[MUSIC] Timaeus, on the material elements. Recall that the Presocratics naturalists such as Thales and Anaximander and Anaximenes and Empedocles and so on. Thought that everything in the world is composed of roughly four kinds of stuffs, that is Earth, Air, Fire, Water. Although they disagreed about which is more basic, or whether there is something more basic than all of them. Plato inherits this picture and like Anaximander, he thinks these elements are modifications of something more basic. Where Anaximander invoked the APEIRON, the indefinite, Plato and the Timaeus invokes what he calls the RECEPTACLE. In his view, it is only when the order of form is imposed on the RECEPTACLE that Earth, air, fire, and water come to be. And true to the principles of the rest of Timaeus' cosmology, the forms that these material elements emulate are good. As Timaeus says, "the god fashioned these four kinds to be as perfect and excellent as possible." And just as in the case of the World's Soul, that goodness is expressed in mathematical or geometrical terms. The basic idea is this, Earth, air, fire, and water are bodies. They're solids, that is three dimensional figures. Solid figures are constructed out of plane figures, that is, two-dimensional figures. And every plane figure is made of triangles, at least if it has straight sides, a simplifying assumption that Timaeus employs in constructing his model. So, if the demiurge is going to construct the most perfect solids, they must be constructed out of the most perfect triangles. Thus the question Timaeus poses is, what are the most perfect triangles? Well, every triangle, he says, can be divided into two right angle triangles. And right angle triangles can be either isosceles or scalene. In the right angled isosceles triangle, the other two angles are equal, 45 degrees each, but in the right scalene triangle, the other two angles can be any combination adding up to 90 degrees. Out of this range of possible structures, Timaeus identifies the triangle with angles of 30, 60, and 90 degrees as the most perfect. For reasons that he's not eager to explain. He says it's too long a story to tell now. Now we can call this sort of triangle, the 30, 60, 90 triangle a half equilateral because if you juxtapose two of them along the longer side of their right angels they from an equilateral triangle. And the equilateral triangle is perfectly proportional. Every side the same length, every angle the same size. Now Timaeus prefers to generate the equilateral triangle out of six of these 30, 60, 90 triangles, rather than two. Presumably, because only in the six piece construct is the resulting equilateral triangle completely symmetrical in relation to its constituent triangles. Every side and every angle of the equilateral is bisected by the sides of the constituent triangles. This is a result you wouldn't get if you used only two constituent triangles. At any rate, after Timaeus has identified the two most perfect triangles, or the two most perfect right angle triangles, he is in the position to construct four perfect solids. That is, three dimensional figures that are perfectly symmetrical. Every face the same as any other, every side the same length, every angle the same size, and every vertex composed of the same number of angles. So out of the first of his perfect right angled triangles, the right isosceles, he constructs the cube. His first step is to construct its face, the square. If we take four right-angled isosceles triangles and juxtapose them with their right angles at the center, we get a square. A perfectly symmetrical two-dimensional figure. Now once again, we may ask why two of these constituent triangles isn't enough? The same question we asked in the case of the half equilaterals. But once again, I think it's the internal symmetry of the resulting square that's important, with only two constituents right angled triangles. Not every vertex of the square will be bisected by the sides of the constituent triangles. Okay, so out of the four triangles that are perfect in their own way, we can generate another perfect plane figure, the square. Next, if we take six of these squares, we can construct a perfect solid, the cube. Timaeus identifies the cube as the basic structure of the element Earth. Now let's see what we can make out of the other perfect triangle. The half equilateral, or 30, 60, 90 triangle. Recall, that out of a six pack of these, Timaeus has constructed another perfect plane figure, the equilateral triangle. What perfect solids can we construct from equilateral triangles? Well in fact, there were three different ones. The simplest of these is the tetrahedron. Who's four faces, tetra is four in Greek, are equilateral triangles. The next is the octahedron which has eight equilateral triangles as its faces. And the largest is the icosahedron which has 20 faces. All of them equilateral triangles. All of these solids are perfectly regular. Every surface is the same size and shape. Every angle, plane or solid is the same as every other. Now Timaeus identifies the smallest, pointiest shape, the tetrahedron, as the structure of elemental fire. The next largest, the octahedron, is the form of air. And the largest of the three, the icosahedron, is the form of water. Plato seems to be operating on the assumption that the denser and more sluggish the element, the more triangles it contains. Note that with these principles of construction, it is possible to specify conversion ratios between fire, water and air. One unit of fire contains 24 half-equilateral triangles, 4 faces times six triangles per face. One unit of air, contains 48 constituent triangles and one unit of water, the icosahedron contains 120. So from one unit of water, a 120, you can get 1 fire and 2 air. And from 1 air you can get 2 fire, and so on. So Plato was thus able to give some mathematical precision to the accepted scientific view in his day, that the elements transform into each other. However, given the particular geometrical model he's working with, where Earth, the cube, is not made of the same constituent triangles as the other three, he has to revise the standard physical theory. Only fire, air, and water convert with each other, Earth does not. Now, we might ask, why didn't Plato assign Earth a structure that is made out of equilateral triangles? The answer, I think, is quite simple. He is working from a menu of perfect solids identified by geometers of his day, and only three of them are constructed out of equilateral triangles. Geometers in Plato's day had in fact had identified five perfect solids. The four we have already seen the cube, the tetrahedron, the octahedron, and the icosahedron, and in addition the dodecahedron, which is constructed out of 12 regular pentagons. These are known as the five platonic solids because of their occurrence here in Plato's Timaeus. The dodecahedron, in fact, makes a cameo appearance in the Timaeus at 55c, as the shape of the universe as a whole, since it approaches the shape of the sphere. So okay, back to the main question. Plato is looking for four perfect solids to structure the four elements on his periodic table, Earth, air, fire, and water, and he goes to the geometers to get his list of perfect solids. Unfortunately, there aren't four perfect solids that can be constructed out of the same plane figure. Three can be constructed out of equilateral triangles, one out of squares, and one out of regular pentagons. So Plato goes for the three that convert, and a fourth, the cube, that does not. But now we might be able to appreciate why Plato has Timaeus insist that all four of the perfect solids he uses are constructed out of triangles. After all, he could've just started with the equilateral triangle and the square as his generating principles, for each of these is perfectly regular. But this wouldn't be good enough, because he wants something to connect the equilateral triangle and the square. He's not satisfied unless he can find a unified theory that generates both the square, the principle of the cube, and the equilateral triangle, the principal of the other three perfect solids. So now, I think we can appreciate the motivation for his opening statements in this section. That any plane figure can be resolved into triangles, that any triangle can be resolved into two right angle triangles. And that the two perfect varieties of the right angle triangle are the right isosceles and the half equilateral. These are the two triangles from which he will construct all four of his perfect solids, not just the three built from the equilateral triangle. Thus, even if on the resulting picture, the perfect solids that structure the elements are not fully convertible with each other. Their structures are generated from a single set of principles that attempt to capture perfection in terms of mathematical and geometrical proportions. Now, we know, today, that this lovely geometrical picture is not an accurate theory of the structure of matter. But I don't think even Plato thought it was. That's another way in which the cosmology of the Timaeus is a likely story. What is essential to that story even if the details are inaccurate, is that the structure of the natural world is mathematical and that this is the source of it's order, an intelligibility. Around 2,000 years after Plato wrote the Timaeus, the Italian astronomer Galileo would write. "The book of nature is written in the language of mathematics, and its symbols are triangles, circles, and other geometrical figures." This is an oft quoted slogan that is the manifesto of the mathematical physics that has become the greatest success stories of modern science. But we can see from reading Plato's Timaeus, that this was hardly a new idea in Galileo's time.