Neopagon

Neopagon is a term used in geometry to describe a generalized polygonal figure that extends the standard concept of a polygon by allowing the sequence of side lengths to vary while maintaining a simple, closed chain of n straight edges. The name combines neo- (new) with pagon, from polygon, and is used to distinguish these figures from classical regular polygons.

Construction and definition: For an integer n ≥ 3, a neopagon consists of n vertices v1, v2, ..., vn connected in order by straight edges to form a simple cycle. Unlike a regular polygon, the side lengths |vi vi+1| need not be equal, and the internal angles at vertices can differ. A common realization is to note that any simple polygon with n sides is a neopagon, but the term is often used when emphasis is on the freedom of side-length variations within a fixed-sides count. A cyclic neopagon is a standard subtype in which all vertices lie on a common circle, but adjacent edges need not be equal.

Variants: An equilateral neopagon has all sides equal in length, though angles may vary; an equiangular neopagon has equal interior angles; a regular neopagon would require both, which reduces to the traditional regular polygon.

Properties: The area is computable by the shoelace formula from vertex coordinates. The space of neopagons with a fixed n and a fixed perimeter is a continuous family, ranging from the regular polygon to extremely irregular shapes. When restricted to cyclic neopagons, area can be investigated via standard formulas for polygons inscribed in a circle.

Applications: In computer graphics, parametric design, and tiling theory, neopagons provide a framework for exploring shape variation while preserving topological type. See also polygon, regular polygon, cyclic polygon, star polygon.