Isogonal figure

In geometry, a polytope such as a polygon or polyhedron, or even a tiling, is called isogonal or vertex-transitive if all its vertices are the same in terms of symmetry. This means that if you pick any two vertices, there is a symmetry movement of the shape that can move one vertex directly to the position of the other while keeping the shape looking the same. Because of this, every vertex has the same surroundings — the same types of faces arranged in the same order and with the same angles between them.

The idea of being isogonal is important in studying symmetrical shapes. It tells us that the shape looks the same from the viewpoint of each vertex. This concept helps mathematicians and scientists understand how shapes repeat their patterns and why certain objects in nature and design show such beautiful regularity. For example, many common solids like cubes and regular pyramids are isogonal because their vertices all experience the same environment.

All vertices of a finite, n-dimensional isogonal figure lie on an (n−1)-sphere. While the term isogonal has traditionally described polyhedra, vertex-transitive is a newer name for the same idea, coming from the study of symmetry groups and graph theory. This shows how different areas of mathematics connect to describe and classify symmetrical patterns in shapes.

Isogonal polygons and apeirogons

All regular polygons, apeirogons, and regular star polygons are isogonal. This means that every corner, or vertex, of these shapes looks the same. The shape can be turned or flipped, and it will still look the same.

Some even-sided polygons, like a rectangle, that have two different edge lengths but still look the same from every corner are also isogonal. These shapes have a special kind of symmetry called dihedral symmetry, which means they can be reflected across lines that cut through the middle of their edges.

Isogonal polyhedra and 2D tilings

An isogonal polyhedron and 2D tiling has just one kind of vertex. When an isogonal polyhedron has all regular faces, it is also called a uniform polyhedron and can be described using a special notation that shows the pattern of faces around each vertex.

Isogonal shapes can be grouped in different ways: regular if all faces and edges look the same, quasi-regular if edges are the same but faces are not, semi-regular if faces are regular polygons but not all the same, and uniform if all faces are regular polygons. There are also special types like semi-uniform, scaliform, and noble based on how their edges and faces are arranged.

N dimensions: Isogonal polytopes and tessellations

These ideas about shapes can also apply to higher dimensions, like in tessellations. All uniform polytopes are isogonal. This includes things like uniform 4-polytopes and convex uniform honeycombs.

The dual of an isogonal shape is an isohedral figure, meaning it moves the same way on its facets.

k-isogonal and k-uniform figures

A polytope or tiling can be called k-isogonal if its vertices are grouped into k different sets that behave the same way under the figure's symmetries. This means that the vertices in each set are equivalent to each other.

A more specific term, k-uniform, refers to k-isogonal figures that are made only from regular polygons. These figures can sometimes be shown with different colors to help visualize their structure, using uniform colorings.