Deltahedron

A deltahedron is a polyhedron whose faces are all equilateral triangles. It was named by Martyn Cundy, after the Greek capital letter delta, which looks like a triangle Δ. Deltahedra are special because every face is an equilateral triangle, making them interesting shapes in geometry.

The simplest deltahedron is the regular tetrahedron, a pyramid with four equilateral triangles as its sides. There are eight different convex deltahedra, meaning they are shapes that bulge outward without any dents. These shapes are useful in chemistry, helping scientists understand how atoms are arranged in certain chemical compounds, as described in the polyhedral skeletal electron pair theory.

Besides the convex ones, there are also many concave deltahedra, which have indentations or dents in their shape. This makes the study of deltahedra a fascinating part of geometry and science.

Strictly convex deltahedron

A convex polyhedron is one where a line between any two points inside it stays entirely inside or on the edge of the shape. For a polyhedron to be strictly convex, no two faces can lie flat on the same plane, and no two edges can be straight line segments of the same line.

There are eight convex deltahedra — polyhedra with all faces as equilateral triangles. Three of these are well-known shapes called Platonic solids, and the other five are called Johnson solids. These include the regular tetrahedron, a pyramid with four triangular faces; the triangular bipyramid, regular octahedron, and pentagonal bipyramid, which are made by joining identical pyramids base-to-base; the gyroelongated square bipyramid and regular icosahedron; the triaugmented triangular prism; and the snub disphenoid. These shapes are important in chemistry, helping to describe the arrangement of atoms in molecules.

Non-convex deltahedron

A non-convex deltahedron is a special kind of deltahedron that is not perfectly shaped like a ball. Instead, it might have faces that lie flat against each other or edges that line up straight. There are infinitely many examples of these shapes! Some famous ones are the stella octangula, the excavated dodecahedron, and the Boerdijk–Coxeter helix.

Scientists have found many different groups of non-convex deltahedra based on how their corners, or vertices, are arranged. For example, the great icosahedron is special because all its vertices are the same type. There are also seventeen deltahedra with two kinds of vertices, and more were found later. Other interesting shapes include the isohedral deltahedron and the spiral deltahedron, which is made by arranging strips of equilateral triangles.