φ

The golden ratio, represented by the Greek letter phi, is the unique positive value who's reciprocal is the same as subtracting 1, and square is the same as adding 1.

A golden rectangle has a side ratio of φ. When a square is added or subtracted from this rectangle, a new golden rectangle is created with the same proportions, on its side.

When a golden rectangle is divided into a rectangle and a square, the new smaller rectangle has the same proportions, but turned sideways. The height of the new rectangle is 1 and the width is 1/φ to give the same ratio. But its width is also φ - 1 because the square with sides 1 was removed from the original width of φ. So this works if 1/φ = φ - 1.

Similarly when a larger φ x φ square is added below the golden rectangle, the resulting larger rectangle also has the same proportions. The width of the new rectangle is φ and the height is φ² to give the same ratio. But its height is also φ + 1 because the square with sides φ was added to the original height of 1. So this works if φ² = φ + 1.

When a golden rectangle is recursively divided into further squares and rectangles, and rotated curves are drawn in each square, a pleasant logarithmic spiral is created. Similarly, a spiral could be seeded with a small rectangle and grown outwards by adding larger squares to it. Various spirals related to this can be found in nature, such as the classic chambered nautilus shell, or the spiraling seeds of sunflowers.

The growth rate of the Fibonacci numbers is φ. To generate the Fibonacci sequence, start with 1, 1, and then repeatedly add the previous two numbers in the sequence to generate the next number: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377... The ratio of a new Fibonacci number to the previous one quickly converges to φ.

1 / 1 = 1.00000
2 / 1 = 2.00000
3 / 2 = 1.50000
5 / 3 = 1.66667
8 / 5 = 1.60000
13 / 8 = 1.62500
21 / 13 = 1.61538
34 / 21 = 1.61905
55 / 34 = 1.61765
89 / 55 = 1.61818
144 / 89 = 1.61798
233 / 144 = 1.61806
377 / 233 = 1.61803

A logarithmic spiral like the one above can also be created from squares with sizes relative to the Fibonacci numbers. In this case the inner-most rectangle is composed of two squares instead of a golden rectangle, which correspond to the two initial Fibonacci numbers 1 and 1. But if these initial squares are small enough, the spiral will look the same because the ratios of square sizes approach φ as it grows.

Like the Fibonacci numbers, any power of φ equals the sum of the two previous powers of φ. This follows from φ² = φ + 1. If the Fibonacci sequence were seeded with 1,φ instead of 1,1 the results would be the powers of φ.

Any integer power of φ can be reduced to a factor of φ plus a constant. These coefficients and constants follow the Fibonacci sequence with the constants shifted by one. The positive powers can be found by summing the two previous powers as described above: φⁿ = φ^n-1 + φ^n-2. The negative powers can be found using φⁿ = φⁿ⁺² - φⁿ⁺¹ and follow a similar pattern, but their coefficients and constants alternate signs.

φ¹ = 1φ + 0
φ² = 1φ + 1
φ³ = 2φ + 1
φ⁴ = 3φ + 2
φ⁵ = 5φ + 3
φ⁶ = 8φ + 5
φ⁷ = 13φ + 8

φ^-1 = 1φ - 1
φ^-2 = -1φ + 2
φ^-3 = 2φ - 3
φ^-4 = -3φ + 5
φ^-5 = 5φ - 8
φ^-6 = -8φ + 13
φ^-7 = 13φ - 21

This shows the quirk that φⁿ + φ^-n is an integer for even n, and φⁿ - φ^-n is an integer for odd n, because the φ is canceled out. More specifically, these give the sum of 2 Fibonacci numbers spaced 2 apart.

If any positive value of x is repeatedly updated with either of these formulas, the new values converge to φ.

Two isosceles triangles have sides with a φ ratio: the acute golden triangle, and the obtuse golden gnomon.

Golden triangles and gnomons can be divided or combined to make new golden triangles and gnomons. Their internal angles are all multiples of 36° so they fit together nicely in multiple ways. This is similar to how golden rectangles and squares are combined, but with even more options. These also both show that 1/φ + 1 = φ.

Golden triangles and gnomons can be recursively divided in many ways, to generate various geometric patterns, spirals, or fractals.

Pentagrams contain many overlapping golden triangles and gnomons, so their segment length ratios are related to φ. A regular pentagon with sides of 1 has diagonals of length φ, the star within has segments of 1/φ, and the inner pentagon has sides of 1/φ². For pentagrams within pentagrams, the segment lengths continue inwards in a reciprocal power-of-φ sequence.

If 3 golden rectangles are rotated to be mutually perpendicular in 3D as shown, their corners coincide with the 12 vertices of a regular icosahedron. These points also align with the faces of a dodecahedron. The dodecahedron is the dual of the icosahedron so its face centers correspond to the icosahedron vertices.

    [ ±φ, ±1, 0 ] for the yellow rectangle corners
    [ 0, ±φ, ±1 ] for the green rectangle corners
    [ ±1, 0, ±φ ] for the blue rectangle corners

If 3 rectangles with a φ² proportion (or 1/φ x φ) are rotated to be mutually perpendicular in 3D, their 12 corners align with the vertices of a regular dodecahedron. The other 8 dodecahedron vertices align with the corners of a cube (not shown). And these 20 points also align with the faces of an icosahedron.

[ ±1/φ, ±φ, 0 ] for the yellow rectangle corners
[ 0, ±1/φ, ±φ ] for the green rectangle corners
[ ±φ, 0, ±1/φ ] for the blue rectangle corners
[ ±1, ±1, ±1 ] for the cube corners

Golden Ratio Tutorial