Squaren

Squaren is a term used in geometry and digital design to describe a family of shapes derived from a square by a defined edge-modification process. The concept is used to explore how simple operations on a familiar form can produce a range of symmetric, polygonal outlines while preserving a core square-like structure.

Construction and characteristics: A squaren begins with a unit square and applies a repeatable edge transformation, often described as replacing each side with a polygonal chain that maintains fourfold rotational symmetry. By adjusting a single parameter, the resulting shape can range from a near-square form to a highly faceted silhouette that approaches a circular contour. The exact geometry depends on the chosen transformation, but common features include equal treatment of all four sides and a predictable relationship between the parameter and the degree of roundedness or angularity.

Applications: Squaren shapes appear in generative art, computer graphics, and puzzle design as a way to study symmetry, tiling behavior, and visual aesthetics on grid-based systems. They provide a compact framework for comparing square-like forms with polygonal approximations of curves and can serve as a testbed for algorithms related to shape subdivision, rendering, and geometric transformations.

History: The term emerged in online geometry and design communities in the 2010s as a playful, informal label for this family of square-derived shapes. Since then, squaren has been used primarily as a conceptual tool rather than a standardized mathematical object.

See also: square, squircle, polygon, fractal, tiling, generative art.