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

Question

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?

Answer 1

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.

Answer 2

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?