Deltahedron


A deltahedron is a polyhedron whose faces are all equilateral triangles. The deltahedron was named by Martyn Cundy, after the Greek capital letter delta resembling a triangular shape Δ.
Deltahedra can be categorized by the property of convexity. The simplest convex deltahedron is the regular tetrahedron, a pyramid with four equilateral triangles. There are eight convex deltahedra, which can be used in the applications of chemistry as in the polyhedral skeletal electron pair theory and chemical compounds. There are infinitely many concave deltahedra.

Strictly convex deltahedron

A polyhedron is said to be convex if a line between any two of its vertices lies either within its interior or on its boundary, and additionally, if no two faces are coplanar and no two edges are collinear, it can be considered as being strictly convex.
Of the eight convex deltahedra, three are Platonic solids and five are Johnson solids. They are:
The number of possible convex deltahedrons was given by. Rausenberger named these solids as the convex pseudoregular polyhedra.
Summarizing the examples above, the deltahedra can be conclusively defined as the class of polyhedra whose faces are equilateral triangles. Another definition by is similar to the previous one, in which he was interested in the shapes of holes left in irregular close-packed arrangements of spheres. It is stated as a convex polyhedron with equilateral triangular faces that can be formed by the centers of a collection of congruent spheres, whose tangencies represent polyhedron edges, and such that there is no room to pack another sphere inside the cage created by this system of spheres. Because of this restriction, some polyhedrons may not be included as a deltahedron: the triangular bipyramid, pentagonal bipyramid, and regular icosahedron.
Most convex deltahedra can be found in the study of chemistry. For example, they are categorized as the closo polyhedron in the study of polyhedral skeletal electron pair theory. Other applications of deltahedra—excluding the regular icosahedron—are the visualization of an atom cluster surrounding a central atom as a polyhedron in the study of chemical compounds: regular tetrahedron represents the tetrahedral molecular geometry, triangular bipyramid represents trigonal bipyramidal molecular geometry, regular octahedron represents the octahedral molecular geometry, pentagonal bipyramid represents the pentagonal bipyramidal molecular geometry, gyroelongated square bipyramid represents the bicapped square antiprismatic molecular geometry, triaugmented triangular prism represents the tricapped trigonal prismatic molecular geometry, and snub disphenoid represents the dodecahedral molecular geometry. The regular icosahedron along with some other deltahedra appears in the geometry of boron hydride clusters.

Non-convex deltahedron

A non-convex deltahedron is a deltahedron that does not possess convexity, thus it has either coplanar faces or collinear edges. There are infinitely many non-convex deltahedra. Some examples are stella octangula, the excavated dodecahedron, and Boerdijk–Coxeter helix.
There are subclasses of non-convex deltahedra. shows that they may be discovered by finding the number of varying vertex's types. A set of vertices is considered the same type as long as there are subgroups of the polyhedron's same group transitive on the set. Cundy shows that the great icosahedron is the only non-convex deltahedron with a single type of vertex. There are seventeen non-convex deltahedra with two types of vertex, and soon the other eleven deltahedra were later added by, Other subclasses are the isohedral deltahedron that was later discovered by both and, and the spiral deltahedron constructed by the strips of equilateral triangles was discovered by.

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