radiiis

Radiiis is a term encountered in geometry to refer to the collective radii of circles in a packing in which circles are tangent according to a fixed rule. It is a neologism rather than a standard term, appearing in some discussions of circle packings, Apollonian gaskets, and related fractal constructions.

Definition and construction: In a circle packing, the radiiis is the multiset of radii of all circles in the pattern. A canonical example is the Apollonian gasket, started from four mutually tangent circles. Descartes' circle theorem relates their curvatures k = 1/r; new circles tangent to three existing ones yield an iterative process that generates further radii, expanding the radiiis.

Properties: The radiiis often shows a hierarchical, self-similar structure with many small circles clustered around larger ones. When initial circles have integer curvatures, the subsequent radii may satisfy algebraic relations. The distribution is a subject of study in discrete and fractal geometry and connects to number theory via curvature relations.

Applications and status: Beyond pure geometry, circle packings and their radii distributions inform models of porous media, wireless networks, and texture synthesis in computer graphics. As a nonstandard term, radiiis is used inconsistently; most literature simply describes the radii set of a packing rather than invoking the term itself.