Isogonal figure

In geometry, a polytope like a polygon or polyhedron is called isogonal or vertex-transitive if all its vertices are the same in terms of symmetry. This means you can move any vertex to another’s spot using a symmetry movement, and the shape will still look 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 for studying symmetrical shapes. It shows us that the shape looks the same from each vertex’s viewpoint. This concept helps mathematicians and scientists understand how shapes repeat their patterns. Many common solids, like cubes and regular pyramids, are isogonal because their vertices all have the same environment.

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

Isogonal polygons and apeirogons

All regular polygons, apeirogons, and regular star polygons are isogonal. This means every corner, or vertex, of these shapes looks the same. You can turn or flip the shape, 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. This 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.

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, and semi-regular if faces are regular polygons but not all the same. There are also special types like semi-uniform, scaliform, and noble.

N dimensions: Isogonal polytopes and tessellations

These ideas about shapes can also work in 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 act the same way under the figure’s symmetries. This means the vertices in each set are the same as 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 us see their structure, using uniform colorings.