close in on it: 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619. The square of any number in the run differs by exactly one from the product of its two neighbours: five squared is 25, and three times eight is 24.

Why it was called divine

Luca Pacioli named it the divine proportion and gave his reasons, each a likeness to God. It is one and only, as God is one. It joins three terms in one proportion, as the Trinity is three in one. It can be expressed by no ratio of whole numbers, as God cannot be defined in words. It is always the same, however small or great the line, as God is unchanging and whole in every place. And as God gave being to the heavens, the fifth essence, so the ratio gives being to the dodecahedron, which Plato gave to the heavens.

Kepler wrote that geometry has two great treasures, the theorem of Pythagoras and the division of a line in extreme and mean ratio; the first he likened to a measure of gold, the second to a precious jewel.

Where it truly grows

One true home of the number is in the turning of leaves and seeds. Divide a full turn in the golden ratio and the smaller part is about 137.5 degrees, the golden angle. Many plants set each new leaf or seed about this far round from the last, so that they pack evenly and no two line up, and the seeds then fall into crossing spirals whose counts are neighbouring numbers of Fibonacci, such as 34 and 55 in the head of a sunflower.