Golden Ratio

The Golden Ratio is well known and has been for a very long time. A lot has been written on its history so this is a very brief summarization and certainly is not meant to be an exhaustive description. The reader is encouraged to seek out more on the history if it’s of interest but, for the purposes here, suffice it to say that the ancient Greeks pondered what is the most well-proportioned way to divide a line into two segments (perhaps for aesthetic purposes). It is believed that they felt the most natural, and perhaps best, way would be to ensure that the proportion (or ratio) of the two segments should match that of the line itself to the segments. It is for this reason that the golden ratio is sometimes referred to as the golden section (there are other names as well such as the divine proportion).

The main focus of this article is to define mathematically the golden ratio. Subsequent articles will explore other aspects of the golden ratio. The following sections are presented:

Golden Ratio Defined

Given a line, divide the line into two segments where one is longer then the other. Letting \(L\) be the length of the longer segement and \(S\) be the length of the shorter segment, then the total length of the line is \(L + S\). The golden ratio (known as \(\phi\)) is when the ratio of the longer length segment to the short length segment equals the ratio of the total line length to the longer segment The following diagram illustrates this idea.

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The Mathematics of the Golden Ratio

To find this golden ratio,

\[ \phi = \frac{L}{S} = \frac{L + S}{L} \] first do some algebraic manipulation:

\[ \begin{align} \frac{L}{S} &= \frac{L + S}{L} \\ \frac{L}{S} &= \frac{L}{L} + \frac{S}{L} \\ \frac{L}{S} &= 1 + \frac{S}{L} \end{align} \]

Since \(\phi = \frac{L}{S}\) (by definition) then \(\frac{1}{\phi} = \frac{S}{L}\). Making these substitutions:

\[ \begin{align} \frac{L}{S} &= 1 + \frac{S}{L} \\ \phi &= 1 + \frac{1}{\phi} \\ \phi &= \frac{\phi + 1}{\phi} \\ \phi^{2} &= \phi + 1 \\ 0 &= \phi^{2} - \phi - 1 \end{align} \]

Recall that when given a degree two polynomial (a quadratic), \(ax^{2} + bx + c\), the solution to \(ax^{2} + bx + c = 0\) is found using the quadratic formula:

\[ \begin{align} x &= \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \end{align} \]

Applying the quadratic formula to \(\phi^{2} - \phi - 1 = 0\)

\[ \begin{align} 0 &= \phi^{2} - \phi - 1 \\ \phi &= \frac{-(-1) \pm \sqrt{(-1)^{2} - 4(1)(-1)}}{2(1)} \\ \phi &= \frac{1 \pm \sqrt{5}}{2} \end{align} \]

As expected, there are two solutions. However, since \(\phi\) represents the ratio of lengths, and lengths are always positive, only the positive solution is applicable:

\[ \begin{align} \phi &= \frac{1 + \sqrt{5}}{2} \\ \phi &\approx \frac{1 + 2.236067978}{2} \\ \phi &\approx 1.61803398874989484820 \end{align} \]

From a purist’s perspective, the true value of \(\phi\) is \(\frac{1 + \sqrt{5}}{2}\), anything else is just an approximation. And since \(\phi\) is an irrational number, the value \(\phi \approx 1.618\) is usually sufficient. Hence, the golden ratio is commonly known as \(\phi = 1.618\).

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Interesting Properties of the Golden Ratio

It was seen that there were two solutions to the quadractic \(\phi^{2} - \phi - 1 = 0\) and that the positive solution, \(\frac{1 + \sqrt{5}}{2}\) was the golden ratio \(\phi = 1.61803398874989484820\). The first thing to note is that \(\phi - 1 = 0.61803398874989484820\) which is the reciprocal of \(\phi\). That is:

\[ \phi - 1 = \frac{1}{\phi} \]

The second thing of note is that the second solution, \(\frac{1 - \sqrt{5}}{2}\), is not only the conjugate to the first but if expanded is (approximately) \(-0.61803398874989484820\). In other words:

\[ \frac{1 - \sqrt{5}}{2} = \frac{-1}{\phi} \]

This can be easily shown algebraically:

\[ \begin{align} \frac{-1}{\phi} &= \frac{-1}{\frac{1 + \sqrt{5}}{2}} \\ \frac{-1}{\phi} &= \frac{-2}{1 + \sqrt{5}} \\ \frac{-1}{\phi} &= \frac{-2(1 - \sqrt{5})}{(1 + \sqrt{5})(1 - \sqrt{5})} \\ \frac{-1}{\phi} &= \frac{-2(1 - \sqrt{5})}{1 - 5} \\ \frac{-1}{\phi} &= \frac{-2(1 - \sqrt{5})}{-4} \\ \frac{-1}{\phi} &= \frac{1 - \sqrt{5}}{2} \end{align} \]

An alternative expression would be:

\[ \begin{align} \frac{-1 + \sqrt{5}}{2} &= \frac{1}{\phi} \\ \frac{\sqrt{5} - 1}{2} &= \frac{1}{\phi} \end{align} \]

This last expression, \(\frac{\sqrt{5} - 1}{2}\), is sometimes referred to as the golden ratio conjugate

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Irrationality of the Golden Ratio

From The Mathematics of the Golden Ratio the golden ratio is:

\[ \phi = \frac{1 + \sqrt{5}}{2} \]

which is an irrational number with an approximation of \(\phi \approx 1.61803398874989484820\). To illustrate the irrationality of the golden ratio, recall from the Interesting Properties of the Golden Ratio it was shown that

\[ \begin{align} \phi - 1 &= \frac{1}{\phi} \\ \phi &= 1 + \frac{1}{\phi} \end{align} \]

From the concept of a continued fraction, \(\phi\) then can be recursively expanded as:

\[ \begin{align} \phi &= 1 + \frac{1}{\phi} \\ \phi &= 1 + \frac{1}{1 + \frac{1}{\phi}} \\ \phi &= 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{\phi}}} \\ \phi &= 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{\dots}}}} \\ \end{align} \]

Suffice it to say for the purposes here (i.e., not going to into excessive detail) that this sequence will continue forever. The golden ratio is an irrational number. Indeed, the golden ratio is perhaps the most difficult of irrational numbers to approximate by a rational number, hence is often referred to as the most irrational of the irrational numbers.

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Golden Rectangle

A golden rectangle is a rectangle whose side lengths are in the golden ratio. Here’s how to construct unit golden rectangle in five easy steps:

  • Step 1: Construct a unit square
  • Step 2: Mark the midpoint of one side and draw a line anchored at this midpoint to the upper corner; using right triangle geometry (Pythagorean Theorem) the length of this diagonal line is \(\frac{\sqrt{5}}{2}\)
  • Step 3: Pull down this line from its anchor point to the horizontal; now have two segments, one of length \(\frac{1}{2}\) and and one of \(\frac{\sqrt{5}}{2}\)
  • Step 4: Extend the original unit square to meet up with this this extended horizontal line
  • Step 5: Now have a rectangle with the longer side of length \(\frac{1 + \sqrt{5}}{2}\) and the shorter side of length \(1\). The ratio of the longer side to the shorter side is \(\phi = \frac{1 + \sqrt{5}}{2}\). Voila! Now have a golden rectangle!

The following diagram illustrates these five steps:

This golden rectangle is one where the ratio of the width to the height is the golden ratio, \(\phi\) - which implies that the ratio of the height to the width is \(\frac{1}{\phi} = 0.618\). A golden rectangle can also be one where the ratio of the height to the width is \(\phi\) (thus the ratio of the width to the height is \(\frac{1}{\phi}\)).

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Golden Spiral

A spiral is a curve which emanates from a point, moving farther away as it revolves around the point. A logarithmic spiral is a spiral whose radius is given by

\[ r = a(e^{b\theta}) \]

where \(\theta\) is the polar angle and \(a\) and \(b\) are constants. Basically, a logarithmic spiral grows such that the angle between a line drawn from the center of the spiral to the point of curvature on the spiral (i.e., the radius) and the tangent line at that point is constant. The following picture illustrates this idea:

A golden spiral is a special form of the logarithmic spiral. The radius of the spiral increases (or decreases) by a factor of the golden ratio(!) with each quarter turn of the spiral, which equates to and increase of the angle by \(\pi/2\). The equation for the golden spiral becomes:

\[ r = a(\phi^{\frac{2}{\pi}\theta}) \]

A golden spiral can be constructed much like that of a Fibonacci Spiral (described in the article Description, Mathematics And Other Fun Facts of the Fibonacci Sequence). In short, the spiral is created via the generation of golden rectangles which expand in size by the golden ratio as shown:

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Golden Ratio Fun Facts

Though the Golden Ratio is fascinating in-and-of-itself and can be found in nature in numerous ways, here are some fun facts associated with the Golden Ratio:

  • As mentioned, the ancient Greeks supposedly pondered what is the most well-proportioned way to divide a line into two segments. One conjecture is that this ponderance was for aesthetic purposes. Along those lines, people have been known to hang pictures on their walls consistent with the golden ratio. That is, the center of the picture is at the exact point where the (plum) line from the ceiling to the floor is segmented such that the ratio of the segments is \(\phi = 1.618\).
  • In the TV show Criminal Minds, a serial killer used the Fibonacci sequence to define the number of victims for each killing event (season 4 episode Masterpiece). A plot of the locations of the killings are observed to follow that of a golden spiral radiating outward. From this the origin of the spiral can be computed, which hopefully would be the home of the killer.
  • It is said that golden rectangles are the most pleasing. As an experiment, here are 15 rectangles all with areas equal to \(\phi = 1.618\). Some of the rectangles are constructed such that the ratio of the width to the height (or vice versa) is \(\phi = 1.618\), thus being true golden rectangles. The other rectangles, though having area equal to \(\phi = 1.618\), their width to height ratios are not \(\phi = 1.618\). Which are the most pleasing rectangles to you?

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