What is Icosahedron in Mathematics?


Icosahedron Meaning and Explanation

Regular twenty aspect is a regular polyhedron composed of 20 equilateral triangles, with a total of 12 vertices, 30 edges, and 20 faces. 
It is one of the five Platonic polyhedrons.

Name:    Regular twenty aspect
Scope of application:    Mathematical Sciences
Edge:    30
Surface:    20

Table of Content
  1.      Definition
  2.      Nature
  3.      Volume formula
  4.      Calculation formula


Definition of Icosahedron

Each face of a regular polyhedron is a congruent regular polygon, and each polyhedral angle is a congruent polyhedral angle. 
The tetrahedron is the least, and the icosahedron is the most.

Regular twenty aspect is a regular polyhedron composed of 20 equilateral triangles, with a total of 12 vertices, 30 edges, and 20 faces.

What is the Nature of Icosahedron?

1. The circumscribed sphere, inscribed sphere and inscribed sphere of the icosahedron all exist, and the centers of the three spheres coincide.
2. The point where the outer, inner, and inner prisms of the icosahedron coincide is called the center of the icosahedron.
3. The straight line passing through the vertex of the icosahedron and the center of the regular polyhedron must pass through the other vertex of the icosahedron, and the distance between the two vertices and the center of the icosahedron is equal.
4. The two points connecting the center of the regular icosahedron are called opposite vertices, the two edges connecting the two pairs of opposite vertices are called opposite edges of the regular icosahedron, and the two faces surrounded by the opposite edges are called positive Opposite of the icosahedron.
5. The opposite edges and opposite sides of the icosahedron are parallel. 

Diagram of Icosahedron

Volume formula

(Where a is the edge length)

Inscribed regular dodecahedron

On a plane, when a regular polygon is connected to a circle, the more the number of sides, the higher the percentage of the circle area.

In three-dimensional space, this rule cannot be generalized-when regular dodecahedron and regular icosahedrons.

 When receiving a ball, the former accounted for about 66.4909%, the latter only accounted for 60.5461%. Some viruses, such as the herpesvirus family, have a capsid of icosahedron.

Regular icosahedron: 20 faces \ 12 vertices \ 30 edges
If the center of the icosahedron is (0,0,0), the radius of the circumscribed sphere is 1, the coordinates of each vertex are {(± m, 0, ± n), (0, ± n, ± m), (± n, ± m, 0)}, where

Calculation formula

Distance from body center to each vertex (radius of circumscribed ball) =
Distance from the center of the body to the center of each face (inscribed sphere radius) =
The distance from the center of the body to the midpoint of each edge (tangent sphere radius) =

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