Monogon

On a circle, a henagon is a tessellation with a single vertex, and one 360 degree arc.

In geometry a henagon (or monogon) is a polygon with one edge and one vertex. It has Schläfli symbol {1}. Since a henagon has only one side and only one interior angle, every henagon is regular by definition.

In Euclidean geometry

In Euclidean geometry, a henagon is usually considered to be an impossible object, because its single edge would extend to infinity. Most authorities do not consider the henagon as a proper polygon in Euclidean geometry, as it is degenerate: its structure is not that of any abstract polygon, and most polygon formulas do not apply to the henagon. For example, the formula for the angle sum of an n-gon, $(n-2)\pi$, does not work for the henagon with n = 1 as the formula would result in an angle sum of $-\pi$ which is meaningless in this context.

In spherical geometry

In spherical geometry, on the other hand, a finite henagon can be drawn by placing a single vertex anywhere on a great circle. Two henagons can be used to construct a dihedron on a sphere, with Schläfli symbol, {1,2}.

Henagonal dihedron (up-arrow represents vertex).

The henagon can be used in spherical polyhedra, for example the henagonal dihedron {1, 2}, the digonal hosohedron {2, 1} and the henagonal henahedron {1, 1}. The henagonal henahedron consists of a single vertex, no edges and a single face (the whole sphere minus the vertex.)

See also

• Regular polygon
• Digon
• Degeneracy (mathematics)

References

• Olshevsky, George. "Monogon". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Herbert Busemann, The geometry of geodesics. New York, Academic Press, 1955