Cats, Pharaohs, and the Golden Ratio

Certain numbers keep showing up in nature. The Golden Ratio,

\phi \approx \displaystyle\frac{1+\sqrt{5}}{2}

is one of them. It shows up in cats, sunflowers, and Egyptian pyramids.

hiero-cat

If you stretch a cat, the ratio of its tail to its back is golden: the tail’s length counts for \approx 0.39 of the whole spine length, making the point where the tail joins its body a Golden Ratio.

golden-cat

The great pyramid at Giza is the last still extant wonders of the ancient world. Originally rising some 145 meters with a volume of some 2.5 million cubic meters of rock, it still represents, as of today, an incredible construction.

Since ancient times, the pyramids were believed to possess strange mystical powers, some of which, apparently, are realized through numerical magic. If one examine carefully the pyramids, many strange things may be found. For example, the most beautiful pyramids have faces that are slanted about 52^\circ, giving these building their harmonious proportions. But Why 52^\circ? Because, of course, of cats. Well, at least, because of the Golden Ratio.

Observe the following:

kheops

Indeed, if one takes the secant of 52^\circ (remember, SOHCAHTOA, so secant is hypotenuse on adjacent side) on a triangle with a base of 1, you find that the hypotenuse is very close to \phi=1.6108\ldots. If you take the tangent, you find that the height of the triangle is \sqrt{\phi} \approx 1.2720\ldots! One cannot doubt the Egyptians of the time of the Fourth Dynasty possessed advanced mathematical knowledge to build such ratios in the great pyramid [3].

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By now, I guess, you figured out that I’m pulling your leg.

Maybe the ratio of a cat’s tail to its whole backbone is approximately the golden ratio, but, guess what? I changed the picture to stretch the cat’s tail just long enough so that the picture would effectively show a golden ratio. Does it make cats golden? Sweet pets, certainly. Golden? Not really.

Kittens give Morbo gases

"Kittens give Morbo gases"

The same is probably true of the great pyramid as well. Of course, the ratio may be there, but maybe it’s for an entirely different reason. The Egyptian pyramids did not appear in their final glorious version at the first try. There’s plenty of evidence that they tried different shapes—flatter, pointier, bent—before settling with the 52^\circ pyramids because they offered the best trade-off of pointiness and stability given their building technology which consisted in little more than piling up bricks.

So did they build pyramid this way because of the golden ratio or does the golden ratio appear because they are built this way? I guess the latter. From what we know of Egyptian mathematics [1,2], they weren’t advanced enough in that era to know about something like \phi. According to the Rhind Papyrus [2], mathematics in the time of pharaohs wasn’t quite as advanced as one would like to believe. Their mathematical system was too primitive to allow this complex thinking. Even if they knew the exact algebraic solution to certain complex problems such as computing the volume of a truncated pyramid (as we see in the Moscow papyrus, which dates a good 700 or 800 years later than the construction of the pyramid, some time in the 11th dynasty) there’s little evidence that they knew much more.

Only the Greeks, more than two thousand years later would begin developing maths enough to know and understand \phi.

Read more on golden sections and pyramids here.

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This psychological proclivity to finding meaningful relations between random objects is called apophenia and is entirely human. So, the point is that it’s not because it’s there, or seems to be there, that it’s there for the reason you think. Worst, you may be deliberately mislead as my example with the cat. Don’t go measure your cat and think it’s defective because it’s somehow not golden (or maybe it is?). This kind of slight bending of the truth to support one’s pet theory is seen very often in all kind of cults and pseudosciences like astrology and numerology. They don’t make sense, but if you stretch a bit your cat, it fits the “theory”.

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So, what does it have to do with computer science, programming, or any of this blog’s usual topics? Little, in fact. I had the idea of the golden cat while daydreaming during a dreadful alphabet soup/ISO presentation some time last week. I thought it was funny and that I should share this non-sense with you.

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The cat picture is adapted from a wikimedia image released under the GNU Free Documentation License and accordingly, is also GPL-ed.

The hieroglyphic cat comes from Vikkistein‘s PhotoStream, and is used with permissions.


1 Richard Gillings — Mathematics in the Time of Pharaohs — Dover, 1972 (reprinted 1982)

2 Gay Robins, Charles Shute — The Rhind Mathematical Papyrus: an Ancient Egyptian Text — Dover, 1987

3 Marius Cleyet-Michaud — Le Nombre d’Or — Presses Universitaires de France, Que Sais-je? n⁰ 1530, 1973

5 Responses to Cats, Pharaohs, and the Golden Ratio

  1. […] c’est signe qu’on change de lecture. Sur mon blog, j’ai déjà parlé des Pharaons, des chats, et du nombre d’or (en anglais) en tirant un peu la patte du lecteur en expliquant qu’en étirant un peu les […]

  2. nicolestelle says:

    So, you’re saying that to attribute meaningful relations between the theory of the golden ratio and any aesthetical transcendence is what’s misleading?

    But what if you don’t atribute any meaning to it, just finding it aesthetic and fun…

    • Steven Pigeon says:

      The maths behind the golden ratio are clear. The post isn’t so much about the golden ratio as about finding meaningful relations in objects that do not contain them. Whether the golden ratio produces aestethically pleasing rectangles is open for debate. In my opinion, it is merely an heuristic elevated to the level of revelation, all thanks to our old friends, the Ancient Greeks.

  3. […] give surprisingly good results in numerical problems (the topic came up while I was discussing the golden ratio and its place in art with friends, trying to make a point that it was mostly heuristic, bordering […]

  4. […] So let’s take an example. Say, , the so-called golden ratio. […]

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