Deltahedron

A deltahedron is a polyhedron composed exclusively of equilateral triangular faces, with each face sharing an edge with every other face. The name derives from the Greek word *delta*, referring to the triangular shape, and *hedron*, meaning "face" in polyhedral terminology. Deltahedra are notable for their geometric symmetry and efficiency in space-filling properties.

Deltahedra can be categorized based on the number of faces they possess, ranging from the simplest tetrahedron (four faces) to more complex structures like the icosahedron (twenty faces). The regular deltahedra are those where all faces are congruent equilateral triangles and all vertices are identical in shape. There are five regular deltahedra: the tetrahedron (4 faces), triangular bipyramid (6 faces), pentagonal bipyramid (8 faces), pentagonal trapezohedron (10 faces), and icosahedron (20 faces). Non-regular deltahedra exist as well, such as the elongated square bipyramid, which has eight triangular faces but is not regular.

Deltahedra are significant in various fields, including crystallography, where they model certain crystal structures, and in mathematics, particularly in the study of convex polyhedra. Their properties make them useful in geometric constructions, such as the design of efficient packaging or the creation of stable frameworks. Additionally, deltahedra appear in molecular chemistry, where they describe the shapes of certain molecules, such as the boron hydrides. Their versatility and aesthetic appeal also make them popular in art and architecture.