Overview
A henagon, also called a monogon, is a theoretical figure defined as a polygon with just one side and one vertex. The name derives from Greek roots meaning "one" and "angle." As a concept it helps clarify the boundaries of what counts as a polygon and is useful in several branches of mathematics even though it cannot be realised as a conventional, nondegenerate polygon in the flat Euclidean plane.
Properties and why it is degenerate in Euclidean geometry
By definition a henagon has a single edge and a single corner (vertex). Because ordinary polygons are chains of straight line segments that meet at distinct vertices, a single straight segment cannot form a closed loop with only one distinct endpoint. For this reason a henagon is considered impossible to draw as a simple polygon in classical geometry or standard Euclidean geometry. Attempts to force a one-sided polygon in the plane either collapse the edge to a point (a degenerate case) or require the side to be a curved loop that returns to its starting point, which departs from the usual line-segment model of a polygon.
Contexts where henagons are meaningful
Despite the Euclidean obstruction, the idea of a one-sided polygon is meaningful in several settings:
- In spherical and other non-Euclidean geometries one can define polygons bounded by one geodesic arc or a closed curve, so analogues of a henagon can occur on curved surfaces. See examples of such shape constructions.
- In combinatorial topology and graph embeddings a face of an embedded graph may be incident to a single edge or loop; this combinatorial face is treated as a monogon for bookkeeping in planar maps.
- In abstract polygon definitions used in algebraic and topological contexts, 1-gons and 2-gons (digons) are permitted as formal objects, useful for counting and classification.
Examples, distinctions, and notable facts
All henagons are trivially regular because there is only one interior angle to consider. The related digon (two sides, two vertices) shares the same Euclidean impossibility but exists naturally on spheres as a lune bounded by two great circle arcs. In polygon theory a henagon is sometimes treated as a degenerate or limiting case; in other frameworks it is a legitimate object with specific combinatorial or topological meaning. For concise introductions to related ideas see discussions of polygons, the notion of a side, and the meaning of a point at infinity when considering degenerate limits.
Because the term appears in several mathematical subfields, context determines whether a henagon is dismissed as impossible, regarded as degenerate, or embraced as a useful formal construct. For further reading see surveys of polygonal definitions and of one- and two-sided figures in non-Euclidean and combinatorial settings: Euclidean. Corner and boundary discussions: shape theory.