Platonic polyhedron is notPlatoThe invention, however, is named after Plato and his followers' research on them. Because they have a high sense of symmetry and order, they are usually called regular polyhedrons. However, here we still call them Plato polyhedrons to avoid confusion with other regular polyhedrons.All faces of a Platonic polyhedron are non self intersecting, with straight segments as side lengthsConvex polygonPlane, each polyhedron has only one regular polygon surface, and there are the same number of faces intersecting at each vertex.Not only do the same number of faces intersect at each vertex, but the sum of the interior angles of all intersecting faces at each vertex will be equal.
The best way to become familiar with Platonic polyhedrons is to construct models and study them through models.In order to construct a model of a Platonic polyhedron, a set of similar unfolded drawings must be drawn on the appropriate material.Students can directly cut out the expanded drawing of polyhedron attached to this material, or enlarge or reduce it and photocopy it on appropriate beautiful paper.If the material is not convenient for photocopying, you can also draw or photocopy the expanded drawing and paste it on the used material.Albrecht Durei, as early as 1525, in his book Unterweisung der Messung Mit dem Zirkel und Richtsheit, gave the expanded diagrams of several polyhedrons.
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It is easy to see that each Platonic polyhedron is convex and intersects the same number, similarity and positive at each vertexConvex polygon。To understand why only five Platonic polyhedrons are quite simple, it is because at least three faces intersect at each vertex to form aStereoscopic figureAnd the angle sum of the faces around each vertex cannot be equal to or more than 360 °, otherwise the resulting faces will be flat or concave.[1]
The regular polygon with the least number of sides is a regular triangle. Three such polygons can make them meet at one vertex. Next, add a fourth face, so that every three faces will meet at one of the four vertices of the figure.Since this figure has four congruent faces, it is calledRegular tetrahedron(TETRAHEDRON)。
Four regular triangles can make them meet at one vertex, and after adding four faces, four faces will meet at the six vertices of the figure.Since this figure has eight sides, it is calledOctahedron(OCTAHEDRON)。
In addition, we can construct a figure where five equilateral triangles can meet at its 12 vertices.Because such a figure has 20 faces.So it is calledIcosahedron。
If six regular triangles meet at a vertex, the sum of the angles of the faces meeting at the vertex is 360 °, and these triangles will form a plane.Therefore, there can only be three kinds of Platonic polyhedrons whose surface is an equilateral triangle.
Next, the polygon to be considered is a square. We can construct a polyhedron where three squares meet at its eight vertices. It is another kind of Platonic polyhedron, commonly known as CUBE. Since it has six faces, it is also called CUBERegular hexahedron（HEXAHEDRON）。
A convex polyhedron cannot be intersected by four squares at each vertex, because the sum of the angles of the faces intersected at each vertex will be 360 °.
The next consideration is that there are five equilateral and five internal angles of 108 °Regular pentagon。A polyhedron can be formed by three regular pentagons intersecting at its 20 verticesRegular dodecahedron(DODECAHEDRON), because it has 12 faces.Usually we also call it positivePentagonal dodecahedron。
FourpentagonIt will not intersect at a vertex to form a convex polyhedron, because the sum of the angles of the faces intersected at a vertex will exceed 360 °
The next consideration isRegular hexagonHowever, if three regular hexagons meet at a vertex, the total angle of these faces will be 360 °, thus forming a plane.It can also be seen that the more faces a polygon has, the larger its internal angle will be. For regular polygons with more than six sides, the sum of their three internal angles will exceed 360 °. Therefore, they cannot be connected together to form a positive convex polyhedron.
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Suppose a regular polyhedron has V vertices, F block faces and E edges;Each face is a positive n-sided shape, and each vertex is connected by the vertices of m block faces.
Since there are F block faces in common, and each face is positive n-sided, after the regular polyhedron is disassembled into F positive n-sided, there should be nF edges;
Since a vertex is connected with m other vertices, after the regular polyhedron is disassembled into F positive n-sided shapes, there should be mV vertices. Since the number of polygon vertices is equal to the number of edges, there are mV edges in common;
Similarly, when multiple n-sided polygons are combined into a regular polyhedron, one edge of each of the two n-sided polygons will be combined into one edge of the regular polyhedron, so after the regular polyhedron is disassembled, there should be 2E edges;
Because E must be a positive integer (m - 2) (n - 2)<4
Because of the basic solid geometry and plane geometry, m>2 and n>2, (m, n) can only be (3, 3), (3, 4), (4, 3), (3, 5) and (5, 3);That is, (V, F, E) can only be (4, 4, 6), (8, 6, 12), (6, 8, 12), (20, 12, 30) and (12, 20, 30).
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Plato regards the "four classical elements" as elements, whose shapes are like four of the regular polyhedron.
atmosphereIt is made of octahedron, and its tiny combination is very smooth.
WhenwaterPut it on a person's hand and it will flow out naturally. It should be composed of many small balls, likeIcosahedron。
SoilIt is different from other elements because it can be stacked asCube。
The remaining useless regular polyhedron——Regular dodecahedronPlato wrote in an unclear tone: "God uses regular dodecahedron to arrange the constellations in the whole sky." Plato's studentAristotleAdded fifth element——ether(Greek: δθ Latin: aith ê r; Latin: aether), and he thinks that the sky is composed of this, but he does not connect the ether with the regular dodecahedron.
Johannes Kepler, following the tradition of establishing mathematical correspondence in the Renaissance, mapped five regular polyhedrons to five planets——MercuryVenus, Mars, Jupiter and Saturn, and they also correspond to the five classical elements.
