Circlei

Circlei is a term used in discussions of geometry and generative art to denote a family of circles indexed by integers. It is not a single standard mathematical object with a fixed definition, but a flexible label used in teaching, puzzles, and code examples to illustrate properties of circles and their arrangements. As a concept, circlei serves as a node for exploring how circles can be organized, nested, or placed in patterns.

In common constructions, Circlei denotes circles with a common center O but radii r_i = r_0/(i+1) for i ≥ 0, yielding a nested sequence of shrinking circles around the same point. In another variant, the centers form a regular geometric arrangement around a fixed point, and the radii follow a chosen sequence; this can produce concentric rings or a spiral-like pattern as i increases. The exact definitions and parameters are chosen to suit the teaching or artistic goal.

Properties and appearances depend on these parameters, affecting tangency, overlap, and density. Circlei is used in algorithmic art, geometry education, and software demonstrations of circle packing, ring formations, and related visual patterns. It is typically presented as an indexed, rather than a single, geometric object.

Etymology and naming: circlei combines the word “circle” with a plural-like suffix “-i,” a labeling convention seen in some mathematical discussions or languages. It is often used informally to denote an indexed family rather than a formal, universally defined object.

See also: circle packing, concentric circles, spiral patterns, Descartes' circle theorem.