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Deltahedron

(Redirected from Deltahedra)
Not to be confused with Deltohedron.

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 largest strictly convex deltahedron is the regular icosahedron.
The (Archimedean) truncated tetrahedron with its hexagons subdivided into triangles. This figure is not a strictly convex deltahedron, since coplanar (adjacent) faces are not allowed within the definition.

The eight strictly convex deltahedra edit

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

Regular strictly convex 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]
Non-regular strictly convex 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, [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 edit

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:

Deltahedra with coplanar faces
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 edit

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 deltahedron
48  

When possible, adding an inverted equilateral pyramid to every face of a polyhedron makes a non-convex deltahedron; example:

 
Excavated dodecahedron
60  

Like all self-intersecting polyhedra, self-intersecting deltahedra are non-convex; example:

Great icosahedron — a Kepler-Poinsot solid, with 20 intersecting triangles:
 

See also edit

References edit

  1. ^ 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.)
  2. ^ Trigg, Charles W. (1978), "An Infinite Class of Deltahedra", Mathematics Magazine, 51 (1): 55–57, doi:10.1080/0025570X.1978.11976675, JSTOR 2689647.
  3. ^ The Convex Deltahedra And the Allowance of Coplanar Faces

Further reading edit

  • Rausenberger, O. (1915), "Konvexe pseudoreguläre Polyeder", Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, 46: 135–142.
  • Cundy, H. Martyn (December 1952), "Deltahedra", Mathematical Gazette, 36: 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 edit

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