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 [原创]亚里斯多德之谜,亚里斯多德声称找到了一种四面体可以填充空间,可是没找到.2400年后俺找到了 |
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亚里斯多德构想出五元素说，柏拉图 四面体(火)， 八面体 (风)，二十面体 (水)，及立方体（地） -- 秀才 - (0 Byte) 2010-4-27 周二, 11:42 (735 reads) |
秀才
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作者：秀才 在 海归商务 发贴, 来自【海归网】 http://www.haiguinet.com
Platonic Solids
A Platonic solid is a 3-dimensional shape built from identical flat figures called regular polygons, with the additional requirement that the same number of polygons should meet at each corner of the solid. A polygon is a closed shape in the plane made up of line segments. It is said to be regular if all the line segments have the same length and all the corner angles are equal. Thus, a 3-sided regular polygon is an equilateral triangle; a 4-sided regular polygon is a square; a 5-sided regular polygons is a pentagon; and so forth.
There are only 5 ways to assemble regular polygons into Platonic solids. When we build a Platonic solid, we must have at least 3 polygons meeting at each corner--two polygons are not sufficient to build a solid angle. Let's start by considering what Platonic solids can be built out of equilateral triangles. If we put 3 triangles together at a corner, they form a base that is another equilateral triangle; we can cap that off with one more triangle to get a 4-sided solid called a tetrahedron. If we put 4 triangles together at a corner, they form a pyramid with a square base. We can put two of these pyramids together along their bases to form an 8-sided solid called an octahedron. If we put 5 triangles together at a corner, it is harder to visualize the final result, but the triangles continue to fit together as we add more, and they finally close up into a 20-sided solid called an icosahedron.
Those are the only Platonic solids that can be formed from triangles. If we try to fit 6 equilateral triangles together, they form a flat surface, since equilateral triangles have 60-degree angles, and 6 of them together will form a full 360-degree angle. And it is impossible to fit more than 6 triangles together, since the angle measurements would add up to more than 360 degrees.
If we try to make Platonic solids out of squares, we should start by trying to put together 3 squares at each corner; this produces a cube. Four squares at a corner will form a flat surface, and more than 4 squares cannot fit at a corner. So the cube is the only Platonic solid made of squares. If we fit 3 pentagons together at a corner, once again it is hard to see the shape that forms. The pentagons do fit together, though, into a 12-sided solid called a dodecahedron. More than 3 pentagons cannot fit together at a corner. Three hexagons (6-sided polygons) fit together to form a flat surface, and more than 3 will not fit together at all. And for any polygon with more than 6 sides, it is impossible to fit together even 3. Thus the 5 solids we have found are the only ones.
Platonic solids were named after Plato, who was one of the first philosophers to be struck by their beauty and rarity. But Plato did more than admire them: he made them the center of his theory of the universe. Plato believed that the world was composed entirely of four elements: fire, air, water, and earth. He was one of the originators of atomic theory, believing that each of the elements was made up of tiny fundamental particles. The shapes that he chose for the elements were the Platonic solids. In Plato's system, the tetrahedron was the shape of fire, perhaps because of its sharp edges. The octahedron was air. Water was made up of icosahedra, which are the most smooth and round of the Platonic solids. And the earth consisted of cubes, which are solid and sturdy. This analysis left one solid unmatched: the dodecahedron. Plato decided that the it was the symbol of the "quintessence," writing, "God used this solid for the whole universe, embroidering figures on it." Plato's description of the universe made a deep impression on his disciples, but it failed to satisfy his most illustrious student, Aristotle. Aristotle reasoned that if the elements came in the forms of the Platonic solids, then each of the Platonic solids should stack together, leaving no holes, since for example water is smooth and continuous, with no gaps. But, Aristotle pointed out, the only Platonic solids that can fill space without gaps are the cube and the tetrahedron, hence the other solids cannot possibly be the foundation for the elements. His argument struck his followers as so cogent that the atomic theory was discarded, to be ignored for centuries.
Although Aristotle's dismissal of Plato's structure was correctly reasoned, his analysis contained a famous error: the tetrahedron does not fill space without gaps. Incredibly, Aristotle's mistake was not discovered for more than 17 centuries. Aristotle was so highly esteemed by his followers that they confined themselves to trying to calculate how many tetrahedra would fit around one corner in space, rather than considering the possibility that the great man was mistaken. In the process, they came up with many conflicting formulas, some absurd. No one seems to have taken the simple step of building some tetrahedra and observing that they do not even fit nicely around a single point, let alone fill all of space without gaps. The error was not set right until the 15th century by Regiomontanus.
Although Plato's and Aristotle's theories were both flawed, they have provided the inspiration for much successful later work. Plato's theory, in which the elements are able to decompose into "subatomic" particle and reassemble in the form of other elements, can be considered a precursor to the modern atomic theory. And Aristotle's question, what kinds of shapes fill space, has proven to be a crucial problem in the study of crystals, in which the atoms are locked into repeating geometric patterns in 3-dimensional space.
作者：秀才 在 海归商务 发贴, 来自【海归网】 http://www.haiguinet.com
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