Descriptions and Examples for the POV-Ray Raytracer by Friedrich A. Lohmüller
Elementary Geometry for Raytracing
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    for Raytracing

    Right-angled Triangle
    Pythagorean Theorem
    Trigonometry Basics
    Law of cosines
    Equilateral Triangle
    Regular Polygon
    Polyhedron
      Tetrahedron
      Octahedron
      Cube & Cuboid
      Dodecahedron
      Icosahedron
      Cuboctahedron
      Truncated Octahedron
      Rhombicuboctahedron
      Truncated Icosahedron
    Circles
      Tangent circles
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   - Geometric 3D Animations

                                               

Regular Icosahedron
Some useful geometrical facts


In the following we write for the square root of a number the expression "sqrt(ZAHL)"
conforming to the syntax used in POV-Ray.


Dimensions of a regular icosahedron
Length of an edge of the icosahedron: a.
The radius of circumsphere:
R = a / 4 * sqrt( 10 + 2*sqrt(5) );
The radius of edgesphere (tangent to edges):
Re = a / 4 *( 1 + sqrt(5) );

The radius of the insphere:
Ri = a / 12 * sqrt(3)* ( 3 + sqrt(5) );

The angle between two faces: ~138.19°
Face_Angle =
degrees(acos(-1/3*sqrt(5)));


The angle between two edges: 108°
Face_Edge_Angle =
degrees(acos(1/4*(1 - sqrt(5))));


        Flolding of a regular icosahedron
Inside the regular icosahedron
Inside the regular icosahedron
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© Friedrich A. Lohmüller, 2009
http://www.f-lohmueller.de