research/AMR-005-0009.md

metadata

id: AMR-005-0009
classification: SOLVED-IN-LITERATURE
wording_corrected: 'no'

AMR-005-0009 — Origami hyperbolic paraboloids (Tabachnikov, Baker's Dozen, Problem 8)

Problem (corrected statement if needed)

From S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Math. J. 1 (2015), §8 (source). The wave1.txt transcription was checked against the source and is faithful; no correction needed.

The classical origami "hyperbolic paraboloid" (hypar) is folded from a square sheet creased along concentric axis-parallel squares (alternating mountain/valley) plus the two diagonals of the sheet, which cut each annular region between consecutive squares into 4 isosceles trapezoids. Demaine, Demaine, Hart, Price and Tachi ("(Non)existence of pleated folds: how paper folds between creases", Graphs Combin. 27 (2011) 377–397) proved that, assuming inextensible paper and straight fold lines, this pleated fold is mathematically impossible as an exact piecewise-linear (PL) origami, and proposed that physical models contain additional "invisible" folds along one diagonal of each elementary trapezoid. Tabachnikov's questions:

1. Assuming invisible straight folds along a chosen diagonal of each elementary trapezoid (2 choices per trapezoid), what is the shape of the piecewise-linear surface obtained by folding?
2. What is obtained from analogous constructions with other patterns of fold lines (e.g., the circular pleated surface of his Fig. 3)?

Status / Literature

• Demaine–Demaine–Hart–Price–Tachi (2011) (verified via the Baker's Dozen reference list and Erik Demaine's publication list): impossibility of the exact pleated fold with only the given creases; they describe triangulation schemes of the trapezoids ("asymmetric" and "alternating asymmetric" triangulations) under which a PL folding exists.
• Liu, Tachi & Paulino, "Invariant and smooth limit of discrete geometry folded from bistable origami leading to multistable metasurfaces", Nature Communications 10, 4238 (2019) (article, DOI 10.1038/s41467-019-11935-x; read in full). This paper essentially answers Question 1 in the homogenized (fine-pleat) limit:
  • They adopt exactly the premise of Tabachnikov's problem: isometric deformation, straight creases, one extra diagonal pleat per trapezoid (the "alternating asymmetric triangulation" of Demaine et al.).
  • They prove that in the limit $w=d/L\to 0$ (panel width $d$ over corrugation length $L$), at every stage of folding, the folded surface is the hyperbolic paraboloid $z = k(x^2-y^2)$, parametrized as $\mathbf X(r,t)=(\pm tr,\ \pm(1-t)r,\ (2t-1)kr^2)$, $0\le t\le 1$; the coefficient $k$ increases monotonically with the folding angle $\rho$ via $\rho(k,p)=\arccos!\big(2/\sqrt{8k^2p^2+1}-1\big)$ ($p$ = intrinsic distance from the center).
  • The limit is independent of which of the two triangulation schemes is used (i.e., of the diagonal pattern), and invariant under non-uniform (graded or random) offsets between squares.
  • They derive the ODE $(2\xi-r\xi')(2r\xi'^3+3r\xi'-2\xi)=0$ for the profile curve $\xi(r)$ of the folded diagonal crease; the branch $2\xi-r\xi'=0$ gives $\xi=kr^2$ (the hypar); the second branch gives a non-saddle shape realizable only after cutting slits (kirigami), hence not relevant.
  • They prove bistability unconditionally under mild convexity assumptions on the folding/bending energy, with two symmetric stable states ($k\leftrightarrow -k$), and verify everything with 3D scans of Mylar models and bar-and-hinge simulations (the single soft folding mode has ~5% of the next modal stiffness — consistent with a 1-DOF mechanism).
  • Experiments confirm the "invisible folds": each trapezoidal panel bends along one dominant diagonal, matching the alternating asymmetric triangulation.
• Supporting/continuing work (verified via Crossref): Filipov & Redoutey, "Mechanical characteristics of the bistable origami hypar", Extreme Mech. Lett. 25 (2018) 16–26; Liu, Johnson & Sung, "Increasing Reliability of Self-Folding of the Origami Hypar", J. Mechanisms Robotics (2022), DOI 10.1115/1.4054310; Liu & Paulino, "Symmetric Self-folding of N-Gon Hypar Origami", Lecture Notes in Mechanical Engineering (2026), DOI 10.1007/978-981-96-8661-2_16 (extends the hypar analysis to N-gon outlines — partial progress on Question 2).
• Question 2 in general (other crease patterns, notably the circular pleat of Tabachnikov's Fig. 3) remains essentially open. The 2019 paper states only that its homogenization framework "can be used as a basis to investigate other corrugated origami shells, such as concentrically pleated patterns with polygonal outlines". For circular rings/pleats see Mouthuy et al., Nat. Commun. 3:1290 (2012) (overcurvature-driven buckling of rings), which addresses smooth curved folds rather than the PL question. No rigorous analogue of the hypar theorem for the circular pleat was found.

Work done

1. Retrieved and verified the exact source statement (Arnold Math. J. site); the wave1.txt wording needs no correction.
2. Located, verified (DOI/Crossref) and read the 2019 Liu–Tachi–Paulino paper, which postdates the problem list and resolves Question 1 in the asymptotic sense.
3. Checked the key algebra myself: for $\mathbf X(r,t)=(tr,(1-t)r,(2t-1)kr^2)$, one has $x^2-y^2=(t^2-(1-t)^2)r^2=(2t-1)r^2$, so indeed $z=k(x^2-y^2)$; and $\xi=kr^2$ annihilates the factor $2\xi-r\xi'$ of their profile ODE.
4. Independent local analysis of the crease pattern (my own computations):
   • Vertex geometry. Take squares $Q_k$ of half-side $k$, $k=1,\dots,n$, spokes along $y=\pm x$. At each interior corner vertex $(\pm k,\pm k)$ the sector angles are $(135^\circ,45^\circ,45^\circ,135^\circ)$; at the center they are $4\times 90^\circ$. Both satisfy Kawasaki's flat-foldability condition ($135-45+45-135=0$), so every vertex is locally flat-foldable — the obstruction proved by Demaine et al. is genuinely global.
   • DOF count after triangulation. Adding one diagonal per trapezoid (and the central vertex) gives a triangulated disc with $V=4n+1$ vertices ($V_b=4$ boundary corners, $V_i=4n-3$ interior), $E=12n-4$ edges ($E_{\mathrm{int}}=12n-8$), $F=8n-4$ triangles (Euler check: $V-E+F=1$ ✓). The standard rigid-origami count $\mathrm{DOF}=E_{\mathrm{int}}-3V_i=(12n-8)-3(4n-3)=1$. So for every choice of diagonals the generic folded hypar is a 1-DOF mechanism: the folded PL surface is not unique but moves along a one-parameter family — matching Liu–Tachi–Paulino's parameter $k$ and their single-soft-mode eigenvalue computation. (This is the usual Maxwell-type generic count, valid when the constraint Jacobian has full rank; I did not prove full-rankness for this pattern.)
   • Hence the 2^{4(n-1)} diagonal patterns each yield (generically) a 1-parameter family of PL foldings; the experiments of Liu–Tachi–Paulino show physical paper selects the alternating asymmetric pattern, and the homogenized limit of any of these families is the same surface $z=k(x^2-y^2)$.

Result

Question 1 is resolved in the literature (post-2015): with one added diagonal fold per trapezoid the pattern becomes a generically 1-DOF triangulated origami, and its folded shape — at any folding depth and independently of the diagonal pattern (among the symmetric schemes) and of the spacing of the squares — converges, as the pleats refine, to the genuine hyperbolic paraboloid $z=k(x^2-y^2)$, with $k$ a monotone function of the folding angle (Liu–Tachi–Paulino, Nat. Commun. 10:4238, 2019). Bistability with two mirror-symmetric stable states is proved in the same work. Question 2 (other fold-line patterns, e.g. circular pleats) is only partially addressed (N-gon hypar extensions; framework remarks) and remains open.

What remains

• A fully discrete, rigorous description of the finite-$n$ PL folded surface for each of the $2^{4(n-1)}$ diagonal patterns (existence/uniqueness of the 1-DOF motion, self-intersection, which branch paper selects); currently only simulation and experiment.
• The circular pleated surface (Tabachnikov's Fig. 3): no theorem analogous to the hypar limit is known; even a conjectural limit shape seems absent from the literature.
• General theory: for which concentric-polygon pleat patterns does the homogenized limit exist and which Monge–Ampère-type PDE/ODE governs the limit shape (the hypar ODE came from an integrability identity $\cos\psi'=\cos\gamma-1$ specific to right-angled symmetry planes).