Deltahedron
![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Icosahedron.svg/220px-Icosahedron.svg.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Triangulated_truncated_tetrahedron.png/220px-Triangulated_truncated_tetrahedron.png)
In geometry, a deltahedron (plural deltahedra) is a polyhedron whose faces are all (congruent) equilateral triangles. The name is taken from the Greek upper case delta letter (Δ), which has the shape of an equilateral triangle. There are infinitely many deltahedra. By the handshaking lemma, each deltahedron has an even number of faces. Only eight deltahedra are strictly convex; these have 4, 6, 8, 10, 12, 14, 16, and 20 faces.[1] These eight deltahedra, with their respective numbers of faces, edges, and vertices, are listed below.
The eight strictly convex deltahedra
There are eight strictly convex deltahedra: three are regular polyhedra and Platonic solids, five are Johnson solids.
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 Triakis octahedron
(Stella octangula)Pentakis dodecahedron Triakis icosahedron Faces 12 24 60
Other non-convex deltahedra can be generated by assembling several regular tetrahedra:
Some non-convex equilateral augmented tetrahedra Image Faces 8 10 12
Like all toroidal polyhedra, toroidal deltahedra are non-convex; example:
A toroidal deltahedron48
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:
- Great icosahedron — a Kepler-Poinsot solid, with 20 intersecting triangles:
See also
- Simplicial polytope — polytope with all simplex facets
References
- ^ Freudenthal, 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 eight strictly convex deltahedra.)
- ^ Trigg, Charles W. (1978), "An Infinite Class of Deltahedra", Mathematics Magazine, 51 (1): 55–57, doi:10.1080/0025570X.1978.11976675, 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.
- Cundy, H. Martyn (December 1952), "Deltahedra", Mathematical Gazette, 36 (318): 263–266, doi:10.2307/3608204, 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.
External links
- Weisstein, Eric W., "Deltahedron", MathWorld
- The eight convex deltahedra
- Deltahedron
- Deltahedron