Monogon
| Monogon | |
|---|---|
 | |
| Rank | 2 |
| Type | Regular |
| Space | Spherical |
| Notation | |
| Schläfli symbol | {1} |
| Elements | |
| Edges | 1 |
| Vertices | 1 |
| Vertex figure | Dyad, length 0 |
| Related polytopes | |
| Dual | Monogon |
| Conjugate | None |
| Abstract properties | |
| Flag count | 2 |
| Net count | 1 |
| Topological properties | |
| Orientable | Yes |
| Properties | |
| Symmetry | A1×I, order 2 |
The monogon is a 1-sided polygon. It is highly degenerate. It can be embedded in a spherical geometry, with a single edge that covers an entire great circle and joins a vertex to itself. It is unique compared to other polygons, by having a vertex figure where two different vertices represent the same element.
Most definitions of a polytope do not allow for monogons, as they violate the diamond condition. However, a monogon can be seen as a map on a surface. It may also be seen as a generalized complex (see below).
Monogonal complex[edit | edit source]
A complex is an properly edge-colored graph. If we generalize this notion to multigraphs, one can build a complex with two vertices joined with edges of two colors. As it turns out, the omnitruncate of this generalized complex gives a digon. This suggests that this complex may be regarded as the true monogon under a more general notion of polytopes.
One surprising consequence of this would be that a monogon would have two flags instead of one. We might think of them as belonging to two halves of a region enclosed by the monogon.
By adding more parallel edges of different colors, one might also build monohedra and higher-dimensional monotopes.