Polyhedron and spherical problems

A regular polyhedron, or Platonic solid, refers to a convex polyhedron whose faces are congruent regular polygons and whose vertices are connected by the same number of faces.

Named source

Plato's solid, another name for regular polyhedron, named after Plato. Plato's friend Teitetus told Plato about these solids, and Plato wrote them in Timaeus. The exercise of regular polyhedron is contained in the volume 13 of the Elements of Geometry. Describe the regular tetrahedron method in the proposition 13. Proposition 14 is a regular octahedron, proposition 15 is a cube, proposition 16 is a regular icosahedron, and proposition 17 is a regular dodecahedron.

Judgment basis

There are three criteria for judging a regular polyhedron.

The faces of a regular polyhedron are composed of regular polygons.

Every vertex of a regular polyhedron is equal.

All sides of a regular polyhedron are equal.

These three conditions must be met at the same time, otherwise it is not a regular polyhedron, such as a pentagonal dodecahedron. Although it is surrounded by twelve pentagons like a regular dodecahedron, it is not a regular polyhedron because its vertex angles are not equal.

Regular polyhedron has a highly symmetrical shape, and each regular polyhedron has the highest symmetry in the point group to which similar polyhedrons belong. Changing the regular polyhedron will lead to the decrease of symmetry. For example, when the dodecahedron belongs to the Ih point group, the symmetry will also be reduced to the Td group.

Existence of regular polyhedron

There are five kinds of regular polyhedron, all of which were discovered by the ancient Greeks: (A is the side length of regular polyhedron in the table)

name

Perspective painting

Rotating perspective view

stereograph

Constitutive plane

face

edge

pinnacle

Geometric data

Attribution point group

Regular tetrahedron equilateral triangle 4 6 4 surface area:

Volume:

Dihedral angle:

Radius of external sphere:

Radius of inscribed sphere:

Double polyhedron: regular tetrahedron

Td group

Cubic (regular hexahedron) square 6 12 8 surface area:

Volume:

Dihedral angle:

Radius of external sphere:

Radius of inscribed sphere:

Double polyhedron: regular octahedron

Oh group

8 12 6 surface area of regular octahedral equilateral triangle;

Volume:

Dihedral angle:

Radius of external sphere:

Radius of inscribed sphere:

Double polyhedron: cube

Oh group

Regular dodecahedron regular pentagon 12 30 20 surface area:

Volume:

Dihedral angle:

Radius of external sphere:

Radius of inscribed sphere:

Double polyhedron: icosahedron

Ih group

Surface area of equilateral icosahedral triangle 20 3012;

Volume:

Dihedral angle:

Radius of external sphere:

Radius of inscribed sphere:

Double polyhedron: regular dodecahedron

Ih group

[Edit] Purpose

Regular polyhedron dice often appear in role-playing games because regular polyhedron dice are fairer.

Regular tetrahedrons, cubes and octahedrons also naturally appear in the crystal structure.

Other structures with similar symmetry can be obtained by chamfering the regular polyhedron. For example, the spatial structure of the famous spherical molecule C 60 is obtained by chamfering the dodecahedron, so we can know that the symmetric group to which the C 60 molecule belongs is also the same Ih group of the dodecahedron.

Regular polyhedron and chamfered regular polyhedron derived from regular polyhedron have good spatial packing properties, that is, they can be closely packed in space, so regular polyhedron or chamfered regular polyhedron box is often chosen as the periodic boundary condition of molecular simulation calculation.

In addition to the regular dodecahedron mentioned above, there is also a polyhedron composed of pentagons (four sides are equal in length)-pentagonal dodecahedron, which is a possible crystal structure of pyrite. The pentagonal dodecahedron is composed of pentagons, but it is not a Platonic body, and its symmetric group is not the Ih group of a regular dodecahedron, but the same Oh group as the cube.

[Editor] Symbolic significance

Kepler gave Plato's three-dimensional model of the solar system in The Mystery of the Universe (1596).

Plato regards the "four classical elements" as elements, and their shapes are like four in a regular polyhedron.

The heat of fire makes people feel sharp and stinging, like a small regular tetrahedron.

Air is composed of octahedron, and it can be felt that its tiny combination is very smooth.

Water will flow out naturally when put into people's hands, so it should be made up of many small balls, like an icosahedron.

Soil is different from other elements because it can be stacked like a cube.

Leaving the useless regular polyhedron-regular dodecahedron, Plato wrote in an ambiguous tone: "God arranged the whole constellation in the sky with regular dodecahedron." [1] Plato's student Aristotle joined the fifth element-ether (Greek: α ι θ? ρ, Latin transliteration: aithê r; Latin: aether), and thought that the sky was made of it, but he did not associate Taihe with the regular dodecahedron.

According to the tradition of establishing mathematics corresponding to the Renaissance, johannes kepler drew five regular polyhedrons to five planets-Mercury, Venus, Mars, Jupiter and Saturn, which also correspond to five classical elements.

[Edit] Prove that there are only five regular polyhedrons.

The properties of all regular polyhedrons related to the number of vertices v, edges e and faces f can be given by the number of edges (edges) p on each face and the number of edges q from each vertex. Because each edge has two vertices and two faces, we have

Another relationship is Euler's formula:

This obvious fact can be proved in many ways. In geometric topology, this is because the Euler characteristic of the sphere is 2. ) the above three equations can solve v, e, f:

Note that swapping p and q will swap f and v, but e will not change.

The theorem that there are only five regular polyhedrons is a classical result. Two proofs are given below. Note that these two proofs only prove that there are at most five kinds of regular polyhedrons, and the existence of these five kinds needs to be given by construction.

[Edit] Geometric Proof

The following geometric discussion is very similar to the proof given by Euclid in Primitive Geometry:

Each vertex of a polyhedron is on at least three faces.

The sum of the angles of these intersecting surfaces (that is, the angles emitted by the vertices) must be less than 360 degrees.

The angles emitted by the vertices of a regular polyhedron are equal, so this angle must be less than 360/3 = 120.

The angles of regular hexagon and regular polygon with more sides are greater than or equal to 120, so the faces on the regular polyhedron can only be regular triangles, squares or regular pentagons. So:

Regular triangle: each angle is 60, so the number of angles emitted by each vertex of a regular polyhedron is less than 360/60 = 6, that is, each vertex can only be on three, four and five faces, corresponding to regular tetrahedron, regular octahedron and regular icosahedron respectively;

Square: each angle is 90, so the number of angles emitted by each vertex of a regular polyhedron is less than 360/90 = 4, that is, each vertex can only be on three faces, corresponding to a cube;

Regular Pentagon: each angle is 108, so the number of angles emitted by each vertex of a regular polyhedron is less than 360/ 108 = 10/3, that is, each vertex can only be on three faces, corresponding to a regular dodecahedron.

[Edit] Topology Proof

Pure topological proof can only use the properties of regular polyhedron. The key is and. Combining the above equations, we get

therefore

Because,

Noting that p and q must be greater than or equal to 3, we can easily find all five groups (p, q):

Source: zh.wikipedia.org/wiki/ Regular Polyhedron