Deltahedron

A deltahedra is a polyhedron, which is limited only by mutually congruent equilateral triangles.

There are 8 convex deltahedra. By assembling two deltahedra can be any number of other deltahedra produce, but are generally not convex, and whose most famous representative is the star tetrahedron.

Convex deltahedra

Since each surface at three edges and vice versa pushes every edge in two areas is considered in a deltahedra always guarantees 3F = 2K. From the Eulerian Polyedersatz E F -K = 2 then result further by eliminating K and F the formulas F = 2 (E -2) and K = 3 (E -2). As a result of the convexity at each corner bump up to five faces, but vice versa, each face has three corners, definitely applies 5E ≥ 3F, from which together with F = 2 ( E-2), the inequality E ≤ 12 (hence F ≤ 20 and K ≤ 30 ) results.

Three of the eight existing convex deltahedra are platonic solids (namely tetrahedron, octahedron and icosahedron ). The remaining five are deltahedra Johnson body.

Formally, one could simple equilateral triangle with two sides, three corners and three edges also be regarded as deltahedra, just missing the equilateral triangle is the property of a body. Starting from the equilateral triangle each deltahedra is extended by adding one corner and three edges to its predecessor. This can be traced clearly with a set of balls and magnetic rods. However, this pattern is broken once. It is not possible to build a convex 18- Flächner of equilateral triangles.

14 - Flächner

The 14- Flächner is constructed as follows: take a regular prism whose base is an equilateral triangle and its three faces are squares. In this three squares you sit now each a pyramid with a square base whose side faces are equilateral triangles that are identical to the equilateral triangles of the base of the prism in size.

16- Flächner

The 16- Flächner is constructed as follows: take a regular antiprism whose base consists of a square and whose eight faces are equilateral triangles. The two square bases to sit now each a pyramid with a square base whose side faces are equilateral triangles that are identical to the equilateral triangles of sides of the antiprism in size.

Non convex deltahedra

Among the non- convex deltahedra include, inter alia, the boat, the cumulative tetrahedron, hexahedron, the cumulative and the star tetrahedron.