geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a deltahedron (

plural The plural (sometimes list of glossing abbreviations, abbreviated pl., pl, or ), in many languages, is one of the values of the grammatical number, grammatical category of number. The plural of a noun typically denotes a quantity greater than the ...

''deltahedra'') is a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...

whose

face The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...

s are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle. There are infinitely many deltahedra, all having an even number of faces by the handshaking lemma. Of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces. 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 solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnso ...

s. The three regular convex polyhedra are indeed Platonic solids. 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

s: 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 In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

, the cube can be deformed into a non-right square prism. There is no 18-faced convex deltahedron. 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

There are infinitely many cases with coplanar triangles, allowing for sections of the infinite

triangular tiling In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...

s. 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 equilateral polygon faces. Each face must be a convex polyiamond such as Polyiamond-1-1

and

, ...The Convex Deltahedra And the Allowance of Coplanar Faces
/ref> Some smaller examples include:

Non-convex forms

There are an infinite number of nonconvex forms. Some examples of face-intersecting deltahedra: * Great icosahedron - a Kepler-Poinsot solid, with 20 intersecting triangles *: Other nonconvex deltahedra can be generated by adding equilateral pyramids to the faces of all 5 Platonic solids: Other augmentations of the tetrahedron include: Also by adding inverted pyramids to faces: * Excavated dodecahedron

References

External links

* {{mathworld , urlname = Deltahedron, title = Deltahedron, mode=cs2
The eight convex deltahedra

* Polyhedra

The eight convex deltahedra

Non-strictly convex cases

Non-convex forms

See also

References

Further reading

External links