Dihedron

A dihedron (pl. dihedra) is a special kind of shape known as a polyhedron. It is made up of two polygon faces that share the same set of n edges. In ordinary three-dimensional space, a dihedron looks flat. But on the surface of a sphere, it can look like a lens.

When we think of a dihedron as a pattern on a sphere, it can be made with two n-sided faces. Each face covers half of the sphere like a hemisphere. The points where these faces meet sit along a great circle. If these points are spaced evenly, the dihedron is called regular.

Each n-sided dihedron has a matching shape called its dual. This matching shape is an n-sided hosohedron. In a hosohedron, n digon faces come together at just two points. Dihedra are also sometimes called bihedra, flat polyhedra, or doubly covered polygons.

As a flat-faced polyhedron

A dihedron is a special shape made from two identical flat polygons stuck back-to-back, so it has no depth. These polygons must be mirror images of each other.

Dihedra are interesting in geometry because they follow special rules about shapes and their surfaces. For example, they can be thought of as very simple members of families of other shapes. A regular dihedron is made from two regular polygons.

Main article: prism Main articles: prism, digon, pyramid Main article: regular polygons Main article: Schläfli symbol

As a tiling of the sphere

A spherical dihedron is made from two spherical polygons that share the same set of n points along a great circle, which works like an equator. Each polygon covers one half of the sphere, called a hemisphere.

A regular spherical dihedron has two regular spherical polygons. The points are spaced the same around the great circle equator. The shape {2,2} is both a hosohedron and a dihedron. It looks the same on both sides.

Apeirogonal dihedron

As the number of sides, n, becomes very large, an n-gonal dihedron turns into an apeirogonal dihedron. This shape is like a flat, two-dimensional pattern that repeats forever in every direction, like a tiling that covers the entire plane.

Ditopes

A regular ditope is a special shape that exists in higher dimensions, much like how a dihedron has two flat sides in our world. It has a unique symbol called a Schläfli symbol, written as {p,...,q,r,2}. This shape has two main parts called facets, connected by something called ridges.

This shape is interesting because it shows how geometry can work in more than just three dimensions. We live in a world with length, width, and height, but these shapes can have even more directions!