Trapezohedron

The trapezohedron or antidiamant or deltoèdre N - gonal is the dual Polyèdre of a Antiprisme N - gonal regular. Its 2 N faces are deltoïdes adequate (or kite). The faces are shifted symmetrically.

The name trapezohedron is misleading since the faces are not Trapèze S, but the alternative term deltoèdre is sometimes confused with the term not connected Deltaèdre .

The part N - gonale of the name does not refer to the faces but to the arrangement of the tops around an axis of symmetry. Dual antiprism N - gonal has two faces N - gonales.

A trapezohedron N - gonal can be broken up into two pyramids N - gonales equal and a Antiprisme N - gonal.

In the texts describing the habitus in mineralogy, the word trapezohedron is often used to refer to the polyhedron known under the name trapezoidal Icositétraèdre.

Forms

  1. the trigonal Trapezohedron - 6 faces (rhombic) - dual: Octaèdre
  2. * a Cube is a particular case of trigonal trapezohedron whose faces are square.
  3. * a trigonal trapezohedron is a particular case of Rhomboèdre with adequate rhombic faces
  4. the Trapézoèdre tétragonal - 8 faces in kite - dual: square Antiprisme
  5. the pentagonal Trapezohedron - 10 faces in kite - dual: pentagonal Antiprisme
  6. the hexagonal Trapezohedron - 12 faces in kite - dual: hexagonal Antiprisme
  7. the heptagonal trapezohedron - 14 faces in kite - dual: heptagonal antiprism
  8. the Trapezohedron octagonal - 16 faces in kite - dual: Antiprisme octagonal
  9. the trapezohedron ennéagonal - 18 faces in kite - dual: antiprism ennéagonal
  10. the Trapezohedron décagonal - 20 faces in kite - dual: Antiprisme décagonal
  • … the trapezohedron n-gonal - 2n faces in kite - dual: Antiprisme n-gonal

In the case of the dual one of a regular antiprism triangular , the kite is rhombic, consequently, these trapezohedrons are also Zonoèdre S. They are called rhombohedron . They are Cube S dimensioned in the direction of a diagonal. They are also Parallélépipède S with adequate rhombic faces.

A particular case of rhombohedron is that whose rhombuses which form the faces have angles of 60° and 120°. It can be broken up into two equal regular tetrahedrons and a regular Octaèdre. Since the parallelepipeds can fill space, as can do it a combination of a regular tetrahedron and octahedral regular a.

Examples

  • the crystalline arrangements of the atoms can be repeated in space with trapezoedric cells.
  • the pentagonal Trapézoèdre is the first solid different from the solids of Plato used like a Dé in the Roleplays such as Donjons and Dragons. Having 10 dimensioned, it can be used in repetition to generate any discrete probability desired basic decimal.

Symmetry

The Group of symmetry of a trapezohedron N - gonal is Dnd of order 4 N , except in the case of a cube, which has a broader group of symmetry Od of order 48, which has four versions of D3d like sub-groups.

The Groupe of rotation is Dn of order 2 N , except in the case of a cube, which has the broader group of rotation O of order 24, which has four versions of D3 like sub-groups.

See too

External bonds

  • Polyhedral actually virtual the encyclopedia of the model polyhedrons
    • vrml (George Binder) <3> <4> <5> <6> <7> <9> <10>
    • Notation of Conway for the polyhedrons To test: " dA' " , where N =3,4,5… " example; dA5" is a pentagonal trapezohedron.
  • Owner out of paper of a trapezohedron tétragonal (square)

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