What do we mean by "Regular" for Orientable Regular Maps?

Introduction

"Regular", as applied to regular maps, has been defined in more than one way. This page considers various definitions that may be used of orientable regular maps. Non-orientable regular maps are not considered here.

We start by describing seven test cases which we will consider as candidates for being "regular". The page leads up to a table with a row for each of the seven, and a column for each of fifteen aspects of regularity. But first we consider edges and flags, and the two symmetry groups associated with any candidate for being a regular map. Then we explain what is shown in each column of the table.

Finally we quote some definitions of "regular" map.

Seven Specimens

The Cube

The cube is regular by all standards. All its faces are regular and interchangeable, and all its vertices are regular and interchangeable.

Its Petrie polygons all have six edges.

It has mirror symmetry. Its rotational symmetry group is S4, with 24 elements (twice its number of edges), and its full symmetry group (including reflections) is S4×C2, with 48 elements.

The image above, and the two below, are copied from Wikipedia.

The Cuboctahedron

The cuboctahedron is not regular (it is "quasiregular"). Its faces are all regular, but it has two different kinds of face. Its vertices are all interchangeable, but are not regular.

Its Petrie polygons all have eight edges. Its holes all have six edges.

It has mirror symmetry. Its rotational symmetry group is S4, with 24 elements (equal to its number of edges), and its full symmetry group (including reflections) is S4×C2, with 48 elements.

The Rhombic dodecahedron

The rhombic dodecahedron is the dual of the cuboctahedron. It is not regular. Its faces are interchangeable, but are not regular, they are rhombi rather than squares. Its vertices are regular, but are not interchangeable: some have three edges, some have four.

Its Petrie polygons all have eight edges.

Like its dual, it has mirror symmetry. Its rotational symmetry group is S4, with 24 elements (equal to its number of edges), and its full symmetry group (including reflections) is S4×C2, with 48 elements.

Tiling of the Torus with Two Squares

This may or may not be regarded as regular. Its faces are interchangeable and regular. Its vertices are interchangeable and regular. However, if we walk "east" along an edge until we return to where we have started we traverse two edges, whereas if we walk "north" we traverse one. So the north-south edges are not interchangeable with the east-west edges, and so rotating it through a right angle about the centre of a face is not a symmetry operation.

Its Petrie polygons all have four edges. One of its holes has two edges, two others have one edge each.

It has mirror symmetry. Its rotational symmetry group is C2×C2, with 4 elements (equal to its number of edges), and its full symmetry group is C2×C2×C2, with 8 elements.

(This is not the only way to tile a torus with two squares.