Descartes' theorem

Descartes' theorem relates the signed curvatures of four mutually tangent circles: their sum squared equals twice the sum of their pairwise products.

Circle packing: An arrangement of circles whose interiors are disjoint, often with circles tangent to one another. The theorem gives a curvature constraint for each four-circle tangent configuration in a packing.
Circle: The set of points in a plane at a fixed distance from a center. Each object in the theorem is a circle described by its radius and curvature.
René Descartes: A French philosopher and mathematician who developed analytic geometry and wrote on geometry and optics. The theorem bears Descartes' name because he stated its circle-curvature relation in correspondence.
Apollonian gasket: A fractal packing formed by repeatedly filling curvilinear gaps between mutually tangent circles. Repeated use of the theorem supplies the curvature of each newly inserted circle.
Apollonian circle packing problem: The study of geometric, arithmetic, and counting properties of recursively generated tangent-circle packings. The theorem generates the packings whose curvature distributions and counts are investigated.
Signed curvature: A circle's curvature is the reciprocal of its radius, assigned a negative sign when it encloses the other circles. The sign distinguishes an enclosing circle from circles inside it in Descartes' equation.
Tangency: A relation in which geometric objects touch at a point without crossing there. The theorem applies only when every pair among the four circles is tangent.
Elizabeth of Bohemia: A seventeenth-century philosopher whose correspondence with René Descartes addressed mathematical and philosophical questions. Descartes communicated the circle theorem in a 1643 letter to Elizabeth of Bohemia.
Apollonian packing: A circle packing generated by recursively inserting circles tangent to existing circles. The theorem makes recursive construction possible without solving each geometry from scratch.
Apollonian group: A group of transformations generated by replacing one circle in a Descartes configuration with the other tangent circle. Its transformations encode repeated applications of the theorem and organize curvature quadruples.

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