The pentagram and the golden ratio
Inside the pentagram lies a smaller pentagon, inside that another pentagram, without end. All its segments stand in the golden ratio, and it was probably on this figure that the Pythagoreans first saw that some ratios cannot be expressed by any number.
The pentagram was the Pythagoreans’ sign of recognition. It is drawn by joining every second vertex of a regular pentagon, in a single stroke. It is the very first star polygon, {5/2}, and so the first figure drawn in one stroke that crosses itself (see Simplexes and star polygons).
The simplex of the pentagon: pentagon and pentagram, together all ten connections between five points.
A figure that contains itself
The five crossing points of the lines form a smaller pentagon in the middle. It stands upside down. Join every second point in it again and a smaller pentagram appears, with another pentagon in its middle, and so on without end. Each stage is a reduced, rotated copy of the one before, smaller by the factor φ² ≈ 2.618. The pentagram is thus a simple fractal, related to the fractal polygons, where overlaps begin from the pentagon on.
d diagonals sidexyd : s = s : x = x : y = φ ≈ 1,618
Pentagrams inside one another, up to the fifth stage with the slider. The two crossings divide the diagonal d into x, y and x. Below, the lengths are laid side by side: d, the side s = x + y, then x and y. Each is φ times the next.
The golden ratio
A diagonal of the pentagon is divided into three pieces by the two other lines it crosses: two equal pieces x on the outside and a shorter piece y in the middle. Between these segments the same ratio always holds:
d : s = s : x = x : y = φ
Here d is the diagonal and s the side of the pentagon. The number φ is called the golden ratio:
φ = (1 + √5) / 2 ≈ 1.6180339887 …
A line is divided in the golden ratio when the whole relates to the larger part as the larger part to the smaller. The pentagram divides each of its lines exactly so, at every point and on every stage.
φ has remarkable properties. Its square is exactly one greater, its reciprocal exactly one smaller:
φ² = φ + 1 ≈ 2.618 and 1 / φ = φ − 1 ≈ 0.618
As a continued fraction φ is the simplest of all numbers, nothing but ones: 1 + 1/(1 + 1/(1 + …)). For that very reason it is harder to approximate by fractions than any other number. The best approximations come from the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21 …, each the sum of the two before:
| Fraction | Value |
|---|---|
| 2 : 1 | 2 |
| 3 : 2 | 1.5 |
| 5 : 3 | 1.667 |
| 8 : 5 | 1.6 |
| 13 : 8 | 1.625 |
| 21 : 13 | 1.6154 |
| 34 : 21 | 1.6190 |
The quotients swing around φ and come ever closer without ever reaching it. Notably, the first fractions of the series, 2 : 1 and 3 : 2, are the octave and the fifth, the intervals of the tetractys.
The secret of Hippasus
The pentagram holds a discovery that shook the Pythagorean world view. If all is number, the side and diagonal of the pentagon should relate like two whole numbers. They do not. Try to measure both with a common unit: subtract the side from the diagonal and you get x, subtract x from the side and you get y, and you find yourself in the smaller pentagram in the middle, facing the same task. The procedure never ends. There is no common measure, the ratio is irrational.
The Pythagorean Hippasus of Metapontum is said to have betrayed this insight and to have perished at sea for it (see Pythagoras – the natural scientist). That he discovered irrationality precisely on the pentagram was suggested by the philologist Kurt von Fritz in 1945. It is not documented, but it is plausible, since the pentagram puts the endless repetition before one’s eyes.
The crosser of boundaries
On this site the pentagram keeps appearing at the same point: where a closed order breaks open.
- As a fraction it is 5 : 2 = 2.5, and 2.5 × 10 = 25 is the number at which the order of the twin primes ends.
- Neither the plane nor space can be filled without gaps by pentagons. The pentagon lies between the tiling polygons square and hexagon (see Circle, triangle and square).
- In the four-dimensional 5-cell it appears as a projection and crosses the boundary of space.
- The dodecahedron of twelve pentagons broke the order of four among the Platonic solids, and icosahedron and dodecahedron are pervaded by the golden ratio (see The Platonic solids).
Symbol of life
Because the golden ratio recurs in plants, in leaf arrangements and flower heads and in the proportions of the human body, Hermeticism regards it as the principle of the living. The five-pointed pentagram was read early on as an image of the human being, with head, arms and legs, and five as the number of the human. The Pythagoreans wrote at its points the letters of the Greek word hygieia, health. In magic it became a protective sign, for Éliphas Lévi the sign of the microcosm. It is worth keeping in mind where the line between mathematics and interpretation runs: the ratios in the pentagram can be proved, their meaning cannot.