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David Terr
Ph.D. Math, UC Berkeley

>> 5.3.3. The Base of Natural Logarithms (e ≈ 2.718)

5.3.2.The Golden Ratio (φ ≈ 1.618)

The golden ratio, also known as the golden mean, is an irrational number with some amazing properties. The Greek letter phi (φ) is used to represent this number. To 10 decimal places, its value is given by 1.6180339887. There is also the following useful formula for the golden mean:

• (5.3.2.1) φ = (1 + √5) / 2.

There is a fascinating relationship between the golden ratio and the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... in which each number in the sequence is the sum of the two previous numbers. It turns out that the ratio of successive terms in the sequence converges to the golden ratio. The following table lists the first several of these ratios as well as their approximate values:

 ratio value 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

The golden ratio appears in geometry, art, architecture, and in nature. In what follows, we will discuss each of these applicatioins.

Geometry

The most famous geometric construction employing the golden ratio is the golden rectangle. This is a rectangle with base length φ and height 1 as shown below:

Figure 5.3.2.2: The Golden Rectangle

What makes the golden rectangle fascinating is that it can be divided into a unit square and a similar golden rectangle, as shown below.

The fact that the smaller rectangle is golden is due to the following amazing mathematical fact:

• (5.3.2.3) 1/φ = φ - 1.

Continuing this process, we arrive at a fascinating sequence of inwardly-spiraling squares and golden rectangles, as shown below. The vertices of these squares and rectangles may then be connected by a smooth curve, known as a logarithmic spiral. Logarithmic spirals occur in nature. For instance, the shape of a cross-section of a shell of a chambered nautilus is very closely approximated by a logarithmic spiral.

Figure 5.3.2.4: The Golden Rectangle and Embedded Logarithmic Spiral

(Image taken from Wikipedia)

Many people say the golden rectangle has the most aesthetically applealing proportions.

Another geometric figure in which the golden ratio appears is the pentagram, or five-pointed star. The ratios of several pairs of line segments in the pentagram is equal to the golden ratio, as shown below.

Figure 5.3.2.5: The Pentagram

(Image taken from Wikipedia)

In the above figure, the ratio of lengths of the red line segment and the green line segment is equal to the golden ratio, as is the ratio of the lengths of the green and blue segments as well as the lengths of the blue and violet segments.

Architecture and Art

The ancient Egyptians are the first known people to use the golden ratio in their architecture, in particular, in the design of the pyramids. The diagram below shows a square pyramid

Figure 5.3.2.6: A Square Pyramid

(Image taken from Wikipedia)

In the above figure, h is the height of the pyramid, b is half the length of the base, and a is the height of a triangular face. The ancient Egyptians constructed the Great Pyramids in such a way that the ratio b : h : a is approximately equal to 1 : √φ : φ. Incidentally, √φ is surprisingly close to 4 / π. This means that the ratio of the perimeter to the height of the Great Pyramids is approximately equal to 2π.

Other architecture which uses the golden ratio include the Parthenon in ancient Greece, the Great Mosque of Kairouan, and the United Nations headquarters building in New York.

Many artists used the golden ratio in their paintings and sculptures, including Leonardo da Vinci and Salvador Dali.

Nature

Many flowers and plants exhibit Fibonacci numbers as well as approximations to the golden ratio. For instance, the pedal arrangement of many sunflowers show groups of 89 pedals spiraling one way and 55 pedals spiraling the other way, and 55 and 89 are consecutive Fibonacci numbers. [2]

References:

1. http://en.wikipedia.org/wiki/Golden_ratio (Wikipedia page on the golden ratio)
2. http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html (Fibonacci Numbers and Nature)

>> 5.3.3. The Base of Natural Logarithms (e ≈ 2.718)