Octahedron

Geometry

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Octahedron polyhedron Both are called octahedrons.

By 6 vertex And 8 Equilateral triangle Constitutive pyramid , there are eight faces, and each face is one Equilateral triangle 。 just as Cube Similarly, the eight faces of octahedron are congruent Regular polygon 。^[1]

Chinese name: Octahedron
Foreign name: octahedron / octahedral
Shape: pyramid

Composition: By 6 vertex And 8 Equilateral triangle constitute
Nature: All eight sides are congruent Regular polygon
Applied discipline: geometry

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five Simplex Name

Name Source

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The geometric term "octahedron" mainly refers to Octahedron , of which there are 8 triangular faces. Generally speaking, "octahedron" is associated with Octahedron , but the most original definition is ignored: polyhedron Both are called octahedrons.

Convex octahedron

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Among all convex octahedrons, topology There are 257 kinds of convex octahedrons with obvious structural differences, including their mirror images. Among them, 2 have 6 vertices, 11 have 7 vertices, 42 have 8 vertices, 74 have 9 vertices, 76 have 10 vertices, 38 have 11 vertices and 14 have 12 vertices.

Common octahedron

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Common octahedrons are Octahedron , hexagonal column Heptagonal pyramid , truncated tetrahedron, regular triangle tent tower, heterogeneous double triangle column, side cone triangle column, triangular anti prism, etc.^[2]

Hexagonal column

Hexagonal column is also called hexagonal prism , is a kind of bottom surface hexagon Of Cylinder 。 All hexagonal columns have 8 faces, 18 sides, and 12 vertices. Regular hexagonal columns represent hexagonal columns whose faces are regular polygons, and each vertex is the common vertex of two squares and one regular hexagon, so they have the property of equal angles of each angle and can be classified as semi regular octahedron.

Heptagonal pyramid

A heptacone is a cone with a heptagon bottom, which has 8 faces, 14 edges and 7 vertices, and its dual polyhedron is itself. A regular heptacone is a heptacone with a regular heptagon bottom.

characteristic

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these ones here polygon Connected horn And edges are the same everywhere. Because of the existence of such regularity Therefore, the plane of each polygon is equally possible. The octahedron is called Platonic solids Five of Regular polyhedron One of these three-dimensional All have this regularity.

Octahedron, consisting of 6 vertex And 8 Equilateral triangle Form, 4 triangles intersect at a vertex. Plato It is believed that octahedron is between tetrahedron (fire) and Icosahedron （ water ）Therefore, it is considered that the element it represents is air. There are 6 octahedrons Secondary rotation axis , via In the opposite side Point; 4 Third rotation axis , via Opposite center ； And 3 Quadratic rotation axis , via the opposite vertex 。 Any of these Rotation axis We say it has octahedron symmetric 。

Simplex Name

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Only in Isometric Appears on crystals. It is formed by 8 parallel Equilateral triangle Same as Crystal plane Composed of Octahedron 。 It passes through the center and connects each pair horn The three lines on the top are perpendicular to each other and equal in length. Intercrystalline included angle 109 °; Each pair of crystal faces is 3 times the same as one in the crystal symmetric Vertical axis; Each crystal face and 3 Crystal axis All octahedrons are equal in length. Simplex The symbol is {111}. magnetite 、 spinel Isocrysts often have this simplex. In addition, on the crystal structure Octahedron is a common Coordination polyhedron Form. However, when used to represent a coordination polyhedron, the said octahedron may not be a regular octahedron but may be distorted

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