Porism

Porism is a term in geometry describing a class of propositions about families of geometric configurations that exhibit a closure property. In a porism, if one instance of a construction exists under given conditions, then a whole family of such constructions exists, and often every admissible starting position yields a closed configuration. The emphasis is on the existence of a complete family of solutions once a single case is known.

The best-known example is Poncelet's porism. It concerns two conic sections in the plane. If there exists an n-sided polygon whose vertices lie on an outer conic and whose sides are tangent to an inner conic, then for every starting point on the outer conic there exists such an n-gon; the chain closes after n steps, yielding infinitely many distinct polygons related by projective transformations. The result is projectively invariant and connects to elliptic functions and integrable systems.

Another famous porism is Steiner’s porism, or Steiner chains. Given two non-concentric circles, a chain of circles can be drawn such that each circle is tangent to both circles. If one closed chain with a fixed number of circles exists, then a continuous family of such chains exists and any starting circle yields a closed chain of the same length. This porism is often analyzed by inversion and is closely related to Poncelet-type statements.

Historically, the term porism comes from Greek porisma, meaning a proposition or result. Porisms have been studied in classical and modern geometry, with connections to projective geometry, pencils of conics, and dynamical systems, including billiards in conics.

See also: Poncelet’s porism; Steiner porism.