Circle packing theorem

Every finite simple planar graph can be represented as the tangency graph of circles in the plane. For a connected planar graph, the representation is unique up to Möbius transformations and reflection.

Planar graph: A graph that can be drawn in the plane without edges crossing except at shared endpoints. The theorem translates this graph structure into circle tangencies.
Paul Koebe: A German mathematician who proved a circle-domain uniformization theorem in 1936. His theorem supplied an early form of the circle-packing result.
Discrete conformal geometry: The study of discrete structures that approximate or encode conformal geometry. Circle packings provide a central discrete model of conformal structure.
Fáry's theorem: The theorem that every finite simple planar graph has a plane drawing with straight edges. It also represents planar graphs geometrically, but uses straight edges rather than circle contacts.
Tangency graph: A graph whose vertices represent geometric objects and whose edges represent pairs that touch. Circle packing realizes the input graph precisely through circle contacts.
E. M. Andreev: A Soviet mathematician known for theorems on hyperbolic polyhedra and circle patterns. His work extended circle-pattern results that connect to the theorem’s modern formulation.
Circle packing algorithm: A computational method for constructing a circle packing from graph data. Algorithms make the theorem’s existence result into explicit geometric coordinates.
Schnyder wood: An orientation and coloring structure on a planar graph that yields useful embeddings and decompositions. It produces coordinate-based planar drawings rather than tangent-circle representations.
Maximal planar graph: A planar graph to which no edge can be added without destroying planarity. Triangulated planar graphs provide the rigid combinatorial cases behind packing uniqueness.
William Thurston: An American mathematician whose work reshaped low-dimensional topology and geometry. He formulated the circle-packing conjecture that inspired extensive study and proof.

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Descartes' theoremCompared with: It concerns existence of packings for planar graphs, rather than curvature relations for four tangent circles.