Geometry

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 perhaps 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).

Points

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 $\varphi^{2} \approx 2.618$. The pentagram is thus a simple fractal, related to the fractal polygons, where overlaps begin from the pentagon on.

Depth Mark segments

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:

$$\frac{d}{s} = \frac{s}{x} = \frac{x}{y} = \varphi$$

Here d is the diagonal and s the side of the pentagon. The number $\varphi$ is called the golden ratio:

$$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887\ldots$$

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.

$\varphi$ has remarkable properties. Its square is exactly one greater, its reciprocal exactly one smaller:

$$\begin{gathered} \varphi^2 = \varphi + 1 \approx 2.618 \\ \frac{1}{\varphi} = \varphi - 1 \approx 0.618 \end{gathered}$$

As a continued fraction $\varphi$ 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:

The quotients swing around $\varphi$ 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.

Steps

The anthyphairesis in the pentagram. The diagonal (red) minus the side (teal) gives the diagonal of the inner pentagon, the side minus this remainder its side. With every step one lands in the next smaller pentagram. The remainders shrink but never reach zero.

Later tradition tells that a Pythagorean betrayed this insight and was punished for it. Modern scholarship often links this with Hippasus of Metapontum, whom Iamblichus, however, has perish at sea because of the dodecahedron (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). Yet five is not left without order: with two rhombs in the golden ratio the plane can be filled after all, only never periodically, and a related, three-dimensional order was found in 1982 in quasicrystals.
• 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).
• Plants calculate with the golden ratio too. Sunflower seeds are set off from one another by the golden angle of 137.5 degrees, and their spirals count Fibonacci numbers.

Symbol of life

Because the golden ratio recurs in plants, in leaf arrangements and flower heads, and some also see it in the proportions of the human body, which is disputed, Hermeticism regards it as the principle of the living. The five-pointed pentagram was read at least since the Renaissance as an image of the human being, with head, arms and legs, and five as the number of the human. According to Lucian the Pythagoreans called it hygieia, health, and in the Renaissance the letters of this word were written at its points. 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.

Sources and further reading

Sources

• Euclid, Elements, around 300 BC, book VI, definition 3, and book XIII
• Luca Pacioli, De divina proportione, Venice 1509
• Lucian, A Slip of the Tongue in Greeting 5 (the pentagram as the sign of health)
• Kurt von Fritz, The Discovery of Incommensurability by Hippasus of Metapontum, Annals of Mathematics 46, 1945
• Dan Shechtman et al., Metallic Phase with Long-Range Orientational Order and No Translational Symmetry, Physical Review Letters 53, 1984

Further reading

• Mario Livio, The Golden Ratio, New York 2002