Deltaeder

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A deltahedron is a polyhedron that is limited exclusively by equilateral triangles that are congruent to one another .

There are 8 convex deltahedra. By putting together two deltahedra, any number of other deltahedra can be created, which, however, are generally not convex and the best-known representative of which is the star tetrahedron .

Convex deltahedron

Deltaeder Illustration Surfaces ( F ) Corners ( E ) Edges ( K )
Tetrahedron Tetrahedron.svg 4th 4th 6th
triangular bipyramid Triangular dipyramid.png 6th 5 9
octahedron Octahedron.svg 8th 6th 12
pentagonal bipyramid Pentagonal dipyramid.png 10 7th 15th
Trigondodecahedron Snub disphenoid.png 12 8th 18th
threefold expanded triangular prism Triaugmented triangular prism.png 14th 9 21st
twice expanded antiprism Gyroelongated square dipyramid.png 16 10 24
Icosahedron Icosahedron.svg 20th 12 30th

Since every surface meets three edges and, conversely, every edge meets two surfaces, 3 F = 2 K is guaranteed for a deltahedron . From Euler's polyhedron set E + F - K = 2, the formulas F = 2 ( E −2) and K = 3 ( E −2) result by eliminating K and F, respectively . Since a maximum of five surfaces meet each corner due to the convexity, but conversely, each surface has three corners, 5 E ≥ 3 F applies in any case , from which, together with F = 2 ( E −2), the inequality E ≤ 12 (hence F ≤ 20 and K ≤ 30).

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

Formally, the simple equilateral triangle with two sides, three corners and three edges could also be understood as a deltahedron, but the equilateral triangle lacks the property of a body. Starting from the equilateral triangle, each deltahedron is expanded to its predecessor by adding a corner and three edges. This can be clearly demonstrated with a set of balls and magnetic bars. However, this scheme is broken once. It is not possible to build a convex 18-surface from equilateral triangles.

14-wing

The 14-surface is constructed as follows: Take a regular prism whose base consists of an equilateral triangle and whose three side surfaces are squares . Place a pyramid with a square base on each of these three squares, the side surfaces of which are equilateral triangles that are identical in size to the equilateral triangle of the base of the prism.

16-wing

The 16-sided is constructed as follows: You take a regular antiprism whose base consists of a square and whose eight sides are equilateral triangles. Place a pyramid with a square base on each of the two square bases, the sides of which are equilateral triangles that are identical in size to the equilateral triangles on the sides of the antiprism.

Not convex deltahedron

The non-convex deltahedra include u. a. the boat , the cumulative tetrahedron, the cumulative hexahedron, and the star tetrahedron .

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