Arnold Mathematical Journal
Volume 11, Issue 4, 2025

Research Contribution DOI:10.56994/ARMJ.011.004.005
Received: 04 Apr 2025; Accepted: 09 Jul 2025

Folding of quadrilaterals, zigzags, and Arnold-Liouville integrability

Anton Izosimov

Abstact.

Key words and phrases:

Mathematics Subject Classification:

1 Introduction

One of the historically first manifestations of integrability is Poncelet’s porism, also known as Poncelet’s closure theorem. Poncelet’s theorem says that if a planar $n$ -gon is inscribed in a conic $C_{1}$ and circumscribed about another conic $C_{2}$ , then any point of $C_{1}$ is a vertex of such an $n$ -gon, see Figure 1. The two arguably most standard proofs of this theorem are based, respectively, on complex and symplectic geometry. The complex proof goes roughly as follows. One can identify the space of tangents dropped from a point on $C_{1}$ to $C_{2}$ with an elliptic curve. The successive sides of a polygon inscribed in $C_{1}$ and circumscribed about $C_{2}$ are points on that curve related to each by a fixed translation. This polygon closes up if and only if the translation vector is a torsion point on the elliptic curve. Whether or not that is the case depends only on $C_{1}$ and $C_{2}$ , but not on the initial point, so all polygons inscribed in $C_{1}$ and circumscribed about $C_{2}$ will close up after the same number of steps [GH77].

Refer to caption — Figure 1: Every point of $C_{1}$ is a vertex of a pentagon inscribed in $C_{1}$ and circumscribed about $C_{2}$ .

The second, symplectic, proof is based on the fact that any two generic conics can be mapped, by a projective transformation, to confocal conics. In the confocal case, a polygon inscribed in $C_{1}$ and circumscribed about $C_{2}$ can be identified with a billiard trajectory in $C_{1}$ . The billiard in a conic is an integrable system, and any two polygons inscribed in $C_{1}$ and circumscribed about $C_{2}$ correspond to trajectories belonging to the same level set of the first integral. Hence, by Arnold-Liouville theorem, if one of the trajectories is periodic with period $n$ , then so is the other one, cf. [LT07].

A lesser known relative of Poncelet’s porism is Darboux’s porism on folding of quadrilaterals. Folding of a vertex of a planar polygon is the reflection of that vertex is the diagonal joining its neighbors, see Figure 2. Darboux’s porism says that if a sequence of alternating foldings of adjacent vertices restores, after $2n$ steps, the initial polygon, then this is the case for any polygon with the same side lengths. For example, folding any polygon with side lengths $1,3,3\sqrt{5},5$ six times, we come back to the initial polygon, see [Izm23, Figure 2].

Just like Poncelet’s porism, Darboux’s theorem can be proved using elliptic curves. Specifically, one shows that the complexified moduli space of quadrilaterals with fixed side length is an elliptic curve. Composition of foldings at adjacent vertices amounts to a translation on that curve. Whether or not a sequence of foldings restores the initial polygon depends on whether the translation vector is a torsion point and is independent on the particular choice of a quadrilateral [Izm23].

What currently seems to be missing in the literature is a symplectic proof of Darboux’s theorem. We provide such a proof in the present paper. Specifically, we show that, in an appropriate sense, Darboux’s folding is Arnold-Liouville integrable, and deduce Darboux’s porism.

Furthermore, we extend these results to Bottema’s zigzag porism [Bot65], which can be stated as follows. Let $C_{a}$ and $C_{b}$ be two circles such that there exists a unit equilateral $2n$ -gon whose odd-indexed vertices lie on $C_{a}$ and even-indexed vertices lie on $C_{b}$ . Then there exist infinitely many such $2n$ -gons. The zigzag porism is equivalent to Darboux’s porism when the circles are coplanar [CH00], but is in fact valid for any two circles in $\mathbb{R}^{3}$ [BHH74]. We construct the underlying Arnold-Liouville integrable system in this more general setting.

2 Arnold-Liouville integrability of folding

Let $\mathcal{P}$ be the space of quadrilaterals $A B C D$ with fixed side lengths, considered up to orientation-preserving isometries. There is abundant literature on the topology of such spaces for polygons with any number of vertices, see [KM95] and references therein. The space $\mathcal{P}$ is a smooth manifold assuming that there is no linear combination of side lengths with coefficients $\pm 1$ which is equal to zero [KM95, Lemma 2]. In the case of quadrilaterals, this manifold is diffeomorphic to a circle or disjoint union of two circles, see [KM95, Theorem 1]. These circles are distinguished by the sign of the oriented area and are interchanged by an orientation-reversing isometry, cf. [KM95, Section 10].

Denote by $F_{B}\colon\mathcal{P}\to\mathcal{P}$ folding of the vertex $B$ . This mapping is well-defined assuming that the vertices $A$ and $C$ cannot come together. This holds provided that the side lengths satisfy at least one of the following conditions: $|AB|\neq|BC|$ or $|AD|\neq|CD|$ . Likewise, let $F_{C}\colon\mathcal{P}\to\mathcal{P}$ be folding of $C$ , and let $F:=F_{C}\circ F_{B}$ be the composition of the two foldings. Darboux’s porism says that if $F^{n}(P)=P$ for some quadrilateral $P\in\mathcal{P}$ , then $F^{n}$ is the identity mapping on $\mathcal{P}$ . We shall prove this by establishing Arnold-Liouville integrability of the mapping $F$ .

Clearly, $F$ cannot be Arnold-Liouville integrable on the space $\mathcal{P}$ of quadrilaterals with fixed side lengths, as the latter space is one-dimensional. So, we consider a bigger space ${\mathcal{P^{\prime}}}$ of quadrilaterals with fixed lengths of the sides $A B, B C, C D$ , again considered up to orientation-preserving isometries. This space is diffeomorphic to a two-dimensional torus and is parametrized by the oriented angles $\angle ABC$ and $\angle BCD$ . The squared length of the side $A D$ is a smooth function of the torus $\mathcal{P}^{\prime}$ . The space $\mathcal{P}$ of quadrilaterals with fixed lengths of all sides is a level set of that function.

Theorem 2.1.

The folding mapping $F=F_{C}\circ F_{B}$ is Arnold-Liouville integrable on the moduli space $\mathcal{P}^{\prime}$ of quadrilaterals $A B C D$ with fixed lengths of the sides $A B, B C, C D$ .

Proof.

Folding does not affect side lengths. In particular, $|AD|^{2}$ is a first integral of $F$ . Furthermore, the map $F\colon\mathcal{P}^{\prime}\to\mathcal{P}^{\prime}$ has an invariant symplectic structure given by

\Omega:=d\angle ABC\wedge d\angle BCD.

To show invariance, consider, for instance, folding of the vertex $C$ depicted in Figure 2. The pullback of the symplectic form $\Omega$ by this map is

F_{C}^{*}\Omega=d\angle ABC^{\prime}\wedge d\angle BC^{\prime}D=d(\angle ABC-2% \angle CBD)\wedge d(2\pi-\angle BCD)=-\Omega-2d\angle CBD\wedge d\angle BCD.

Furthermore, since the side lengths $|BC|$ and $|CD|$ are fixed, the angle $\angle CBD$ is a function of the angle $\angle BCD$ and is independent of the angle $\angle ABC$ . So, $d\angle CBD\wedge d\angle BCD=0,$ implying

F_{C}^{*}\Omega=-\Omega,

i.e., the form $\Omega$ is anti-invariant under a single folding, and hence invariant under $F$ . ∎

3 Darboux’s porism

4 A remark on polygons with more vertices

5 The zigzag porism

Let $C_{a}$ and $C_{b}$ be two circles in $\mathbb{R}^{3}$ . A zigzag between $C_{a}$ and $C_{b}$ is an equilateral polygon whose odd-indexed vertices lie on $C_{a}$ and even-indexed vertices lie on $C_{b}$ . The zigzag porism says that if there exists a closed $2n$ -gonal zigzag between $C_{a}$ and $C_{b}$ , then any zigzag between $C_{a}$ and $C_{b}$ with the same edge length is also a closed $2n$ -gon [Bot65, BHH74], see Figure 4.

A zigzag between two circles $C_{a}$ , $C_{b}$ may be built by iterating the zigzag map $Z\colon C_{a}\times C_{b}\to C_{a}\times C_{b}$ which sends a pair $A\in C_{a}$ , $B\in C_{b}$ to a pair $A^{\prime}\in C_{a}$ , $B^{\prime}\in C_{b}$ such that $|A^{\prime}B^{\prime}|=|A^{\prime}B|=|AB|$ , see Figure 5. This map is a composition of two involutions, namely $(A,B)\mapsto(A^{\prime},B)$ , where $|A^{\prime}B|=|AB|$ , and $(A^{\prime},B)\mapsto(A^{\prime},B^{\prime})$ , where $|A^{\prime}B^{\prime}|=|A^{\prime}B|$ . Observe that, in the case when the circles $C_{a},C_{b}$ are coplanar, these involutions are just foldings of the quadrilateral $O_{a}ABO_{b}$ , where $O_{a}$ , $O_{b}$ are centers of $C_{a},C_{b}$ , at $A$ and $B$ respectively, see Figure 6. So, the planar case of the zigzag porism is equivalent to Darboux’s porism [CH00]. Here we show that the integrability result carries over to the spatial situation:

Theorem 5.1.

The zigzag map $Z$ is Arnold-Liouville integrable for any circles $C_{a}$ , $C_{b}$ in $\mathbb{R}^{3}$ .

Proof.

By definition, the map $Z\colon(A,B)\mapsto(A^{\prime},B^{\prime})$ preserves the squared distance between $A$ and $B$ . So, it suffices to find an area form on $C_{a}\times C_{b}$ invariant under $Z$ . Let $\phi_{a},\phi_{b}\in\mathbb{R}/2\pi\mathbb{Z}$ be standard angular parameters on $C_{a}$ , $C_{b}$ . We will prove that the form $d\phi_{a}\wedge d\phi_{b}$ on $C_{a}\times C_{b}$ is preserved by $Z$ . To that end, it suffices to establish anti-invariance of that form with respect to the involutions whose composition gives $Z$ . Furthermore, since those involutions are related to each other by interchanging the roles of the circles $C_{a},C_{b}$ , it is sufficient to consider the involution $(A,B)\mapsto(A^{\prime},B)$ defined by the condition $|A^{\prime}B|=|AB|$ , where $A,A^{\prime}\in C_{a}$ . Let $\hat{B}$ be the orthogonal projection of $B$ onto the plane containing $C_{a}$ . Then $|A\hat{B}|=|A^{\prime}\hat{B}|$ , see Figure 7. Here $O_{a}X$ is the reference direction used to define the angular coordinated $\phi_{a}$ on $C_{a}$ . We have

\angle XO_{a}A+\angle XO_{a}A^{\prime}=2\angle XO_{a}\hat{B}.

So, the sum on the left only depends on the position of the point $B$ but not $A$ . Therefore, in coordinates $\phi_{a},\phi_{b}$ , the involution $(A,B)\mapsto(A^{\prime},B)$ has the form

(\phi_{a},\phi_{b})\mapsto(f(\phi_{B})-\phi_{a},\phi_{b})

for a certain smooth function $f$ . So, the form $d\phi_{a}\wedge d\phi_{b}$ is indeed anti-invariant under this involution. ∎

In terms of the map $Z$ , the zigzag porism says that if an orbit of $(A,B)\in C_{a}\times C_{b}$ under $Z$ is $n$ -periodic, then the same holds for any $(A^{\prime},B^{\prime})\in C_{a}\times C_{b}$ with $|A^{\prime}B^{\prime}|=|AB|$ . This is derived from Theorem 5.1 in the same way as Darboux’s porism is obtained from Theorem 2.1.

Acknowledgments

References

Folding of quadrilaterals, zigzags, and Arnold-Liouville integrability

Abstact.

Key words and phrases:

Mathematics Subject Classification:

1 Introduction

2 Arnold-Liouville integrability of folding

Theorem 2.1.

Proof.

3 Darboux’s porism

Theorem 3.1 (Darboux’s porism).

Remark 3.2.

Proof of Theorem 3.1.

4 A remark on polygons with more vertices

5 The zigzag porism

Theorem 5.1.

Proof.

Acknowledgments

References

Contents