Pseudoicosahedron
Encyclopedia
| Pseudoicosahedron | |
|---|---|
Four views of the pseudoicosahedron, with eight equilateral triangles (red and yellow), and 12 blue isosceles triangles. |
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| Faces Face (geometry) In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube... |
20 triangles:
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| Edges Edge (geometry) In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects.... |
30 (6 short + 24 long) |
| Vertices Vertex (geometry) In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:... |
12 |
| Symmetry group | Th, [4,3+], (3*2), order 24 Td Tetrahedral symmetry 150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group... , [3,3]+, (332), order 12 |
| Dual polyhedron Dual polyhedron In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another... |
Pyritohedron Pyritohedron In geometry, a pyritohedron is an irregular dodecahedron with pyritohedral symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular, and the structure has no fivefold symmetry axes... |
| Properties | convex Convex set In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object... |
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 pseudoicosahedron is a twelve-sided polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
that can be regarded as a particular form of distorted regular icosahedron containing tetrahedral symmetry
Tetrahedral symmetry
150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...
. The 20 triangular faces are divided into two groups of 8 equilateral triangles and 12 isosceles triangles.
Symmetry
If the 8 equilateral triangles are geometrically identical, the pseudoicosahedron has pyritohedral symmetry (3*2), with order 24. A lower tetrahedral symmetryTetrahedral symmetry
150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...
(332), exists as well, seen as the 8 triangles marked (colored) in alternating pairs of four, with order 12.
Cartesian coordinates
The coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the truncated octahedronTruncated octahedron
In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....
with alternated vertices deleted.
This construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector (φ, 1, 0), where φ is the golden ratio
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
.