research/AMR-005-0013.md

---
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?