research/AMR-005-0013.md

metadata

id: AMR-005-0013
classification: SOLVED-IN-LITERATURE
wording_corrected: 'no'

AMR-005-0013 — The Schwartz–Tabachnikov dodecagon configuration theorem

Problem (corrected statement if needed)

The AMR statement is accurate; here is the precise mathematical content. This is Problem 11 ("New Configuration Theorems of Projective Geometry") of S. Tabachnikov, A Baker's Dozen of Problems, Arnold Math. J. 1 (2015), based on R. E. Schwartz and S. Tabachnikov, Elementary Surprises in Projective Geometry, Math. Intelligencer 32(3) (2010), arXiv:0910.1952.

For an $n$-gon $P={p_1,\dots,p_n}$ in $\mathbb{RP}^2$, the $k$-diagonal map $T_k : \mathcal{C}n \to \mathcal{C}n^*$ sends $P$ to the polygon in the dual plane whose vertices are the consecutive $k$-diagonals ${\overline{p_1p{k+1}}, \overline{p_2p{k+2}}, \dots}$; each $T_k$ is an involution, $T_1$ is projective duality, and $T_{abc} = T_a\circ T_b\circ T_c$, etc.

Statement (Figure 5 of the Baker's Dozen): If $P$ is a dodecagon inscribed in a conic, then $T_{31313}(P)$ is circumscribed about a conic; equivalently (dualizing the last step), $T_{131313}(P)$ is again inscribed in a conic.

This was the one case of the Schwartz–Tabachnikov configuration theorems for which the authors had only numerical evidence: the brute-force symbolic check (vertices on $y=x^2$, determinantal identities) was estimated at $>10^{12}$ monomials, beyond Mathematica. A cyclically relabeled equivalent form ($\sigma(i)=5i \bmod 12$): $T_{535353}(P)$ is inscribed.

Status / Literature

• Schwartz–Tabachnikov (2010), arXiv:0910.1952. Eight configuration theorems; all proved by symbolic computation except the starred dodecagon case (Theorem 4(iii) there).
• Tabachnikov (2015), Baker's Dozen, §11. Restates the dodecagon statement as an open problem: "Find a proof."
• Tabachnikov (2016), Projective configuration theorems: old wine into new wineskins, arXiv:1607.04758 (survey chapter; published version: in Fifty Years of Mathematics / EMS volume, DOI 10.1007/978-3-030-13609-3_9). This states explicitly (I verified the full TeX source):

  "Fedor Nilov proved Theorem [dodecagon case] using a planar projection of hyperboloid of one sheet. Unfortunately, none of these proofs were published." So the problem was resolved by F. Nilov, but the proof was never written up.

• Izosimov (2016), Pentagrams, inscribed polygons, and Prym varieties, ERA-MS 23 (2016), arXiv:1607.03558. Gives a conceptual algebro-geometric proof of the related Schwartz–Tabachnikov theorem $E_k=O_k$ for inscribed polygons (self-duality $M(z)=(M(z^{-1})^{-1})^t$ of the scaled monodromy), and explicitly lists "obtain an algebraic geometric explanation of [the Elementary Surprises] results" as still open. I verified his full text does not prove the dodecagon configuration theorem.
• Glick, The Devron property (2014), arXiv:1312.6881 (checked full text) and Ramassamy / Affolter Miquel-dynamics papers (arXiv:1709.05509, 1808.04227; checked full texts) do not contain the dodecagon theorem.
• arXiv full-text searches ("dodecagon" AND "conic", "T_31313", "Schwartz-Tabachnikov") return no published proof of the dodecagon theorem. Nilov's own publication list (Semantic Scholar) contains no paper on it.

Bottom line: the conjecture is a theorem (Nilov, unpublished, reported in the authoritative 2016 survey by the conjecturer himself), but as far as I can determine no proof has ever appeared in print. Hence SOLVED-IN-LITERATURE with a caveat.

Work done

1. Retrieved and read the Baker's Dozen source (AMJ site) and the underlying paper arXiv:0910.1952 in full, fixing the exact statement (it is the starred case of Theorem 4 there).
2. Pulled the full TeX sources of Tabachnikov's 2016 survey, Izosimov 2016, Glick 2014, Ramassamy 2018, Affolter 2018 and grepped them for the dodecagon statement; cross-checked with arXiv API full-text searches and Semantic Scholar citation/author queries for F. Nilov.
3. Analysis of why the naive approaches fail / what a conceptual proof must do:
   • The statement is a polynomial identity in the 9 cross-ratios parameterizing inscribed 12-gons mod $PGL_3$; direct expansion is intractable ($>10^{12}$ terms, per Schwartz–Tabachnikov).
   • The word $w=31313$ is palindromic, so $T_{31313}$ is an involution; the space of inscribed 12-gons mod projectivities is 9-dimensional, as is the space of circumscribed 12-gons, so the statement is a birational "porism-type" correspondence, not a dimension accident.
   • Plausible reconstruction of Nilov's argument (speculation, labeled as such): a one-sheeted hyperboloid $H\subset\mathbb{RP}^3$ is doubly ruled; projecting $H$ from a point to a plane sends the two rulings to two families of lines tangent to conics, and plane sections of $H$ to conics. A 12-gon inscribed in a conic can be lifted to 12 points on $H$; the iterated diagonal intersections in $T_{31313}$ lift to incidence constructions among lines of the two rulings, and the final concyclicity reduces to elementary regulus geometry (the same "skewers"/hyperboloid technology Tabachnikov and Nilov–Skopenkov use elsewhere). This is consistent with the survey's one-line description but I did not verify the details.
   • Alternative conceptual route (open per Izosimov): the dodecagon theorem should follow from algebro-geometric properties of the pentagram-map spectral curve of inscribed polygons (the Prym variety), but no one has carried this out.

Result

The problem (including the headline dodecagon statement) is solved: all eight Schwartz–Tabachnikov configuration theorems are theorems. The dodecagon case — the only one open at the time of the source list — was proved by Fedor Nilov using a planar projection of a one-sheeted hyperboloid; this is documented in Tabachnikov's 2016 survey (arXiv:1607.04758, §"Configurations"), which also notes that the proof was never published. No published proof of the dodecagon theorem appears to exist as of this search (checked: arXiv full-text search, Semantic Scholar, citing literature of both source papers, Nilov's own publications).

What remains

• A published proof of the dodecagon theorem: either Nilov's hyperboloid argument written up, or an independent one. The statement remains a perfectly good target for a clean geometric or computer-algebra proof (modern Gröbner-basis / resultant software might now handle the $>10^{12}$-term identity).
• A conceptual algebro-geometric explanation (Prym varieties / integrable systems), explicitly posed as open by Izosimov (2016).
• The conjecture that the Schwartz–Tabachnikov list is exhaustive — no further "surprises" of this form for $n>12$ — remains unproved.
• Generalizations: which palindromic words $w$ in ${T_k}$ have the property that $T_w$ maps inscribed $n$-gons to circumscribed ones?