Rhombic Dodecahedron
The rhombic dodecahedron is a convex Polyèdre with 12 rhombic faces . It is a dual solid of a Solide of Archimedes or a Solide of Catalan. Its dual is the Cuboctaèdre.
Properties
It is the dual polyhedron Cuboctaèdre and a Zonoèdre. The large diagonal of each face is exactly √2 time the length of the small diagonal, thus, the acute angles of each face measure 2 tan− 1 (1/√2), or roughly 70,53°.
Being the dual one of a Solide of Archimedes, the rhombic dodecahedron is uniform faces, which means that the Groupe of symmetry of the solid transitively acts on the whole of the faces. In elementary terms, this means that for two unspecified faces has and B, it exists a rotation or a reflection of the solid which lets it occupy the same area of space by moving the face has towards the face B.
The rhombic dodecahedron is one of the nine convex polyhedrons with uniform faces, the others being the five solid of Plato, the Cuboctaèdre, the Icosidodécaèdre and the rhombic Triacontaèdre.
The rhombic dodecahedron can be used for to pave a space with three dimensions. It can be piled up to fill a space as the Hexagone S fill the plan; the cells in a network have a form similar to the rhombic dodecahedron cut by half.
This paving can be seen like the Diagramme of Voronoï of a cubic lattice to face centered. The Abeille S use the geometry of the rhombic dodecahedrons to form their honeycombs starting from the paving of the cells, each one of it is a hexagonal prism covered with half of a rhombic dodecahedron.
The rhombic dodecahedron forms the hull of the first projection by tops of a Tesseract towards 3 dimensions. There exist exactly two congruent manners of breaking up a dodecahedron rhombic into four Parallélépipède S, which gives 8 possible parallelepipeds. The 8 cells of the tesseract under this projection are precisely these 8 parallelepipeds.
Cartesian coordinates
The eights summon where three faces meet on their obtuse angles for coordinated Cartesian- (±1, ±1, ±1)
The six tops where the four faces meet on their acute angles are given by the permutations of
- (0, 0, ±2)
See too
References
- Robert Williams, The Geometrical Foundation off Natural Structure: With Book Source off Design, 1979, ISBN 0-486-23729-X
External bonds
- rhombic Dodecahedron – on the site MathWorld
- polyhedrons actually virtual the encyclopedia of the Polyhedrons
- rhombic dodecahedral Calendar – To build a rhombic dodecahedral calendar without adhesive.
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