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Plato takes 120 of the 30-60-90 triangles to form the icosahedron (twenty of these groupings of six):
"[55b] And the third solid is composed of twice sixty of the elemental triangles conjoined, and of twelve solid angles, each contained by five plane equilateral triangles, and it has, by its production, twenty equilateral triangular bases."
Plato takes four of these groupings of six to form the tetrahedron:
"And when four equilateral triangles are combined so that three plane angles [55a] meet in a point, they form one solid angle, which comes next in order to the most obtuse of the plane angles. And when four such angles are produced, the first solid figure is constructed, which divides the whole of the circumscribed sphere into equal and similar parts."
I think that part of Plato's argument is that the simplest possible plane is a triangle: therefore, this shape is properly considered the most basic building block for extending into three-dimensional space. Before we go 3D, Plato starts with a 30-60-90 triangle and -- from six of those -- and builds an equilateral triangle.
[54d] "In the next place we have to explain the form in which each Kind has come to exist and the numbers from which it is compounded. First will come that form which is primary and has the smallest components, and the element thereof is that triangle which has its hypotenuse twice as long as its lesser side. And when a pair of such triangles are joined along the line of the hypotenuse, and this is done thrice, by drawing the hypotenuses [54e] and the short sides together as to a center, there is produced from those triangles, six in number, one equilateral triangle."
It seems the tradition of keeping dodecahedrons for intellectual stimulation rather than practical use continued on, as seen in this antique "zodiac paperweight."
"Pleasure is the reason I do everything."
Yes, I fully agree. Our pleasure should be our continuous guide. We have pleasure, but we need to act in a way that ensures we continue to have it.
I think of pleasure as something like a compass. The boat is afloat, and it is a pleasure to be sailing: but we are always traveling through the seas of time, and we must steer in a coherent direction—or else risk sailing into storms or crashing against reefs.
Many people have goals beyond their own pleasure — some give too much of themselves, others take too much for themselves, while almost all seek unnecessary change of some sort — and this leads them into many unpleasant and unnecessary circumstances!
Cicero was certainly correct that the Timaeus is obscure—and I agree that the reader of Plato develops a sympathy for this obscurity, as Plato does seem to sincerely wish to be as clear as possible. In fact, the Timaeus is really the same geometric explanation repeated three different times, gradually introducing the complexity.
This sort of 'clarity through repetition' or 'clarity through saying your point in different ways' seems to be something that Epíkouros also employs in On Nature.
In this case I don't gather that its reconstructed, but rather intended to set off certain sections of text as being referenced rather than being the words of Diogenes himself, but I'm just not sure.
That is correct, this is a well preserved section. There are a few letters missing here and there, but what is in quotes is present in the ancient Greek.
I gather you have to get to the combinations stage before you can properly start talking about accidents/events/qualities that are perceptible to us
Yes, I think the biggest challenge is the different terms we have in English. For the qualities you are referring to, Lucretius calls "events" and Epíkouros calls "symptoms."
In terms of whole natures: For the atoms, the only necessary qualities are shape, three-dimensionality, and weight. For the void, its only quality is to never be able to be subjected to any influence in any way.
All other qualities are emergent and the contingency or necessity of the qualities is from the perspective of the compounds that exhibit them.
When the distinction between atoms and compounds is not significant, I wanted to highlight that Epíkouros uses the term "What is corporeal (τό Σωματικὸν)" for both together:
"...as both what is corporeal and the void exist..."
(Epíkouros, Peri Phýseōs, Book 28, P.Herc. 1479 fr. 1 col. 1)
Peace signified harmonious relations with neighbors while Safety meant the security of the man as a citizen
Throwing this in to show that "Peace" was also used in the context of macropolitics, in the same sense we use it: "war vs. peace."
"…For in [law courts], as the saying goes, they risk their neck whenever they serve as assemblymen, and whenever they judge cases, they pay attention to what's being said because they fear their oath -- but in the assemblies and displays of the sophists, they do not care at all, either about an oath (because they have not sworn to judge correctly) nor about whether what's being said is beneficial to the city or not (because the speech is not about war and peace [Εἰρήνη], about which we must sometimes vote, or if it does happen to be about war and peace [Εἰρήνη] or some other thing that they deliberate in their assemblies, the speech at that moment is not about anything pressing at all)."
(Philódēmos, On Rhetoric Book 3, P.Herc. 1506, col. 50, line 33 – col. 51, line 21)
Yes, thank you Cassius. Referring to William DeWitt Hyde as DeWitt is far too confusing. I've corrected my post.
I wanted to include this striking quote from William DeWitt Hyde (From Epicurus To Christ, pgs. 22-24):
"Perhaps we are inclined to look down on Epicurus's ideal as a low one. Well, if it is a low ideal, it is all the more disgraceful to fall below it. And of us do fall below it every day of our tense and restless lives. Let us test ourselves by this ideal, and answer honestly the questions it puts to us.
How many of us are slaving all day and late into the night to add artificial superfluities to the simple necessities? How many of us know how to stop working when it begins to encroach upon our health; and to cut off anxiety and worry altogether? How many of us measure the amount and intensity of our toil by our physical strength; doing what we can do healthfully, cheerfully, joyously, and leaving the rest undone, instead of straining up to the highest notch of nervous tension during early manhood and womanhood, only to break down when the life forces begin to turn against us?
Every man in any position of responsibility and influence has opportunity to do the work of twenty men. How many of us in such circumstances choose the one thing we can do best, and leave the other nineteen for other people to do, or else to remain undone?
How many of us have ever seriously stopped to think where the limit of healthful effort and endurance lies, unless insomnia or dyspepsia or nervous prostration have laid their heavy hands upon us and compelled us to pause? Every breakdown from avoidable causes, every stroke of work we do after the border-land of exhaustion and nervous strain is crossed, is a crime against the teaching of Epicurus."
If you can tap into the power of the heavens
Yes, I want to rescind my dismissiveness. If these were any other shape, it would be very tempting to think they just had a practical use. But given that the dodecahedron was an important part of the key to transcending metempsychosis, and as such was almost an object of worship in the early Academy (as Kochiekoch said), it does indeed suggest these objects were not for (some unknown) practical use.
The fact that there were also icosahedral versions seems to solidify the Platonic connection.
The 1712 edition of Creech, published by Jacob Tonson
The next year, Jacob Tonson and John Watts published another version with engravings by Guernier, but this time by with the text edited by Michael Maittaire (French classicist) and funded by Richard Mead (doctor to the crown and patron of the classics).
I recently got a 1713 edition of this for only £40. The text is Latin-only, but does contain one page of English, which is more-or-less an Enlightenment era copywrite.
ANNE R. = Anne Regina = Queen Anne
I agree, Eikadistes. I feel as though the circumstantial evidence we have is sufficient to accept the theory.
In general outline, it seems Plato, as he aged, became less skeptical and more Pythagorean. The old academy would have been the most influenced by Pythagoreanism. Of course, later on, the more skeptical new academy more closely aligned with skepticism seen in Plato's earlier and public works.
The old academy is the least known, least public, and the most "mystic." This Pythagorean-influenced Platonism would have been the version of Platonism that Epikouros was arguing against.
One of his key criticisms was directed at the Pythagorean doctrine of the immortality and transmigration of the soul.
Epikouros was undeterred by the Pythagorian threat that, if he built his science upon sensation, his spirit would be metempsychotized into a bird.
"And the tribe of birds are derived by transformation, growing feathers in place of hair, from men who are harmless but light-minded—men, too, who, being students of the worlds above, suppose in their simplicity that the most solid proofs about such matters are obtained by the sense of sight." [Timaeus, 91d fin.]
The SFOTSE and leaping pig hats are great!
I may be pushing it, but are full-brim hat types a possibility?
I wanted to highlight this section:
"God began by first marking them out into shapes by means of forms and numbers. And that God constructed them, so far as He could, to be as fair and good as possible, whereas they had been otherwise,—this above all else must always be postulated in our account. Now, however, it is the disposition and origin [53c] of each of these Kinds which I must endeavor to explain to you in an exposition of an unusual type; yet, inasmuch as you have some acquaintance with the technical method which I must necessarily employ in my exposition, you will follow me.
[53c fin.] In the first place, then, it is plain I presume to everyone that fire and earth and water and air are solid bodies; and the form of a body, in every case, possesses depth also. Further, it is absolutely necessary that depth should be bounded by a plane surface; and the rectilinear plane is composed of triangles. [53d] Now all triangles derive their origin from two triangles, each having one angle right and the others acute; and the one of these triangles has on each side half a right angle marked off by equal sides, while the other has the right angle divided into unequal parts by unequal sides. These we lay down as the principles of fire and all the other bodies, proceeding according to a method in which the probable is combined with the necessary; but the principles which are still higher than these are known only to God and the man who is dear to God."
Within (2) What always becomes, our realm of sensation, Plato says it is most probable that everything is made out of triangles. This is at the basis of his geometry that Epíkouros attacks, but it is the basis for some other absurd ideas: triangles in your stomach and blood breakdown food, this becomes less efficient as you get older -- because your triangles are getting increasingly more dull (81c), this is, in fact, exactly what aging is.
The elements are built from and can be broken back down into triangles. If it seems to you that triangles alone might not be sufficient as a basis for everything we see in the word -- you have not considered that these triangles have different angles and come in different sizes (57d).
If so, atoms are constantly moving, and in that sense, void is constantly relocating
I would not think that the void can relocate. Void can in no way be altered. In fact, that malleability is what makes Plato's receptacle so different from the void:
"This all takes place at once, like the rotation of a wheel, because there is no such thing as a void." (79c)
"There is no void, these things push themselves around into each other, all things move by exchanging places." (80c)
Plato is very clear that the "(3) The receptacle, or what everything comes to be in" is not void, because void does not exist.
"There is no void into which anything that is moving could enter" (79b)
