# How does Plato's description in the Timaeus relate to the golden ratio?

In the Timaeus, Plato states

For whenever in any three numbers, whether cube or square, there is a mean, which is to the last term what the first term is to it; and again, when the mean is to the first term as the last term is to the mean - then the mean becoming first and last, and the first and last both becoming means, they will all of them of necessity come to be the same, and having become the same with one another will be all one.

Many sources claim that he is describing the golden ratio but I am having trouble parsing the sentence into a formula for the golden ratio. Any ideas?

• To me, he is only describing geometric mean (first /mean = mean/last => mean = sqrt(first*last)). He later goes on to say that a single mean is fine for the plane but solids must be described with two means, so I don't think this is an argument about the golden ratio but rather about using proportions to describe 2d and 3d objects. – user1301930 Aug 19 '16 at 2:21

My edition, translation by Donald J. Zeyl, includes a footnote on the page containing sections 31c-32a. He gives an example $2,4,8,$ where $4$ is our geometric mean.

In sections 55b-55c, he mentions the regular solids; the dodecahedron is

One other construction, a fifth, still remained, and this one the god used for the whole universe, embroidering figures on it.

In the lengthy Introduction, we are told that Plato's contemporary

Theaetetus had constructed the five regular solids..

https://en.wikipedia.org/wiki/Theaetetus_%28mathematician%29

The golden ratio is evident in a regular pentagon.

It was Aristotle who equated the dodecahedron with a fifth element, the ether, after earth, air, fire, water. This book is called De Caelo. He also says Anaxagoras misuses the name. I will try to put the little squiggles on, $$\alpha \iota \theta \eta \rho$$ The derivation was suggested by Plato in Cratylus.

Neither mentioned Bruce Willis or Milla Jovovich.

• Thanks! Very interesting. So looks like the only ties to the golden ratio in Timaeus is the mention of the fifth regular solid (the dodecahedron), which is linked to the golden ratio by way of construction of the pentagon. I wonder if there are any explicit references to the golden ratio before Euclid's definition of the "extreme and mean ratio". The reason I am interested is because I am curious if the Parthenon is actually based on the golden ratio or if it is an example of finding a pattern that isn't actually there. – user1301930 Aug 19 '16 at 4:48
• @user1301930 That is one of those things where nobody knows for sure. There have been far too many supposed applications of the golden ratio in art, where some fairly arbitrary points on, say, a human portrait, are selected. There is a bit more of a chance with a fixed building. If so, however, that is just one aspect of what was an enormous undertaking. – Will Jagy Aug 19 '16 at 4:55
For whenever in any three numbers, whether cube or square, there is a mean


I take this to refer to an average, and within the context of this dialogue the terms "cube" and "square" refer to geometric objects.

which is to the last term what the first term is to it


I interpret this phrase as saying a square can be made a cube, and vice versa given the similarities.

and again, when the mean is to the first term as the last term is to the mean


This section is focused on the ability to derive an average that is equivalent between the squares and cubes just mentioned.

- then the mean becoming first and last, and the first and last both becoming means, they will all of them of       necessity come to be the same, and having become the same with one another will be all one.


This last statement resolves the argument that an average of a square can be equivalent to the average of a cube, using a function which divides until the averages are equal to one another.

I don't see how this applies to the phi ratio, and since no articles were linked to show how it can be interpreted to refer to the phi ratio I believe the example used is more generally simple geometric methods.

If I'm not mistaken, this dialog of Plato's showed Socrates demonstrate "anamnesis" by having someone "remember" mathematics they said they did not actually know?

• Sorry, I think I was mistaken about many sources, just en.wikipedia.org/wiki/Timaeus_(dialogue)#Golden_ratio The Timaeus is also mentioned in the wikipedia article about the Golden Ratio (en.wikipedia.org/wiki/Golden_ratio#Timeline ) so was curious what part of the Timaeus alludes to the Golden Ratio. – user1301930 Aug 19 '16 at 3:35
• I studied as a philosophy major during my undergrad and took a seminar class on Plato. It looks like the timeframe for the discovery of the golden ratio was around 500 - 490 BC, Plato is generally accepted by philosophers to have been born around 430, so he could have been aware of the ratio. That said, I don't see this as a reference to that. – user304051 Aug 19 '16 at 3:40
• The reference to the wiki on the golden ratio mentions that the dialog covers the platonic solids, but doesn't explicitly say that the quote in your question is specifically about the golden ratio. The source cited for the reference in the golden ratio wiki also only points to the dialog, not any expert or discussions about it. Since there was no citation given at all in the wiki about this dialog, my guess is that the golden ratio wiki led to the source being used in the wiki about the dialog. Either way, I would like to see actual peer reviewed articles discussing this. – user304051 Aug 19 '16 at 3:46
• Yes, I agree that this quote isn't a reference to the golden ratio, and should probably be removed from the Wikipedia article about Timaeus, especially since there are no sources to back it up. – user1301930 Aug 19 '16 at 4:25