Platonic solids

Platonic solids

This animation demonstrates the five regular three-dimensional (or Platonic) solids, the best known of which is the cube.

Mathematics

Keywords

Platonic solid, tetrahedron, cube, octahedron, dodecahedron, icosahedron, dual, Pythagoras, Aristotle, geometry, solid geometry, mathematics

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In Euclidean geometry, a Platonic solid is a regular, convex polyhedron composed of congruent faces that are regular polygons, with the same number of faces meeting at each vertex.

There are five Platonic solids, each is named according to its number of faces.

Tetrahedron: 4 faces
Hexahedron: 6 faces
Octahedron: 8 faces
Dodecahedron: 10 faces
Icosahedron: 20 faces

Polyhedra are associated into pairs called duals. In the dual of a polyhedron its vertices correspond to the faces of the other: the centers of each face are connected to the centers of the neighboring faces. The duals of regular polyhedrons are also regular polyhedrons, therefore Platonic solids are arranged into pairs.

You can view the dual of the solids by clicking the button ´Dual´.

A dodecahedron is a regular polyhedron composed of congruent, regular pentagonal faces.

Faces: 12
Edges: 30
Vertices: 20
Dual: icosahedron
Angle between two faces: ~116°33’55.84”
Number of edges meeting at each vertex: 3
Space diagonals: 100

Prepare a dodecahedron with unit edge lengths (Animation).
Take a cube in which the length of the edges is the golden ratio (t). Take also 3 congruent rectangles in which the shorter edges are 1 unit long, the longer edges 1+t long. Place the rectangles inside the cube in a way that their centers fall on the center of the cube and any two of them intersect at right angles, with the plane of all three being parallel to one pair of faces of the cube. Now connect the vertices of the rectangles with the two nearest vertices of the cube. These lines, together with the shorter edges of the rectangles form the frame of a dodecahedron with unit edge lengths.

An icosahedron is a regular polyhedron composed of congruent, regular triangular faces.

Faces: 20
Edges: 30
Vertices: 12
Dual: dodecahedron
Angle between two faces: ~138°11’22.87”
Number of edges meeting at each vertex: 5
Space diagonals: 36

Prepare an icosahedron with unit edge lengths. (Animation)

Take 3 congruent golden rectangles. These are rectangles in which the ratio of the length of the edges is the golden ratio: a+b : a = a : b. Let the shorter edge of these golden rectangles be of unit length, then the longer edges will be t long (where t is the golden ratio). Place the rectangles in a way that their centers coincide and any two of them intersect at right angles. Now connect the vertices of each rectangle with the two nearest vertices of the 2 other rectangles. These lines, together with the shorter edges of the rectangles form the frame of an icosahedron with unit edge lengths.

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