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# Deltahedron

In geometry, 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 4, 6, 8, 10, 12, 14, 16 and 20 faces.[1] 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
ImageNameFacesEdgesVerticesVertex configurationsSymmetry group
tetrahedron4644 × 33Td, [3,3]
octahedron81266 × 34Oh, [4,3]
icosahedron20301212 × 35Ih, [5,3]
Johnson deltahedra
ImageNameFacesEdgesVerticesVertex configurationsSymmetry group
triangular bipyramid6952 × 33
3 × 34
D3h, [3,2]
pentagonal bipyramid101575 × 34
2 × 35
D5h, [5,2]
snub disphenoid121884 × 34
4 × 35
D2d, [2,2]
triaugmented triangular prism142193 × 34
6 × 35
D3h, [3,2]
gyroelongated square bipyramid1624102 × 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.[2] 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

Some smaller examples include:

Coplanar deltahedra
ImageNameFacesEdgesVerticesVertex configurationsSymmetry group
Augmented octahedron
Augmentation
1 tet + 1 oct
101571 × 33
3 × 34
3 × 35
0 × 36
C3v, [3]
4
3
12
Trigonal trapezohedron
Augmentation
2 tets + 1 oct
121882 × 33
0 × 34
6 × 35
0 × 36
C3v, [3]
612
Augmentation
2 tets + 1 oct
121882 × 33
1 × 34
4 × 35
1 × 36
C2v, [2]
2
2
2
117
Triangular frustum
Augmentation
3 tets + 1 oct
142193 × 33
0 × 34
3 × 35
3 × 36
C3v, [3]
1
3
1
96
Elongated octahedron
Augmentation
2 tets + 2 octs
1624100 × 33
4 × 34
4 × 35
2 × 36
D2h, [2,2]
4
4
126
Tetrahedron
Augmentation
4 tets + 1 oct
1624104 × 33
0 × 34
0 × 35
6 × 36
Td, [3,3]
464
Augmentation
3 tets + 2 octs
1827111 × 33
2 × 34
5 × 35
3 × 36
D2h, [2,2]
2
1
2
2
149
Edge-contracted icosahedron1827110 × 33
2 × 34
8 × 35
1 × 36
C2v, [2]
12
2
2210
Triangular bifrustum
Augmentation
6 tets + 2 octs
2030120 × 33
3 × 34
6 × 35
3 × 36
D3h, [3,2]
2
6
159
triangular cupola
Augmentation
4 tets + 3 octs
2233130 × 33
3 × 34
6 × 35
4 × 36
C3v, [3]
3
3
1
1
159
Triangular bipyramid
Augmentation
8 tets + 2 octs
2436142 × 33
3 × 34
0 × 35
9 × 36
D3h, [3]
695
Hexagonal antiprism2436140 × 33
0 × 34
12 × 35
2 × 36
D6d, [12,2
]
12
2
2412
Truncated tetrahedron
Augmentation
6 tets + 4 octs
2842160 × 33
0 × 34
12 × 35
4 × 36
Td, [3,3]
4
4
1812
Tetrakis cuboctahedron
Octahedron
Augmentation
8 tets + 6 octs
3248180 × 33
12 × 34
0 × 35
6 × 36
Oh, [4,3]
8126

## 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:

Other augmentations of the tetrahedron include:

Also by adding inverted pyramids to faces:

• Excavated dodecahedron

60 triangles 48 triangles Excavated dodecahedron A toroidal deltahedron

• Simplicial polytope - polytopes with all simplex facets

## References

[1]
Citation Linkopenlibrary.orgFreudenthal, H; van der Waerden, B. L. (1947), "Over een bewering van Euclides ("On an Assertion of Euclid")", Simon Stevin (in Dutch), 25: 115–128 (They showed that there are just 8 convex deltahedra. )
Sep 29, 2019, 10:12 AM
[2]
Citation Link//www.jstor.org/stable/2689647Trigg, Charles W. (1978), "An Infinite Class of Deltahedra", Mathematics Magazine, 51 (1): 55–57, JSTOR 2689647.
Sep 29, 2019, 10:12 AM
[3]
Citation Linkwww.interocitors.comThe Convex Deltahedra And the Allowance of Coplanar Faces
Sep 29, 2019, 10:12 AM
[4]
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[5]
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[6]
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[7]
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[8]
Sep 29, 2019, 10:12 AM
[9]
Citation Linkwww.interocitors.comThe Convex Deltahedra And the Allowance of Coplanar Faces
Sep 29, 2019, 10:12 AM
[10]