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Pyritohedron

 

 
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Pyritohedron
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a pyritohedron is an irregular dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
. Like the regular dodecahedron, it has twelve identical pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
al faces, with three meeting in each of the 20 corners. However, the pentagons are not regular, and the structure has no five-fold symmetry axes; instead, it has a particular kind of tetrahedral symmetry
Tetrahedral symmetry

A regular tetrahedron has 12 rotational symmetries, and a total of 24 symmetries including transformations that combine a reflection and a rotation....
. It is one of the two common crystal forms of pyrite
Pyrite

The mineral pyrite, or iron pyrite, is an iron sulfide with the chemical formula ironsulfur2. This mineral's metallic Lustre and pale-to-normal, brass-yellow hue have earned it the nickname fool's gold due to its resemblance to gold....
, the other one being cubical.

A regular dodecahedron can be formed from a cube in the following way: The top square in the cube is replaced by a "roof" composed of two pentagons, joined along the top of the roof.






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a pyritohedron is an irregular dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
. Like the regular dodecahedron, it has twelve identical pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
al faces, with three meeting in each of the 20 corners. However, the pentagons are not regular, and the structure has no five-fold symmetry axes; instead, it has a particular kind of tetrahedral symmetry
Tetrahedral symmetry

A regular tetrahedron has 12 rotational symmetries, and a total of 24 symmetries including transformations that combine a reflection and a rotation....
. It is one of the two common crystal forms of pyrite
Pyrite

The mineral pyrite, or iron pyrite, is an iron sulfide with the chemical formula ironsulfur2. This mineral's metallic Lustre and pale-to-normal, brass-yellow hue have earned it the nickname fool's gold due to its resemblance to gold....
, the other one being cubical.

A regular dodecahedron can be formed from a cube in the following way: The top square in the cube is replaced by a "roof" composed of two pentagons, joined along the top of the roof. The diagonals in the pentagons parallel to the top of the roof coincide with two opposite sides of the square. The other five squares are replaced by a pair of pentagons in a similar way. The pyritohedron is constructed by changing the slope of these "roofs".

External links

  • VRML models and animations of Pyritohedron and its stellation
    Stellation

    Stellation is a process of constructing new polygons , new polyhedron in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again....
    s.