This sequence of articles prove the following theorem, first enunciated by Roger Penrose about 70 years ago: In \(\mathbb{P}^2\), if regular 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 have a common point, then there exists an eighth conic which completes the cube and satisfies the same conditions (i) and (ii). This conic is unique if not all the common points fof condition (ii) are the same.

One proof is based elementary projective geometry in \(\mathbb{P}^5\), the parameter space of conics.

Another proof is based on the following analogue theorem: In \(\mathbb{P}^3\), if regular quadrics are assigned to seven of the vertices of a combinatorial cube such that (i) quadrics connected by an edge are in ring contact, and (ii) the ring planes associated to a cube face have a common axis, then there exists an eighth quadric which completes the cube and satisfies the same conditions (i) and (ii). This quadric is unique if not all common axes of condition (ii) are the same. A proof for this eight-quadric theorem is directly in \(\mathbb{P}^5\). An alternative proof is based on elementary projective geometry in \(\mathbb{P}^9\), the parameter space of quadrics.

• Publication at the London Mathematical Society Newsletter (September 2025).
• Russell Arnold, Albert Chern, Morten Eide, Charles Gunn, Thomas Neukirchner, and Roger Penrose. "Penrose's eight-conic theorem via Penrose's eight-quadric theorem."Preprint (arXiv)
• Slides
  (Presented Sep 2, 2024, in "Discrete Geometric Structure 2024")

• Intersection of 3 cylinders
• 3 Cones
• Brianchon
• Poncelet
• Monge
• Monge-like
• Pascal