The Penrose 8-Conic Theorem #
The Penrose 8-conic theorem is a meta-theorem of projective geometry:
If conics are assigned to seven of the eight vertices of a combinatorial cube such that conics joined by an edge are in double contact, and the chords of contact in each face share a common point, then there exists an eighth conic completing the cube under the same conditions.
By letting vertices degenerate into line pairs, point pairs, double lines or double points, it specializes to a host of classical theorems — Pappus, Pascal, Brianchon, Salmon, Desargues, Poncelet’s porism, and many more.
This section catalogs those special cases. Each page carries a live, interactive configuration you can manipulate directly in the browser.
Source: R. Arnold, A. Chern, M. Eide, C. Gunn, T. Neukirchner, R. Penrose, “Penrose’s 8-Conic Theorem: Revealed after 70 Years”, LMS Newsletter 516 (2025). Preprint: arxiv.org/abs/2409.17150.