Deltahedron
![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Icosahedron.svg/220px-Icosahedron.svg.png)
In
The eight strictly convex deltahedra
There are eight strictly
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] |
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, [4,2+] |
![]() |
Triaugmented triangular prism | 14 ![]() |
21 | 9 | 3 × 34 6 × 35 |
D3h, [3,2] |
![]() |
Gyroelongated square bipyramid | 16 ![]() |
24 | 10 | 2 × 34 8 × 35 |
D4d, [8,2+] |
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 non-regular deltahedra belong to the class of Johnson solids: non-uniform strictly convex polyhedra with regular polygons for faces.
A deltahedron retains its shape: even if its edges are free to rotate around their vertices (so that the angles between them are fluid), they don't move. Not all polyhedra have this property: for example, if some of the angles of a cube are relaxed, it can be deformed into a non-right square prism or even into a rhombohedron with no right angle at all.
There is no 18-faced strictly convex deltahedron.[2] However, the edge-contracted icosahedron gives an example of an octadecahedron that can either be made strictly convex with 18 irregular triangular faces, or made equilateral with 18 (regular) triangular faces that include two sets of three coplanar triangles.
Non-strictly convex cases
There are infinitely many cases with coplanar triangles, allowing for convex sections of the infinite triangular tiling. 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 polygonal faces. Each face must be a convex polyiamond, such as ,
,
,
,
,
,
,
, ...[3]
Some small examples include:
Image | Name | Faces | Edges | Vertices | Vertex configurations | Symmetry group |
---|---|---|---|---|---|---|
![]() |
Augmented octahedron Augmentation 1 tet + 1 oct |
10 ![]() |
15 | 7 | 1 × 33 3 × 34 3 × 35 0 × 36 |
C3v, [3] |
4 ![]() 3 ![]() |
12 | |||||
![]() |
Trigonal trapezohedron Augmentation 2 tets + 1 oct |
12 ![]() |
18 | 8 | 2 × 33 0 × 34 6 × 35 0 × 36 |
D3d, [6,2+] |
6 ![]() |
12 | |||||
![]() |
Augmentation 2 tets + 1 oct |
12 ![]() |
18 | 8 | 2 × 33 1 × 34 4 × 35 1 × 36 |
C2v, [2] |
2 ![]() 2 ![]() 2 ![]() |
11 | 7 | ||||
![]() |
Triangular frustum Augmentation 3 tets + 1 oct |
14 ![]() |
21 | 9 | 3 × 33 0 × 34 3 × 35 3 × 36 |
C3v, [3] |
1 ![]() 3 ![]() 1 ![]() |
9 | 6 | ||||
![]() |
Elongated octahedron Augmentation 2 tets + 2 octs |
16 ![]() |
24 | 10 | 0 × 33 4 × 34 4 × 35 2 × 36 |
D2h, [2,2] |
4 ![]() 4 ![]() |
12 | 6 | ||||
![]() |
Tetrahedron Augmentation 4 tets + 1 oct |
16 ![]() |
24 | 10 | 4 × 33 0 × 34 0 × 35 6 × 36 |
Td, [3,3] |
4 ![]() |
6 | 4 | ||||
![]() |
Augmentation 3 tets + 2 octs |
18 ![]() |
27 | 11 | 1 × 33 2 × 34 5 × 35 3 × 36 |
{Id,R} where R is a reflection through a plane |
2 ![]() 1 ![]() 2 ![]() 2 ![]() |
14 | 9 | ||||
![]() |
Edge-contracted icosahedron | 18 ![]() |
27 | 11 | 0 × 33 2 × 34 8 × 35 1 × 36 |
C2v, [2] |
12 ![]() 2 ![]() |
22 | 10 | ||||
![]() |
Triangular bifrustum Augmentation 6 tets + 2 octs |
20 ![]() |
30 | 12 | 0 × 33 3 × 34 6 × 35 3 × 36 |
D3h, [3,2] |
2 ![]() 6 ![]() |
15 | 9 | ||||
![]() |
Triangular cupola Augmentation 4 tets + 3 octs |
22 ![]() |
33 | 13 | 0 × 33 3 × 34 6 × 35 4 × 36 |
C3v, [3] |
3 ![]() 3 ![]() 1 ![]() 1 ![]() |
15 | 9 | ||||
![]() |
Triangular bipyramid Augmentation 8 tets + 2 octs |
24 ![]() |
36 | 14 | 2 × 33 3 × 34 0 × 35 9 × 36 |
D3h, [3,2] |
6 ![]() |
9 | 5 | ||||
![]() |
Hexagonal antiprism | 24 ![]() |
36 | 14 | 0 × 33 0 × 34 12 × 35 2 × 36 |
D6d, [12,2+] |
12 ![]() 2 ![]() |
24 | 12 | ||||
![]() |
Truncated tetrahedron Augmentation 6 tets + 4 octs |
28 ![]() |
42 | 16 | 0 × 33 0 × 34 12 × 35 4 × 36 |
Td, [3,3] |
4 ![]() 4 ![]() |
18 | 12 | ||||
![]() |
Tetrakis cuboctahedron Octahedron Augmentation 8 tets + 6 octs |
32 ![]() |
48 | 18 | 0 × 33 12 × 34 0 × 35 6 × 36 |
Oh, [4,3] |
8 ![]() |
12 | 6 |
Non-convex forms
There are an infinite number of non-convex deltahedra.
Five non-convex deltahedra can be generated by adding an equilateral pyramid to every face of a Platonic solid:
Equilateral pyramid-augmented Platonic solids Image Name Triakis tetrahedron Tetrakis hexahedron Stella octangula)Pentakis dodecahedron Triakis icosahedron Faces 12 24 60
Other non-convex deltahedra can be generated by assembling several
Like all
When possible, adding an inverted equilateral pyramid to every face of a polyhedron makes a non-convex deltahedron; example:
Like all self-intersecting polyhedra, self-intersecting deltahedra are non-convex; example:
See also
- Simplicial polytope — polytope with all simplex facets
References
- 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 eight strictly convex deltahedra.)
- JSTOR 2689647.
- ^ The Convex Deltahedra And the Allowance of Coplanar Faces
Further reading
- Rausenberger, O. (1915), "Konvexe pseudoreguläre Polyeder", Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, 46: 135–142.
- JSTOR 3608204.
- Cundy, H. Martyn; Rollett, A. (1989), "3.11. Deltahedra", Mathematical Models (3rd ed.), Stradbroke, England: Tarquin Pub., pp. 142–144.
- Gardner, Martin (1992), Fractal Music, Hypercards, and More: Mathematical Recreations from Scientific American, New York: W. H. Freeman, pp. 40, 53, and 58–60.
- Pugh, Anthony (1976), Polyhedra: A visual approach, California: University of California Press Berkeley, ISBN 0-520-03056-7, pp. 35–36.