家 » 五大柏拉图实体

五大柏拉图实体

-350

Theaetetus
Plato (for philosophical association)

(generate image for illustration only)

柏拉图实体是仅有的五个凸正多面体：正多面体具有全等的正多边形面，每个顶点上有相同数目的面相交。这五种实体是四面体（4 个面）、立方体（6 个面）、八面体（8 个面）、十二面体（12 个面）和二十面体（20 个面）。自古以来，人们一直在研究它们的对称性和特性。

The Platonic solids represent a unique and finite set of three-dimensional shapes defined by their high degree of symmetry. To be a Platonic solid, a polyhedron must be convex and regular. This means all its faces must be identical (congruent) regular polygons, and the same number of faces must meet at every vertex. The proof that only five such solids can exist is a classic result in geometry. It relies on the fact that the sum of the angles of the faces meeting at any vertex must be less than 360 degrees; otherwise, the shape would flatten out. By systematically checking all regular polygons (triangles, squares, pentagons, etc.) and how many can meet at a vertex, one finds only five possibilities.

The five solids are:1. **Tetrahedron**: 4 triangular faces, 3 meeting at each vertex.2. **Cube (Hexahedron)**: 6 square faces, 3 meeting at each vertex.3. **Octahedron**: 8 triangular faces, 4 meeting at each vertex.4. **Dodecahedron**: 12 pentagonal faces, 3 meeting at each vertex.5. **Icosahedron**: 20 triangular faces, 5 meeting at each vertex.No regular polygon with six or more sides can be used, as the angle at each vertex is 120 degrees or more, and three such faces meeting at a point would sum to 360 degrees or more.

These shapes were known to the ancient Greeks, with the mathematician Theaetetus providing a mathematical description and proof of their existence. They are named “Platonic” because the philosopher Plato associated them with the classical elements in his dialogue *Timaeus*: the tetrahedron with fire, the cube with earth, the octahedron with air, the icosahedron with water, and the dodecahedron with the cosmos or aether. This philosophical connection elevated their status beyond mere geometric curiosities. Later, Johannes Kepler attempted to model the orbits of the planets using nested Platonic solids, a testament to their perceived fundamental importance in the structure of the universe.