Penrose's eight-conic theorem

Abstract

This article proves the following theorem, first enunciated by Roger Penrose about 70 years ago but never published: In $\mathbb{R}P^{2}$, if conics are assigned to seven of the vertices of a combinatorial cube such that (i) conics connected by an edge are in double contact, and (ii) the chords of contact associated to a cube face meet in a common point, then there exists an eighth conic such that the completed cube satisfies (i) and (ii). The theorem turns out to be a remarkable generalization of many well-known theorems of projective geometry -- Pappus, Desargues, Pascal, Brianchon, Monge, and Poncelet are the best-known ones. This archetypal principle provides a unifying framework in which the myriad specializations of the theorem and their interrelationships can be grasped as an organic whole, enriching the field of projective geometry and opening new vistas for research. The article begins with a series of motivational examples. It then gives a geometric proof assuming that the conics are regular, followed by an algebraic one that removes this restriction. The geometric proof is obtained as a slice of an analogous theorem for quadrics in $\mathbb{R}P^{3}$; the algebraic one is based on the determinants of a special matrix associated to the configuration of conics.

Figures (19)

• Figure 1: Pappus's theorem (340 AD) (left) and its dual theorem (right).
• Figure 2: Pascal's theorem (1640) (left) and its dual theorem---Brianchon's theorem (1807) (right).
• Figure 3: Salmon's theorem (1848) and its dual theorem.
• Figure 4: The main theorem---the eight-conic theorem---first enunciated by Roger Penrose in around 1950. The right figure illustrates an instance of the configuration where all 8 conics are ellipses, shown next to the corresponding cube graph of conics and double contact relationships.
• Figure 5: The cube graph of the 8-conic theorem (left) and an example of 4 conics $\mathbf{{S}}_0,\mathbf{{S}}_2,\mathbf{{T}}^1,\mathbf{{T}}^3$ satisfying condition (ii) (right).

Theorems & Definitions (33)

• Lemma 1: Regular pencil
• proof
• Lemma 2
• Lemma 3
• Remark 4
• Lemma 5
• proof
• Lemma 6
• Theorem 7: 8-conic theorem
• Theorem 8: 8-quadric theorem