Deltaèdre

A deltaèdre is a Polyèdre of which all the faces are equilateral triangles. The name is resulting from the capital letter of the Greek delta (Δ), who with the shape of an equilateral triangle. There exists an infinity of deltaèdres, but of those, only eights are convex S, having four, six, eight, ten, twelve, fourteen, sixteen and twenty faces. The number of faces, edges and Sommet S is listed below for each of the eight convex deltaèdres.

The deltaèdre should not be confused with the Deltoèdre S (spelled with a " o"), the polyhedrons whose faces are kite.

Eight convex deltaèdres

Only three deltaèdres are solid of Plato (polyhedral in whom the number of faces meeting at each top is constant):

  • the deltaèdre with 4 faces (or Tétraèdre), in which three faces meets at each top
  • the deltaèdre with 8 faces (or Octaèdre), in which four faces meet at each top
  • the deltaèdre with 20 faces (or Icosaèdre), in which five faces meet at each top

In the deltaèdre with 6 faces, certain tops are of degree 3 and some of degree 4. In the deltaèdres with 10,12,14 and 16 faces, certain tops are degrees 4 and some of degree 5. These five deltaèdres irregular belong to the class of the solid of Johnson: the convex polyhedrons whose faces are regular polygons.

The deltaèdres maintain their form, even if the edges are free to turn around their tops, i.e. the angles between the edges are fluid. The polyhedrons do not have all this property: for example, if you slacken certain angles of the Cube, the cube can be deformed in a square prism not right.

Not-convex forms

There exists an infinite number of not-convex forms.

Some not-convex examples of deltaèdres:

Others can be generated by adding equilateral pyramids to the faces of these five regular polyhedrons:

  1. equilateral Triakitétraèdre
  2. equilateral Tétrakihexaèdre
  3. equilateral Triakioctaèdre (spangled Octangle)
  4. equilateral Pentakidodécaèdre
  5. equilateral Triaki-icosahedron

Moreover, by adding pyramids reversed to the faces:

  • Third stellation of Wenninger of the Icosahedral

External bonds

  • MathWorld
  • eight convex deltaèdres
  • Deltaèdre

References

  • H. Martyn Cundy Deltahedra. Maths. Gas. 36,263-266, DEC 1952. * H. Martyn Cundy and A. Rollett Deltahedra. §3.11 in Mathematical Models, 3rd ED. Stradbroke, England: Tarquin Pub., pp. 142-144, 1989.
  • Charles W. Trigg Year Infinity Class off Deltahedra , Mathematics Magazine, vol. 51, No 1 (Jan., 1978), pp. 55-57 * Mr. Gardner Fractal Music, Hypercards, and More: Mathematical Re-creations , Scientific American Magazine. New York: W.H. Freeman, pp. 40,53, and 58-60, 1992.
  • A. Pugh Polyhedra: In Visual Approach. Berkeley, CA: University off California Close, pp. 35-36, 1976.

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