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Discrete geometry studies finite or discrete configurations of geometric objects, such as points, lines, circles, polytopes, and packings. Typical questions count incidences, optimize separation or coverage, or determine which combinatorial patterns can be realized geometrically.

For example, $n$ distinct points in the Euclidean plane determine at most $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ different lines, because every determined line contains a pair of points. Equality holds when no three points are collinear; collinearity merges several pairs into the same line. This simple argument connects a geometric degeneracy to a change in a counting problem.

Other problems ask how many equal disks can touch a given disk without overlapping, or how a polygon can be triangulated. Convex geometry, graph theory, and computational geometry supply complementary tools: coordinates constrain realizability, while a combinatorial description records which objects meet.

Willem Abraham Wythoff, also spelled Willem Abraham Wijthoff, worked on combinatorial game theory and the geometry of polytopes. His1907 modification of Nim became Wythoff's game. His work also included a published proof of the triangle-free graph bound associated with Willem Mantel.^[ref] The game illustrates how a small change to legal moves can replace a binary-arithmetic strategy by one organized through Beatty sequences and the golden ratio. His geometric work concerns the symmetry and construction of higher-dimensional polytopes, rather than only recreational games.

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