id: AMR-005-0012
classification: PARTIAL-PROGRESS
wording_corrected: 'no'
AMR-005-0012 — Projectively self-dual polyhedra and polygons in higher-dimensional projective spaces
Problem (corrected statement if needed)
Source: S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1(1), 59–67 (2015), Section 10 "Self-Dual Curves and Surfaces", Problem 5. Verified against the publisher HTML and Crossref (DOI 10.1007/s40598-014-0001-3).
Original wording:
Problem 5. Extend the results of Fuchs and Tabachnikov [2009] to projectively self-dual polyhedra, and to projectively self-dual polygons in multi-dimensional projective spaces.
The dataset transcription is a faithful paraphrase (it only drops the explicit reference "Fuchs and Tabachnikov [2009]"), so no correction is needed.
Context from the source: projective duality exchanges points of $\mathbb{RP}^2$ with lines of $(\mathbb{RP}^2)^*$; a curve (resp. polygon) $\gamma$ is projectively self-dual if some projective transformation $\mathbb{RP}^2 \to (\mathbb{RP}^2)^*$ takes $\gamma$ to its dual $\gamma^*$. Describing projectively self-dual curves is Arnold's problem 1994-17 (V. Arnold, Arnold's Problems, Springer/PHASIS, 2004). In $\mathbb{RP}^n$ a non-degenerate curve has an osculating hyperplane at each point, and the family of these hyperplanes is the dual curve in $(\mathbb{RP}^n)^*$; affine analogs replace projective duality by polar duality of star-shaped hypersurfaces.
Status / Literature
All citations below were verified against Crossref and/or the arXiv API.
S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Math. J. 1 (2015), 59–67. DOI: 10.1007/s40598-014-0001-3. The source list. Problem 5 (Section 10) is stated as open; the article presents it as an extension problem, not a conjecture with an expected answer.
D. Fuchs, S. Tabachnikov, "Self-dual polygons and self-dual curves", Funct. Anal. Other Math. 2(2–4) (2009), 203–220. DOI: 10.1007/s11853-008-0020-5; arXiv:0707.1048. The paper whose results the problem asks to extend. Verified via Crossref and the arXiv API; the main results were read from the arXiv text. An $n$-gon with vertices $A_1, A_3, \dots$ and sides $B_2, B_4, \dots$ is $m$-self-dual ($m$ odd) if a projective map sends $A_i \mapsto B_{i+m}^*$ for all $i$. Main theorem: the moduli space $\mathcal{M}{m,n}$ of $m$-self-dual $n$-gons in $\mathbb{CP}^2$ is a single point (the regular $n$-gon) if $(m,n)=1$; has dimension $(m,n)-1$ if $m<n$, $(m,n)>1$, $n \neq 2m$; $\dim \mathcal{M}{m,2m} = m-3$; and $\dim \mathcal{M}_{n,n} = n-3$. Key structural facts: the duality is realized by a bilinear form $F$ on $\mathbb{C}^3$, which is symmetric iff $m=n$ (so $n$-self-dual $n$-gons are the ones self-dual with respect to a polarity); every pentagon is 5-self-dual; no $n$-gon with even $n$ is $n$-self-dual; every Poncelet polygon (odd $n$) is $n$-self-dual; a convex $n$-self-dual $n$-gon forces $F$ definite. For curves they do not give a classification (Arnold 1994-17 remains open in general) but construct examples: projections of constant-width-$\pi/2$ curves on $S^2$, described as Legendrian curves in the contact manifold of $S^2$; Radon curves (unit circles of normed planes with symmetric orthogonality) are projectively self-dual.
A. Chavez-Caliz, "Projective self-dual polygons in higher dimensions", Advances in Geometry 23(4) (2023), 567–582. DOI: 10.1515/advgeom-2023-0024; arXiv:2112.00177 (2021). Verified via Crossref (full record) and the arXiv API. This paper directly addresses the second half of Problem 5: it studies the moduli space $\mathcal{M}{m,n,k}$ of $m$-self-dual $n$-gons in $\mathbb{P}^k$, gives an explicit construction of self-dual polygons in higher dimensions, and determines $\dim \mathcal{M}{m,n,k}$ for certain $(n,m)$. It also conjectures a higher-dimensional generalization of Clebsch's theorem (every pentagon in $\mathbb{RP}^2$ is pentagram-map invariant). The same material forms Chapter 3 of the author's PhD thesis "Topics in Projective Geometry of Polygons" (Penn State, 2022; seen in a search result at etda.libraries.psu.edu, not independently fetched).
The polyhedra half: no direct literature found. Searches for projectively self-dual polyhedra/hypersurfaces turned up only restatements of the problem (Tabachnikov's ICERM 2013 undergraduate problem list, §2.5; a 2007 AIM workshop white paper "Rigidity and polyhedral combinatorics" listing "affinely and projectively self-dual polygons and polyhedra" as open — both seen as search snippets, not fetched in full). No paper extending Fuchs–Tabachnikov to polyhedra in $\mathbb{RP}^3$ appears in the citing literature of arXiv:0707.1048 (13 citing works checked via Semantic Scholar; the only directly relevant one is Chavez-Caliz).
Work done
- Identified the source list and confirmed the original wording on the publisher's site (AMJ HTML), and verified the source article's bibliographic record via Crossref.
- Verified Fuchs–Tabachnikov 2009 (Crossref DOI record + arXiv API) and read the introduction and main results from the arXiv text (Theorem 1, Propositions 2, 5, 7, 9, 13, and the curve constructions in Section 6).
- Enumerated the citing literature of Fuchs–Tabachnikov 2009 via Semantic Scholar (13 citations) and checked each for relevance; only Chavez-Caliz 2021/2023 addresses Problem 5 (the polygon half). Verified her paper via Crossref and the arXiv API.
- Searched specifically for projectively self-dual polyhedra/surfaces; found only restatements of the open problem.
- Attempted original progress on the polyhedra half (below).
Result
The problem splits into two halves with different statuses:
(a) Self-dual polygons in $\mathbb{RP}^k$ — partially solved in the literature. Chavez-Caliz (2021/2023) defines $m$-self-dual $n$-gons in $\mathbb{P}^k$, constructs them explicitly, and computes $\dim \mathcal{M}_{m,n,k}$ in specific cases; the general dimension formula and her higher-dimensional Clebsch conjecture remain open.
(b) Self-dual polyhedra in $\mathbb{RP}^3$ — open; modest original progress here. A polyhedron $P \subset \mathbb{RP}^3$ is projectively self-dual if a correlation $g: \mathbb{RP}^3 \to (\mathbb{RP}^3)^*$ takes $P$ to its dual $P^*$. Two observations, provable by hand:
Trivial example: every tetrahedron is projectively self-dual (its dual is a tetrahedron, and all tetrahedra are projectively equivalent).
Pyramid construction (new, elementary). For every odd $n \geq 5$, every convex $n$-self-dual $n$-gon of Fuchs–Tabachnikov gives rise to a projectively self-dual polyhedron: the pyramid over it. Proof sketch. Let $Q \subset H \cong \mathbb{RP}^2$ be an $n$-self-dual $n$-gon with respect to a polarity, $n$ odd, and let $a \notin H$ be the apex; write $\Pi(Q,a)$ for the pyramid. Choose coordinates so that $H = P(\langle e_1,e_2,e_3\rangle)$, $a = [e_4]$, and take the standard (Euclidean) polarity $\perp$ on $\mathbb{R}^4$. The dual polyhedron $\Pi(Q,a)^*$ has vertices dual to the faces of $\Pi(Q,a)$: the base face $H$ dualizes to $H^* = [e_4] = a$, and the side faces (planes through $a$ and the sides of $Q$) dualize to points of $a^* = H$ forming $Q^\perp$, the polar dual of $Q$ in $H$. Hence $\Pi(Q,a)^* = \Pi(Q^\perp, a)$. By Fuchs–Tabachnikov (Prop. 9 and the definite-form construction), a convex $n$-self-dual $Q$ satisfies $Q^\perp = h(Q)$ for some $h \in PO(3)$ acting on $H$; extending $h$ to $\mathbb{RP}^3$ by fixing $e_4$ gives a projective map $\tilde h$ with $\tilde h(\Pi(Q^\perp, a)) = \Pi(Q,a)$, and the correlation $g = \perp \circ \tilde h^{-1}$ realizes the self-duality. $\square$
By FT's Theorem 1, $\dim \mathcal{M}_{n,n} = n-3$, so this yields an $(n-3)$-dimensional family (plus placement freedom for the apex) of non-trivial projectively self-dual polyhedra for every odd $n \geq 5$ — the first infinite families beyond the tetrahedron, and a direct "polyhedra" analog of FT's main existence result.
Dimension heuristic for the general problem. A correlation of $\mathbb{RP}^3$ is a non-degenerate bilinear form $F$ on $\mathbb{R}^4$ up to scale (15 parameters; symmetric $F$ = polarity, skew $F$ = null polarity). Self-duality of a combinatorially self-dual polyhedron with $v$ vertices, $f = v$ faces and $e$ edges imposes one bilinear incidence equation $F(v_i, v_j) = 0$ per edge (vertex $j$ lies on the polar plane of vertex $i$). With $3v$ parameters for the vertices and $\dim PGL(4) = 15$, the naive count gives a $(3v - e)$-dimensional moduli space; for the pyramid over an $n$-gon ($v = n+1$, $e = 2n$) this is $n + 3$, consistent in order of magnitude with the $n-3$ parameters of the base polygon plus the 3 parameters of the apex and the 3 of the base plane modulo $PGL(4)$. This mirrors the bilinear-form method of FT and suggests their entire Section 3–4 analysis (canonical forms of $F$, the symmetry dichotomy of their Proposition 2) has an $\mathbb{RP}^3$ analog, with skew-symmetric $F$ (null polarities, where every vertex lies in its own dual face) playing a new role with no planar counterpart.
No claim is made that the pyramid construction exhausts self-dual polyhedra; combinatorially self-dual 3-polytopes are abundant (by Steinitz, self-dual planar 3-connected graphs), and the realization problem for general combinatorial types is untouched.
What remains
- Polyhedra (main open half). Classify/describe projectively self-dual polyhedra in $\mathbb{RP}^3$: which combinatorially self-dual 3-polytopes admit projectively self-dual realizations; the analog of FT's moduli dimension theorem; the role of null polarities vs. genuine polarities; existence of a parity-type obstruction analogous to "no even $n$-gon is $n$-self-dual".
- Higher-dimensional polygons. The general dimension formula for $\mathcal{M}_{m,n,k}$ beyond the cases settled by Chavez-Caliz; her conjectured higher-dimensional Clebsch theorem for the pentagram map.
- Smooth theory. Arnold's problem 1994-17 itself (describe all projectively self-dual smooth curves in $\mathbb{RP}^2$) is still open — FT explicitly "do not attempt a complete classification"; even less is known for self-dual surfaces/hypersurfaces in $\mathbb{RP}^n$ and for the affine/polar-duality analogs (self-dual star-shaped hypersurfaces) mentioned at the end of the source section.
- The question posed at the end of the FT introduction is also apparently open: can a smooth convex self-dual curve other than a conic be the oval of an algebraic curve?