Penrose's eight-conic theorem via Penrose's eight-quadric theorem
| Autor: | Arnold, Russell, Chern, Albert, Gunn, Charles, Neukirchner, Thomas, Penrose, Roger |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | This article proves the following theorem, first enunciated by Roger Penrose about 70 years ago: In $\mathbb{R}P^2$, 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 meet in a common point, then there exists a unique eighth conic such that the completed cube satisfies (i) and (ii). The proof is based on the following analogous theorem, which is also proved: In $\mathbb{R}P^3$, 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 meet in a common axis, then there exists a unique eighth quadric such that the completed cube satisfies (i) and (ii). Comment: 21 pages, 12 figures |
| Databáze: | arXiv |
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