Urgent! ! ! Senior two math!

Regular tetrahedron arccos( 1/3)? =? 70 32′

Regular hexahedron (square) 90

Octahedral arccos (-1/3) =10928'

Regular dodecahedron? Arccos(- radical 5/5) = 1 16 34'

Icosahedron? Arccos(- radical 5/3) =13811'

Refer to the following website (please delete *) and search for the word "regular polyhedron" to get more relevant data about regular polyhedron.

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Regular polyhedron or Platonic solid? 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 three dimensions, and Plato wrote them in Timaious. 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.

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

2.？ Every vertex of a regular polyhedron is equal.

3.？ 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 polyhedrons, all of which were discovered by the ancient Greeks.

(See figure for geometric data)

use

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.

Besides the regular dodecahedron mentioned above, there is also a polyhedron composed of regular triangles-pentagonal dodecahedron, which is a possible crystal structure of pyrite. The pentagonal dodecahedron is also composed of a regular triangle, but it is not a Platonic body, and its symmetric group is not an Ih group of a regular dodecahedron, but an Oh group that is the same as a cube.

symbolic meaning

Plato regarded the four elements as atoms, and their shapes were like four of the regular polyhedrons.

*? 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." (Timayus 55) Plato's student Aristotle added the fifth element-Aithê r? (Greek:? 'Αιθ? ρ, Latin: ether, Chinese:? Ether), and think that the sky is made of this, but he didn't connect the Taihe 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.

external links

*? Regular polyhedron plane expansion diagram

*? 360-degree stereoscopic panorama of regular polyhedron

*? There are only five proofs of regular polyhedron.

*? Polyhedral paper model? regular polyhedron