8.3.6 The Golden Ratio

Photo of nautilus shell

Have you ever heard of the golden ratio?

[ To learn more about the golden ratio and play with the proportion, visit this site. ] One way to express the golden ratio is one plus Square root of five divided by two, which is approximately 1.61803398. This value describes the ratio found in many contexts, such as number patterns, geometry, and nature.

Videoclip iconSome people claim that measurements in paintings, aesthetically pleasing architecture, and even the human body reflect the golden ratio! Check out this short video to see some examples and claims of where the golden ratio seems to appear:

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Some artists and architects have deliberately used the golden ratio in their work, but often the ratio of measurements is only approximately equal to the golden ratio. Even some of the claims made in the video you have just watched have been disputed: see, for instance, mathematician Keith Devlin’s article, “The Myth That Will Not Go Away.” [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]

When you see a claim that two measurements are in the golden ratio, be critical. Check the measurements and work out their ratio. You may get an answer that is approximately the value of the golden ratio, but this might have occurred by chance.

There have been many claims that the golden ratio occurs in the proportions of the human body. You can test this claim in the next activity!

Notepad iconActivity: The Golden Ratio and the Human Body: A Match Made in Heaven?

Leonardo do Vinci “Vitruvian Man” picture

For this activity, you will need a measuring tape and a buddy.

(a) Stand with one arm out to your side. Have your buddy measure the distance from your middle finger tip to your elbow. Record this value. Using the same unit of measurement, let your friend measure from your wrist to your elbow. Record this value, too. Now, calculate the ratio of the first measurement to the second.

(b) Repeat part (a) with measurements for your friend.

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(a) and (b)

The ratios are probably somewhere between 1.5 and 1.7. However, there are variations in the proportions of the human body for different people, so you may have values different to these. (The value should be greater than 1 though, so if it isn’t, check your calculation!)

(c) It has been claimed that the ratio you calculated in part (a) and part (b) will reflect the golden ratio. Do you agree?

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(c) It is unlikely that the ratios are exactly equal to the golden ratio, 1.618. The ratio depends on how precisely you made the measurements and where from. For example, which part of the wrist did you use? If you repeated the measurements, would you get the same results? Sometimes a small change in the measurement can appear to give a ratio closer to the golden ratio. Although each of your ratios may be approximately equal to the golden ratio, this may just be a coincidence!

8.3.5 Percentage Points

8.3.7 Sweet or Savory?