Deltahedron Totally Explained

Everything about Deltahedron totally explained

A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta (Δ), which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having four, six, eight, ten, twelve, fourteen, sixteen, and twenty faces. The number of faces, edges, and vertices is listed below for each of the eight convex deltahedra.
The deltahedra shouldn't be confused with the deltohedra (spelled with an "o"), polyhedra whose faces are geometric kites.

The eight convex deltahedra

Name	Faces	Edges	Vertices	Vertex configurations	Symmetry group
regular tetrahedron	4	6	4	4 × 3³	$T_d$
triangular dipyramid	6	9	5	2 × 3³ 3 × 3⁴	$D_$
regular icosahedron	20	30	12	12 × 3⁵	$I_h$

Only three of the deltahedra are Platonic solids (polyhedra in which the number of faces meeting at each vertex is constant). These are:

the 4-faced deltahedron (or tetrahedron), in which three faces meet at each vertex
the 8-faced deltahedron (or octahedron), in which four faces meet at each vertex
the 20-faced deltahedron (or icosahedron), in which five faces meet at each vertex

In the 6-faced deltahedron, some vertices have degree 3 and some degree 4. In the 10-, 12-, 14-, and 16-faced deltahedra, some vertices have degree 4 and some degree 5. These five irregular deltahedra belong to the class of Johnson solids: convex polyhedra with regular polygons for faces.
Deltahedra retain their shape, even if the edges are free to rotate around their vertices so that the angles between edges are fluid. Not all polyhedra have this property: for example, if you relax some of the angles of a cube, the cube can be deformed into a non-right square prism.

Non-convex forms

There are an infinite number of nonconvex forms.
Some examples of non-convex deltahedra:

Great icosahedron - a Kepler-Poinsot solid. Others can be generated by adding equilateral pyramids to the faces of all 5 regular polyhedra:

Equilateral triakis tetrahedron

Equilateral tetrakis hexahedron

Equilateral triakis octahedron (stella octangula)

Equilateral pentakis dodecahedron

Equilateral triakis icosahedron Also by adding inverted pyramids to faces:

Wenninger's third stellated icosahedra

Great icosahedron
(20 intersecting triangles)

stella octangula
(24 triangles)

Third stellation of icosahedron
(60 triangles)

A toroidal deltahedron
(48 triangles)

Further Information

Get more info on 'Deltahedron'.

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