Deltahedron

From Wikipedia, the free encyclopedia
Jump to: navigation, search
The largest strictly-convex deltahedron is the regular icosahedron
This is a truncated tetrahedron with hexagons subdivided into triangles. This figure is not a strictly-convex deltahedron since coplanar faces are not allowed within the definition.

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.(Freudenthal 1947) The number of faces, edges, and vertices is listed below for each of the eight convex deltahedra.

The deltahedra should not be confused with the deltohedra (spelled with an "o"), polyhedra whose faces are geometric kites.

The eight convex deltahedra[edit]

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.jpg tetrahedron 4 6 4 4 × 33 Td, [3,3]
Octahedron.svg octahedron 8 12 6 6 × 34 Oh, [4,3]
Icosahedron.jpg icosahedron 20 30 12 12 × 35 Ih, [5,3]
Johnson deltahedra
Image Name Faces Edges Vertices Vertex configurations Symmetry group
Triangular dipyramid.png triangular bipyramid 6 9 5 2 × 33
3 × 34
D3h, [3,2]
Pentagonal dipyramid.png pentagonal bipyramid 10 15 7 5 × 34
2 × 35
D5h, [5,2]
Snub disphenoid.png snub disphenoid 12 18 8 4 × 34
4 × 35
D2d, [2,2]
Triaugmented triangular prism.png triaugmented triangular prism 14 21 9 3 × 34
6 × 35
D3h, [3,2]
Gyroelongated square dipyramid.png 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.[1] 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[edit]

There are infinitely many cases with coplanar triangles, allowing for sections of the infinite triangular tilings. The coplanar triangular faces can be merged into rhombic, trapezoidal, hexagonal, or other equilateral polygon faces.[2] If the sets of coplanar triangles are considered a single face (called a triamond[3]), a smaller set of faces, edges, and vertices can be counted. Triamond faces used must be convex, including: Polyiamond-1-1.svg, Polyiamond-2-1.svg, Polyiamond-3-1.svg, Polyiamond-4-2.svg, Polyiamond-4-3.svg, Polyiamond-5-1.svg, Polyiamond-6-1.svg and Polyiamond-6-11.svg, ...

Some smaller examples include:

Coplanar deltahedra
Image Name Faces Edges Vertices Vertex configurations Symmetry group
Augmented octahedron.png Augmented octahedron
Augmentation
1 tet + 1 oct
10 Polyiamond-1-1.svg 15 7 1 × 33
3 × 34
3 × 35
0 × 36
C3v, [3]
4 Polyiamond-1-1.svg
3 Polyiamond-2-1.svg
12
Gyroelongated triangular bipyramid.png Trigonal trapezohedron
Augmentation
2 tets + 1 oct
12 Polyiamond-1-1.svg 18 8 2 × 33
0 × 34
6 × 35
0 × 36
C3v, [3]
6 Polyiamond-2-1.svg 12
Tet2Oct solid.png Augmentation
2 tets + 1 oct
12 Polyiamond-1-1.svg 18 8 2 × 33
1 × 34
4 × 35
1 × 36
C2v, [2]
2 Polyiamond-1-1.svg
2 Polyiamond-2-1.svg
2 Polyiamond-3-1.svg
11 7
Triangulated monorectified tetrahedron.png Triangular frustum
Augmentation
3 tets + 1 oct
14 Polyiamond-1-1.svg 21 9 3 × 33
0 × 34
3 × 35
3 × 36
C3v, [3]
1 Polyiamond-1-1.svg
3 Polyiamond-3-1.svg
1 Polyiamond-4-3.svg
9 6
TetOct2 solid2.png Elongated octahedron
Augmentation
2 tets + 2 octs
16 Polyiamond-1-1.svg 24 10 0 × 33
4 × 34
4 × 35
2 × 36
D2h, [2,2]
4 Polyiamond-1-1.svg
4 Polyiamond-3-1.svg
12 6
Triangulated tetrahedron.png Tetrahedron
Augmentation
4 tets + 1 oct
16 Polyiamond-1-1.svg 24 10 4 × 33
0 × 34
0 × 35
6 × 36
Td, [3,3]
4 Polyiamond-4-3.svg 6 4
Tet3Oct2 solid.png Augmentation
3 tets + 2 octs
18 Polyiamond-1-1.svg 27 11 1 × 33
2 × 34
5 × 35
3 × 36
D2h, [2,2]
2 Polyiamond-1-1.svg
1 Polyiamond-2-1.svg
2 Polyiamond-3-1.svg
2 Polyiamond-4-2.svg
14 9
Double diminished icosahedron.png Edge-contracted icosahedron 18 Polyiamond-1-1.svg 27 11 0 × 33
2 × 34
8 × 35
1 × 36
C2v, [2]
12 Polyiamond-1-1.svg
2 Polyiamond-3-1.svg
22 10
Triangulated truncated triangular bipyramid.png Triangular bifrustum
Augmentation
6 tets + 2 octs
20 Polyiamond-1-1.svg 30 12 0 × 33
3 × 34
6 × 35
3 × 36
D3h, [3,2]
2 Polyiamond-1-1.svg
6 Polyiamond-3-1.svg
15 9
Augmented triangular cupola.png triangular cupola
Augmentation
4 tets + 3 octs
22 Polyiamond-1-1.svg 33 13 0 × 33
3 × 34
6 × 35
4 × 36
C3v, [3]
3 Polyiamond-1-1.svg
3 Polyiamond-3-1.svg
1 Polyiamond-4-3.svg
1 Polyiamond-6-11.svg
15 9
Triangulated bipyramid.png Triangular bipyramid
Augmentation
8 tets + 2 octs
24 Polyiamond-1-1.svg 36 14 2 × 33
3 × 34
0 × 35
9 × 36
D3h, [3]
6 Polyiamond-4-3.svg 9 5
Augmented hexagonal antiprism flat.png Hexagonal antiprism 24 Polyiamond-1-1.svg 36 14 0 × 33
0 × 34
12 × 35
2 × 36
D6d, [12,2+]
12 Polyiamond-1-1.svg
2 Polyiamond-6-11.svg
24 12
Triangulated truncated tetrahedron.png Truncated tetrahedron
Augmentation
6 tets + 4 octs
28 Polyiamond-1-1.svg 42 16 0 × 33
0 × 34
12 × 35
4 × 36
Td, [3,3]
4 Polyiamond-1-1.svg
4 Polyiamond-6-11.svg
18 12
Triangulated octahedgon.png Tetrakis cuboctahedron
Octahedron
Augmentation
8 tets + 4 octs
32 Polyiamond-1-1.svg 24 18 0 × 33
12 × 34
0 × 35
6 × 36
Oh, [4,3]
8 Polyiamond-4-3.svg 12 6

Non-convex forms[edit]

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:

Also by adding inverted pyramids to faces:

Great icosahedron.png
Great icosahedron
(20 intersecting triangles)
Stella octangula.png
Stella octangula
(24 triangles)
Third stellation of icosahedron.png
Excavated dodecahedron
(60 triangles)
Toroidal polyhedron.gif
A toroidal deltahedron
(48 triangles)

See also[edit]

External links[edit]

References[edit]

  1. ^ Trigg, Charles W. (1978), "An Infinite Class of Deltahedra", Mathematics Magazine 51 (1): 55–57, JSTOR 2689647 .
  2. ^ The Convex Deltahedra And the Allowance of Coplanar Faces
  3. ^ http://www.interocitors.com/polyhedra/Triamonds/
  • 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 8 convex deltahedra. )
  • H. Martyn Cundy Deltahedra. Math. Gaz. 36, 263-266, Dec 1952. [1]
  • H. Martyn Cundy and A. Rollett Deltahedra. §3.11 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 142–144, 1989.
  • Charles W. Trigg An Infinite Class of Deltahedra, Mathematics Magazine, Vol. 51, No. 1 (Jan., 1978), pp. 55–57 [2]
  • M. Gardner Fractal Music, Hypercards, and More: Mathematical Recreations, Scientific American Magazine. New York: W. H. Freeman, pp. 40, 53, and 58-60, 1992.
  • Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7.  pp. 35–36