Golden spiral

In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to φ, the golden ratio.

Specifically, a golden spiral gets wider by a factor of φ every quarter-turn it makes, which means it gets wider by a factor of φ4 (about 6.854) every full turn.

Approximate and true Golden Spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of a larger square to the next smaller square is in the golden ratio. (A Fibonacci spiral is not shown, but could be constructed from a similar "whirling rectangle diagram", in which the ratios of the rectangles were based on the terms in the Fibonacci series, rather than phi.)

Fibonacci number

In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation:

That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers also denoted as Fn, for n = 0, 1, … , are:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418…

Sometimes this sequence is considered to start at F1 = 1, but it is more common to include F0 = 0. The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci, although they had been described earlier in India.



A tiling with squares whose sides are successive Fibonacci numbers in length.


A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above.

Golden rectangle

A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1:φ, that is, approximately 1:1.618.

A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle, that is, with the same proportions as the first. Square removal can be repeated infinitely, which leads to an approximation of the golden spiral.

According to astrophysicist and math popularizer Mario Livio, since the publication of Luca Pacioli's Divina Proportione in 1509, when "with Pacioli's book, the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use," many artists and architects have proportioned their works to approximate the form of the golden rectangle, which has been considered aesthetically pleasing.

Constructing a golden rectangle

A method to construct a golden rectangle. The resulting dimensions are in the ratio 1:φ, the golden ratio.

1. Construct a simple square
2. Draw a line from the midpoint of one side of the square to an opposite corner
3. Use that line as the radius to draw an arc that defines the height of the rectangle
4. Complete the golden rectangle