 Deltahedron
Get Deltahedron essential facts below. View Videos or join the Deltahedron discussion. Add Deltahedron to your PopFlock.com topic list for future reference or share this resource on social media.
Deltahedron

In geometry, a deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (?), which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces. The number of faces, edges, and vertices is listed below for each of the eight convex deltahedra.

## The eight convex deltahedra

There are only eight strictly-convex deltahedra: three are regular polyhedra, and five are Johnson solids.

Regular deltahedra
Image Name Faces Edges Vertices Vertex configurations Symmetry group tetrahedron 4 6 4 4 × 33 Td, [3,3] octahedron 8 12 6 6 × 34 Oh, [4,3] icosahedron 20 30 12 12 × 35 Ih, [5,3]
Johnson deltahedra
Image Name Faces Edges Vertices Vertex configurations Symmetry group triangular bipyramid 6 9 5 2 × 33
3 × 34
D3h, [3,2] pentagonal bipyramid 10 15 7 5 × 34
2 × 35
D5h, [5,2] snub disphenoid 12 18 8 4 × 34
4 × 35
D2d, [2,2] triaugmented triangular prism 14 21 9 3 × 34
6 × 35
D3h, [3,2] gyroelongated square bipyramid 16 24 10 2 × 34
8 × 35
D4d, [4,2]

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.

There is no 18-faced convex deltahedron. However, the edge-contracted icosahedron gives an example of an octadecahedron that can either be made convex with 18 irregular triangular faces, or made with equilateral triangles that include two coplanar sets of three triangles.

## Non-strictly convex cases

There are infinitely many cases with coplanar triangles, allowing for sections of the infinite triangular tilings. If the sets of coplanar triangles are considered a single face, a smaller set of faces, edges, and vertices can be counted. The coplanar triangular faces can be merged into rhombic, trapezoidal, hexagonal, or other equilateral polygon faces. Each face must be a convex polyiamond such as , , , , , , and , ...

Some smaller examples include:

## Non-convex forms

There are an infinite number of nonconvex forms.

Some examples of face-intersecting deltahedra:

Other nonconvex deltahedra can be generated by adding equilateral pyramids to the faces of all 5 regular polyhedra:

triakis tetrahedron tetrakis hexahedron triakis octahedron(stella octangula) pentakis dodecahedron triakis icosahedron     Other augmentations of the tetrahedron include:

Also by adding inverted pyramids to faces:

60 triangles 48 triangles Excavated dodecahedron A toroidal deltahedron