Theory of Plato's Atomic Structure
Date: Mon, 7 Oct 1996 21:50:22 -0400
Subject: Theory of Plato's Atomic Structure
[ You wouldn't know it by viewing my output now, but I hated to write back in high school and college. Most of my academic career was devoted to picking courses that didn't involve 8:30AM classes, labs, or term papers. I could do very well with class participation, homework assignments, and tests, thank you very much. Not meeting on Friday or Monday was also a plus, so I could take a long weekend via Amtrak discount fare to visit my sweetie at Bryn Mawr.
It is a reflection on how interested I was in Ancient History that I took a course that met Monday Wednesday Friday at 8:30, and had two term papers. They are relevant to the discussion about Plato, so... "Sherman, set the wayback machine for 1980, and a very grumpy Union College student hammering away at a manual typerwriter, with copious amounts of white-out ready to hand!" ]
The Physics of Plato's Atomic Structure
Greek Philosophy
Professor Peterson
From prehistoric times, Man has wondered about the nature of the universe - when and why it was created, how it was made, where it came from, and what it is made of. The ancient Greeks hypothesized that everything was built up from four Primordial Elements: Earth, Air, Fire, and Water. However Plato thought up an even more basic system, and presents it, amidst much hemming and hawing such as "...another regarding the question from another point of view, will be of another mind" (55d) or "I have given the results of my thoughts" (52d) in his Socratic Dialogue Timaeus. Here, Plato discusses a simpler system, one composed of only two discrete particles. According to Plato, the basic units of the universe are made of triangles. Although he does hint that some, perhaps the Pythagoreans, know differently when he says "the principles which are prior to these God only knows, and he of men who is friend to God" (52d), he talks no further on the subject for the rest of the dialogue. Every rectilinear surface is composed of triangles, so they are his starting point.
Plato then asks which triangles, out of the infinite number possible, are the most basic. He chooses right triangles, saying, "all triangles are originally of two kinds, both of which are made up of one right and two acute angles." (53c-d) The geometric way to show this is to draw a line from an angle perpendicular to its opposite side. For example:
[ I'm not going to draw it in ascii, maybe I'll draw one and put it on my web page. -Carl ]
All of the right triangles can be divided into two classes: the 45 degree iscosceles right triangle, which has both acute (less than 90 degrees) equal, and the rest - which do not. He then looks for a second triangle among the second class, for reasons that which soon become apparent, and settles on what is know as the 30-60-90 triangle, where one acute angle is twice the second:
[ image]
A reason why this particular triangle was singled out, aside from the practical reasons of having the theory work out, is that the ratio of its sides is 1-2-3, the simplest after the 1-1-2 of the isosceles right triangle.
Using his two types of triangles, Plato constructs the building blocks of the universe. The isosceles triangle is arranged in a group of four, with each 90 degree angle at their joining, forming a square:
[ image]
"Six of these [equilateral quadrangles are] united [to] form eight solid [three dimentional] right angles, each of which is made by the combination of three plane right angles, the figure of the body thus composed is a cube." (55b-c) He then moves to the scalene triangle for the remaining three solids. However, they do not directly make up the plane faces. Instead, six 30-60-90 triangles form a larger equilateral triangle:
[ image]
four triangles form the tetrahedron, eight make the octahedron, and twenty the icosahedron.
Having constructed four of the five regular solids, Plato assigns the elements to them, saving the fifth (the dodecahedron) for a special purpose. He bases his assignment on the properties of the elements. Therefore, Earth, the most stable and unchangable of the four, is assigned the six-sided cube. The cube not only stacks the best, but also is composed of a different triangle from the first, so that it could not change into the other elements.
Conversely, Fire, Water, and Air, being built of the same 30-60-90 triangle, can be changed from one to another. "To Water we assign that one of the remaining forms [tetrahedron, octahedron, icosahedron] which is the least movable, and the most moveable of them to Fire, and to Air that which is intermediate." (56a) Also, the smallest body goes to the element which moves around the most, or Fire. According to the above two principles, Air is assigned the octahedron, and Water the icosahedron.
However, Plato's arguement on the stability of the regular solids has met with some disagreement. After two years of experimenting with polyhedra dice, it appears that the least moveable is the tetrahedron. In other words, the four-sided die is least likely to roll from one face ot the next, because of the acute angle between them, whle the the twenty-sided die, or D20 for short, is very likely to roll into places that it is difficult to retrieve it from. Further, it is remarkably easy to induce the D20 to rotate on an axis like a toy top, but it is singularly difficult to spin a tetrahedron.
On the other hand, Plato may have felt that there were no gravity, or that his atomic particles would not be affecte by it. In so, then the least massive solid, the one enclosing the smallest volume, would indeed be the least stable.
Plato briefly mentions the dodecahedron saying, "There was yet a fifth combination which God used in the deliniation of the univers with the figures of animals." (55c) The regular twelve-sided solid does not fit into the triangle theory, because its sides are pentagons, and the Pythagoreans had not found a way to construct it from triangles. Therefore, Plato assigns it to the heavens, where it will not interact with the four elements. The animals referred to are either those in the Zodiac, which seems strange because the sun does not travel all through the heavens, or those found throughout the sky. In that case, God divided up the celestial sky into twelve areas, for ease in naming the constellations.
In Timaeus, Plato talks about an atomic system of creation, based on triangles. The triangles compose the four Elements: Earth, Air, Fire, and Water, which in turn make up everything else. His system is very mathematical, fitting the ideology of the Pythagoreans that 'All is Number'. Although he says often that his system is not Unchangeable Truth, engraved in stone, it seems rational, and was probably accepted, if not initially proposed, by the Pythagoreans.
[A. Excellent - Prof. Peterson]
The Chemistry of Plato's Atomic Structure
or, Plato's Playtoys
Greek Philosophy
Prof. Peterson
From ancient times, Man has strived to learn the basis of the universe. Nuclear physists today talk about strange colored sub-atomic particles, trying to develop a theory that will unify the known types of energy. In ancient times, men believed that all matter was composed of four elements: Earth, Air, Fire, and Water. Plato, in his dialogue Timaeus, reduces the four elements to two triangles, in much the same way that the early scientists reduced the ninty-two elements to protons, neutrons, and electrons. After constructing the elements from combinations of triangles, Plato discusses the interactions of his atomic elements. Plato talks about how the elements are transmuted into each other, after reminding the listener that his is "the most probable conclusion." (56c) Earth is never reformed into another element, because it is constructed of an isoceles triangle, but the other three are transformed according to specific formulas. Water, with twnty sides to an atom, can be changed into one atom of fire, with four sides each, and two of Air, with eight sides. (20 * 1) = (1 * 4) + (2 * 8). Also, one Air can be transformed into two Fires, and vice versa: (1 * 8) = (2 * 4). Similarly two and one half Air are transmuted to one Water: (2.5 * 8) = (1 * 20). Plato mentions experimental observations to back his calculations with: "Air, again, when inflamed, becomes fire, and, again, fire, when condensed and extinguished, passes once more into the form of air, and, once more, air, when collected and condensed, produces cloud and mist.... (49c)
Also, his proportions agree with modern chemistry's law that the ratio of chemical elements in a compound are small whole numbers. Fire is the usual cause of change. "When one of the other elements is fastened upon by fire, and is cut by the sharpness of its angles and sides, it coaleses with the fire, and then ceases to be cut by them any longer." (57a) Eventually, the smaller quantity of the various jumbled elements is decomposed into its original 30-60-90 right triangles, and is then reformed in the shape of the larger amount, much like a small crystal causes the solution it is placed in to crystalize. However, when Earth is decomposed, its isosceles triangles drift about in space until they are able to reform in peace.
Plato talks about the entropy of his mathematical universe. Before the creation of today's universe, before an Unmoved Mover moved, the elements were "full of pwoers which were neither similar nor equally balanced [and] was never in any part in any state of equipose." (52d-e) Chaos was complete, and everything was tossed hither and yon. Eventually, as like sought like, the elements were arranged in four piles, with the heaviest farthest from the lightest, "a condition one may expect to find whenever God is absent." (53b)
However, God was not absent for long, and shuffled the piles [vectoring useful energy - Prof. Peterson], moving relatively small quantities of elements into each other's heaps. Therefore, men never see pure elements. What is underneath us is merely an ore of the element Earth, with impure water on top of it, while a mixture of Air and the other elements is what we breath, but mostly pure Fire can be seen whenever a hole in the air occurs, and we see lightning.
The elements tend towards their original piles but as one Element is transformed into another, it moves towards a different place in the cosmos, "continually causing a perpetual motion of the elements in all time." (58c) In this manner, total entropy, or the death of all motion in the universe, is averted.
Plato now introduces imperfection into his mathematical unvierse, and talks about subvarieties of his elements. To forestall the arguement that only four primal elements could not create the many different substances in the visible world, he says that there are "varieties in the strucutre of the two original triangles." (57c) Not all triangles are created equal, some are bigger than others. "Hence, when they are mingled with themselves and with one another there is an endless variety of them...." (57d) However, one can now say that there exists a tetrahedron of Fire that has more volume than an icosahedron of Water composed of smaller triangles, and contradict Plato's earlier intention of assigning Fire the smallest polyhedra.
Plato says there are "diverse kinds of fire," (58c) and mentions three: flame, as from a candle, light which does not burn, and the red glow coals emit. Flame probably has the smallest triangles, in order to have the sharpest corners and angles, while pure light would have the largest, with red glow in between, as it both illuminates and burns.
Air too, has many variants "which arise from the inequality of the triangles: " (58d) pure aether and misty darkness, although breathing atmosphere is probably a third.
Plato divides Water into two kinds: liquid, or fluids, and fusile, or the metals. Liquid is formed from irregular triangles and unequal icosahedra, and pours easily because some fire tetrahedra have been mixed in, and the bases roll easily. When the fire escapes to the surrounding air, the change in pressure causes the fluid to solidify. Depending on the amount of compression, and the location, hail, ice, snow, or hoarfrost is formed. (59d) Plato distinguishes four more admixtures of water and fire: wines, oils, honeys, and the acidic plant saps. He says, "they are the numerous kinds of water which have been mingled with one another and are distilled through plants which grow in earth," (59e) so perhaps these fluids have some Earth in their substance.
Fusile Water is "formed of large and uniform particles" (59d) of Water, and when Fire (ie, heat) is added, it too melts and flows on the ground. The most compact and uniform icosahedra form gold, which has been hardened by pressure from the surrounding Earth and Water. Very compressed gold is transformed into adamant, a black ore, which may be onyx or obsidian. The third metal Plato mentions is copper. He explains the verdigris that forms on old copper by saying that "there is an alloy of Earth mingled with it which, when the two parts grow old and are disunited, shows itself seperately and is called rust." (59c)
Earth is also discussed in depth. Stone or rock is formed from an initial mixture of Earth and Water, which changes to Air and is violently expelleed. The pressure on the remaining Earth causes the hardness of the rock. Perhaps volcanos are the visible signs of the expelled air; there certainly is enough new rock coming from them. Rock formed by this process that has triangles and cubes of unequal size become clay, and if the rock is heated by applying Fire, all of the remaining Water is driven out by coalesing with the Fire, and pottery is formed.
Plato says that if not all of the Water is driven out by Fire, then "a certain stone of a black color is formed." (60d) A fifth class of Earth is salts and sodas, which are fine particles of Earth mixed with large amounts of Water. As soda "is used for purging away oil and dirt" (60d) Plato may be describing soap. Finally, he talks about substances which are insoluble, but are dissolved by Fire or heating, such as parrafin or glass.
In Timaeus, Plato talks about the structure of the universe. Using a strict mathematical approach, he describes how one element is changed to another. By observing the world, he deduces that there must exist a God who has created the cosmos, keeping it from Chaos. Finally, he looks at some different chemical transformations that occur, and lists how some of the compounds around us are formed.
[ A. Clear and to the point. -Prof. Peterson.]
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Carl Hommel , notelrac@net1plus.com
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