Simpligons

Simpligons are a geometric concept introduced by John Conway, a mathematician and logician, in the 1950s. They represent a generalization of polygons, which are two-dimensional shapes with straight edges. Simpligons are similar to polygons but have vertices that are not necessarily straight lines.

A simplignon is a flat shape with the general form of a finite sequence of bounded lines with only a finite number of ways to cut, or a set of straight lines in the plane with a fixed number of vertices and edges. The concept of a simplignon is more flexible than traditional polygons because it allows for inclusion of points of non-zero length, vertices, and spaces between the them.

Conway initially explored simpligons as a part of his work on Atkin's theorem, a result in number theory with various subsequent applications. The main advantage of simpligons is their ability to handle degenerate cases, such as collapsing shapes and rounded or polygonal domains. The combinatorial method involved in defining a simplignon makes it an unique area of study, standing somewhat outside the multivariable geometry tradition. Despite the promising related area of 3D space, the original 2D research saw very little direct follow up in the subsequent 20 years.

The name "simplignon" itself reflects the shape's connection to the concept of simplicity. This step in geometric abstraction can be associated with the concept's name "Simplex" which has other connections to this field of mathematics. No significant errors in earlier conventional treatments of polygons are implied with this flexibility.