Dihedron

A dihedron (pl. dihedra) is a special kind of shape known as a polyhedron, which 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 when you think about it on the surface of a sphere, it can take the shape of a lens. This idea is important in mathematics, especially in areas that study spaces shaped like spheres.

When we look at a dihedron as a pattern that covers a sphere, it can be made with two n-sided faces, each covering 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, which 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, showing how they can appear in different mathematical contexts.

As a flat-faced polyhedron

A dihedron is a special kind of 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 fit 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 formed by two spherical polygons that share the same set of n vertices along a great circle, which acts like an equator. Each of these polygons covers one half of the sphere, known as a hemisphere.

A regular spherical dihedron has two regular spherical polygons with vertices equally spaced around the great circle equator. The special shape called {2,2} is both a hosohedron and a dihedron, and it is self-dual, meaning 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 can be thought of as a flat, two-dimensional pattern that repeats forever in every direction, like a tiling that covers the entire plane.

Ditopes

A regular ditope is like a special shape in higher dimensions, similar to how a dihedron is a shape with two flat sides in our world. It has a special symbol called a Schläfli symbol, written as {p,...,q,r,2}. This shape has two main parts called facets, and they share something called ridges, which help connect the different parts of the shape together.

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