Apeirogon

An apeirogon is a special kind of shape in geometry. The word comes from Ancient Greek: ἄπειροv apeiron meaning "infinite" or "boundless," and γωνία gonia meaning "angle." So, an apeirogon is like a polygon but with an infinite number of sides!

In math, apeirogons are part of a bigger group called infinite polytopes. Sometimes, people talk only about the regular apeirogon, which has a special kind of balance and symmetry called an infinite dihedral group of symmetries. These shapes help mathematicians explore ideas about space and infinity.

You might also hear about an book titled Apeirogon (novel), but in math and geometry, an apeirogon is all about understanding shapes that go on forever!

Definitions

Geometric apeirogon

Imagine you have a point and you move it forward and backward forever in the same direction. Each step you take is the same length, and if you connect these points with straight lines, you create something called a regular apeirogon. It’s like a polygon that goes on forever instead of closing up like a circle.

Hyperbolic pseudogon

There is also something called a regular pseudogon, which works like the regular apeirogon but on a curved surface instead of a flat one.

Abstract apeirogon

An abstract apeirogon is a special way to describe shapes using sets of points and lines. It has rules about how these points and lines connect, and it can be turned into a real shape in space.

Symmetries

The regular apeirogon, a shape with an infinite number of sides, has special patterns of symmetry called the infinite dihedral group. These symmetries come from two reflections, which when combined, move each point of the shape to the next in a repeating pattern.

When we think of shapes in math, a "flag" means picking one part of each size all connected together. For a flat, two-dimensional shape like the apeirogon, its symmetries naturally let any such flag match any other. We can also picture this shape in regular space, where its symmetries match exact distance-preserving moves.

Moduli space

The moduli space of a faithful realization of an abstract polytope is a convex cone of infinite dimension. For the abstract apeirogon, this realization cone has uncountably infinite algebraic dimension and cannot be closed in the Euclidean topology.

Classification of Euclidean apeirogons

In geometry, an apeirogon is a special kind of shape with an endless number of sides. In spaces with more than two dimensions, regular polygons can be thought of as combinations of simpler shapes. This idea also helps us understand apeirogons.

In two dimensions, these shapes look like endless zigzags, formed by combining a simple line with a two-sided polygon. In three dimensions, they appear as spirals, where points are evenly spaced along a curly path, made by mixing a line with a flat polygon.

Generalizations

Main articles: Apeirotope and Apeirohedron

An apeirogon is a special kind of shape with an endless number of sides. In higher dimensions, we can think of shapes called apeirohedra, which are like the 3D version of an apeirogon. These ideas can be expanded even further into what are called n-apeirotopes, which are the infinite versions of shapes in any number of dimensions.