The necklace of a rational right triangle: exact counting and complete periodicity

Introduction

The question

A billiard trajectory in a triangle that meets a side perpendicularly retraces its path. The set of such trajectories therefore organises the dynamics: fixing a side $\sigma$ of a triangle $T$ and shooting perpendicularly from each point of $\sigma$, one obtains a partition of $\sigma$ into finitely or infinitely many cells — the maximal subintervals of $\sigma$ on which the orbit’s combinatorics (the sequence of sides it meets) is constant, equivalently the components of $\sigma$ minus the launch points whose orbit meets a vertex. Two cells may carry the same orbit, launched from mirror points; they are counted separately. When $T$ is rational the partition is finite, and one may ask for the number of cells,

as a function of the triangle. For a general rational triangle this is an unsolved counting problem. This paper solves it — as an exact identity, at every triangle in a family, and in a strong enough form to pin the individual counts — for right triangles, and it does so by producing an explicit model of the unfolded surface rather than by estimating anything.

The family is the following. Let $T$ be a right triangle with acute angle $\alpha$, legs $L_1$, $L_2$ and hypotenuse $H$, and write

Directions are measured in units of $\pi/2Q$, in which the three sides have directions $s(L_2)=0$, $s(L_1)=Q$, $s(H)=P$; a class is a pair $(P/Q,e)$ with $e\in\{0,1\}$ the direction parity.

There is no standing parity hypothesis: every statement below carries its own scope, and the scopes are collected in § 1.5. The model of § 2, the counting identity of § 3, the interval model of § 4, the rigidity theorems of §§§ 6.1, Theorem 6.7 on the pole prongs, the overlap theorem and cylinder count of §§§ 5.2 and § 7.2, the three sufficient conditions of § 7.1, the bounds of § 8 and the sweep law of § 12 hold at every $Q$, both parities. What is developed at odd $Q$ only is the PAIRED/LONE dichotomy — at even $Q$ a side supplies both arcs of $\operatorname{Fix}(\iota)$ or neither, so the two signatures there are $(h,v)=(1,1)$ and $(2,2)$ and the dichotomy does not exist — together with the statement that reads it, the Veech-locus case of $CP\Rightarrow p=0$ (§6.3).

The model: a necklace of kites

Everything below is read off one picture: an explicit presentation of a surface Apisa identified [2 §1.2, §7], and making it explicit is the paper’s first contribution.

Unfolding $T$ in the sense of Katok–Zemlyakov [ZK75] gives a translation surface $S_\alpha$; quotienting by the holonomy involution $\tau$ gives a half-translation surface $B=S_\alpha/\tau$, a meromorphic quadratic differential on $\mathbb{CP}^1$ in the stratum $Q(P-2,Q-P-2,-1^{Q})$ (at $P\in\{2,Q-2\}$ the entry $0$ is a marked point). Concretely (Proposition 2.1):

$B$ is a cyclic chain of $Q$ kites. A kite is two copies of $T$ glued along both legs — a bigon with corners $2\alpha$ and $\pi-2\alpha$ and one interior simple pole. The $Q$ kites are glued along hypotenuse edges, all $O$-corners meeting at a single cone point of angle $P\pi$ and all $A$-corners at one of angle $(Q-P)\pi$.

The kite is where the right angle enters, and it enters as an equivalence rather than a convenience (§ 1.3). The chain carries an explicit transverse coordinate: the interface between the kites at positions $t$ and $t+1$ has transverse measure $\operatorname{ell}_t=\sin(\pi c_t/2Q)$, with $c_t$ running through one parity class modulo $2Q$ in steps of $2P ({xref}Proposition 2.2`). Two features of the model are easy to get wrong. The kites glue in steps of $P$, not of $1$: kite $K_j$ has its hypotenuse edges at directions $\{P+2j,-P+2j\}$, so matching them forces $j'=j\pm P\pmod Q$, and the naive $+1$ reading is right only at $P=1$ and $P = Q - 1$. And on the class $e = 1$ the interface directions carry an offset, without which every interface of that class is misstated.

The class involution $\iota$ reverses the necklace, so it acts as a dihedral reflection of the $Q$-cycle, and its axis meets the cycle in two places, each a kite or an interface. Writing $n_{\mathrm{fix}} \in\{0,1,2\}$ for the number of iota-fixed kites and $m$ for the number of kites the quotient carries in its interior,

In every case the quotient $D = B/\iota$ is a chain of $m$ whole kites, capped at each end either by a half-kite (at an iota-fixed kite, contributing a boundary simple pole) or by an iota-fixed interface. At odd $Q$ — one cap of each kind — the phrase one triangle with both legs on the boundary, followed by $m$ kites in a row is literally right, and it is the picture to hold. The first-return map of the class direction on this chain is a finite composition of interval maps, which we write down explicitly and show to close up (Theorem 4.1); the rest of the paper is the exploitation of that closed form.

Why a right triangle: the hypothesis is not a simplification

It is worth saying at the outset what the right angle buys, because the honest answer makes the hypothesis a feature of the construction rather than a restriction on it.

A kite is the triangle doubled along the two sides meeting at some angle $\gamma$, so that vertex becomes an interior point of cone angle $2\gamma$. A half-translation surface admits only cone angles that are integer multiples of $\pi$ — a quadratic-differential singularity of order $k \ge-1$ has cone angle $(k+2)\pi$ — so $2\gamma= m\pi$ with $0 < \gamma< \pi$ forces $\gamma= \pi/2$, and then $m = 1$: the vertex is a simple pole. No other triangle makes this construction a half-translation surface at all. The orders then sum to $(P-2) + (Q-P-2)-Q = -4$, which forces genus zero, by Gauss–Bonnet for quadratic differentials ([28] Proposition 7.1). So the right angle is what buys the sphere, and the sphere is what buys the explicit interval model. In the parametrisation $(r\pi/n, s\pi/n, t\pi/n)$ with $r+s+t = n$ used by Kenyon and Smillie [19], this family is $n = 2Q$, $(r,s,t) = (P,Q-P,Q)$, and the right angle is exactly the condition $t = n/2$.

What this does not say is that other rational triangles are intractable. The Katok–Zemlyakov unfolding always exists; what fails off the right-angled family is the kite, hence the necklace, hence the interval sequence. Indeed several of the results below use none of it: the counting identity of Theorem 3.4 is an involution count, the total-length identity of Theorem 3.9 is a boundary double count valid on any completely periodic surface, and the reversibility theorems of § 6.1 read only that the return map is reversible and that the pole is the fold’s fixed point. Each generalises once a separatrix diagram is supplied — and producing one off this family is the work.

Relation to previous work

The stratum this paper works in is a well-studied one. Three programmes — Masur–Veech volumes [1], configurations of saddle connections [4], and the statistics of cylinder decompositions [DGZZ20a] — have been carried out over genus-zero strata of quadratic differentials, and all three are asymptotic or statistical: they compute averages over a stratum, growth rates, or almost-everywhere statements. This paper is exact and individual: it fixes one triangle and one direction class, and computes.

That contrast is not by itself what is new. Exact per-direction results are available for lattice (Veech) surfaces, where a Teichmüller curve and its cusps supply the rigidity; McMullen’s analysis of billiards in regular polygons [22] is the sharpest such statement. What distinguishes the present results is the object: $B$ is a lattice surface only on the locus $\min(P,Q-P) \le2$ [19 Thm 2], and the theorems below hold at every coprime $(P,Q)$, with no lattice, no cusp and no congruence invariant available off that locus. The specialisation to right triangles (§ 1.3) is what makes this possible.

Saddle connections and separatrix diagrams. Boissy’s classification of configurations of $\hat{h}$omologous saddle connections on $\mathbb{CP}^{1}$ [4], the genus-zero case of Masur and Zorich’s classification for quadratic differentials [26] (itself the quadratic analogue of Eskin, Masur and Zorich’s [13]), admits a connection joining two poles only in one configuration, accompanied by a longer closed connection that cuts off a half-pillowcase — the shape of the reduction step (ZZ) of § 6.2. It is a resemblance and not an import: the classification assumes that every saddle connection parallel to a member of the collection is $\hat{h}$omologous to it, and in a completely periodic direction of our surfaces almost none are. The separatrix diagram of § 3 is Kontsevich and Zorich’s [20], in the form of [DGZZ19, App. A]; a direction all of whose leaves are closed or saddle connections is a Jenkins–Strebel direction [29], and that is our hypothesis of complete periodicity. What [DGZZ19, DGZZ20a, DGZZ20b] do with these diagrams is count square-tiled surfaces — in genus zero in [DGZZ20a], in general strata in the other two — and ours are not square-tiled.

Counting cylinders. Naveh’s and Lindsey’s bounds on the number of invariant components [27, 21] apply to $S_{\alpha}$ as a translation surface, unmarked — at $P=1$, odd $Q$, Lindsey’s bound gives $(Q-1)/2$ cylinders — but they do not see the marked right-angle images or the branch partition. Closer, for the regular $n$-gon with $n$ odd, McMullen [22] shows that in every periodic direction the geodesics through the Weierstrass points are one zero connection and $(n-1)/2$ closed geodesics, with lengths proportional to $\lvert\sin(2\pi\ell/n)\rvert$; the decomposition into $g$ cylinders of equal modulus goes back to Veech [32]. On the locus $\min(P,Q-P)=2$ this is $n=Q$, and $(n-1)/2$ is the value $\lfloor Q/2\rfloor$ that § 7.2 records for a lone class. We regard this agreement as a check, and do not claim the shape of the count as new. What is new is where it holds: off the lattice locus, where no cusp is available, uniformly in $(P,Q)$; the lone/paired dichotomy, and with it the branch $\lceil Q/2\rceil$, has no counterpart in the lattice picture. Nor does McMullen’s count supply our values on the locus: it counts trajectory types rather than branches of one beam (Proposition 5.1). The values in § 10.2 are ours.

The chain of rhombi. The cell of § 2 and its cyclic gluing are not new. Troubetzkoy reduces billiards in the triangle to billiards in the rhombus obtained by reflecting it in its legs [31 §2]; Hooper describes the invariant surface of a right triangle as a chain of rhombi, each glued to its two neighbours, with removable singularities at their centres and an involution acting by a half-turn about each centre [18 §3]; and Cipra, Hanson and Kolan [7] replace the right triangle by “a rhombus consisting of four copies of the triangle”, for the reason § 2 gives — the right-angle vertex is “a removable singularity” — and follow a trajectory through a chain of such rhombi. Their width $\lvert\sin((2n+1)\alpha)\rvert$ for the interface between consecutive rhombi is Proposition 2.2’s $\ell_t$ at $\varepsilon=0$ — the same expression and the same object. The rhombus is the lift of our kite to $S_\alpha$: the kite is the rhombus folded by Hooper’s half-turn. The quotient $B$ itself is Apisa’s — the pull-back of the pillowcase double of $T$ under $z\mapsto z^Q$ [2 §1.2, §7]. What we add, beyond these, is the explicit kite-chain presentation of $B$ — the gluing in steps of $P$, the transverse measures, and the class $\varepsilon=1$ with the offset of Proposition 2.2, which has no counterpart in the rhombus chain — and what it makes available: § 3’s counting identity and § 6.1’s rigidity. The earlier work on the perpendicular problem itself — numerical, irrational and elementary — is surveyed in [Orph] §1.5.

What is conditional

Two hypotheses appear, each carried on the face of every statement that uses it. Complete periodicity (CP) of the direction class is a genuine hypothesis: it fails at $8/15$ (§ 11). Hypothesis (OB) (orphan branch) is the existence of an orphan branch: the $\perp L_1$ beam of $T(P'/Q)$ has one, where $P' := P$ if $P$ is odd and $P' := Q-P$ if $P$ is even (§ 5.2). It is consumed by Corollary 5.5, and through it in the proof of Corollary 7.11 — to separate the two zeros on the classes with two perpendicular beams off the Veech locus — and in Remark 7.12, Remark 8.3 and the last clause of Proposition 9.3. (OB) holds at every prime-power $Q$ ([Orph] Theorem 4.11), when $Q$ is twice or four times a prime and at the edges of every row ([Orph] Theorems 4.18, 4.24, Corollary 4.20), at $P'=3$ (Corollary 10.9, and [Orph] Theorem 4.11 at $Q=7$), at $P'=Q-2$ (Theorem 10.3), at $P'=5$, $Q\equiv1\pmod{5}$, $Q\ge16$ (Theorem 10.11), and at the centres of [Meth] Theorems 2–3: $\min(P,Q-P)\ge2$ and either $\gcd(P\pm2,Q)=1$ or $Q$ with at most one odd prime factor. It is conjectured everywhere ([Orph] Conjecture 4.12).

At the centres of [23] Theorems 2–3 complete periodicity, $p = 0$ and (OB) all hold, so every row of the table holds outright there ([23] Corollaries 4.1 and 5.13; there $p = 0$ is Corollary 7.11 applied to [23]’s complete periodicity); [23] Corollary 7.15 adds further certified classes.

The computation enters only as a certificate at named centres — complete periodicity decided by the exact flow decomposition of Appendix A — or as data (§ 13).

The parity of $Q$. There is no standing parity hypothesis. The PAIRED/LONE labels exist only at odd $Q$: at even $Q$ a side of $T$ supplies both arcs of Fix(iota) or neither, so $h = v$ and the two signatures are $(1, 1)$ and $(2, 2)$. §6.3 is the one odd-$Q$-only section (apart from statements on rows where coprimality forces $Q$ odd). Three statements have parity in their content, and quoting the odd-$Q$ half alone is a scope error: the shape of $D$ and the triple $(n_{\mathrm{fix}}, m)$ of § 1.2; the right-hand side of the covering law, $Q \mp1$ at odd $Q$ and $Q$ at even; and the cylinder count of Corollary 7.14, whose two branches collapse to $C = Q/2$ at even $Q$.

What this paper imports. From [28]: Theorem 4.6, the turnaround type of an orphan branch, and Theorem 4.7, that there is at most one; Theorems 4.11, 4.18, 4.24 and Corollary 4.20, existence of the orphan, as cases of (OB); Lemma 2.1, Theorem 2.2 and Lemma 4.2 (the retrace criterion and the orphan); Lemmas 3.1 and 4.9 and Proposition 3.4 (the development and finiteness); Corollary 4.14; Corollary 5.1 and Theorem 5.3 (parity, and the two ends of a double normal); the leg’s avoidance of minimal components ([28] §2.1); and the cone data, Propositions 7.5–7.6, Lemma 7.8 and the conventions of [28] §§6–7 — each cited where used. [28] uses nothing from here, so the dependence runs one way. Nothing is imported from [24], which is deposited after this paper: the two results of it that Corollary 12.5 needs, [24] Corollary 4.1 and Proposition 4.3(1)–(4), are reproved as Lemma 12.4 from Hooper’s theorem [18]. [24] is cited only for context, and [23] only for the status of hypotheses (§§1.5, 5.2, 8, 11, 13, Remarks 10.10, 12.6).

Organisation

§2 constructs the necklace and its coordinates. §3 proves the edge count, the counting identity and the covering law, identifies the edges of the separatrix diagram with the billiard prongs, and shows that each pole–pole edge pins a cylinder. §4 gives the interval model and proves that it closes. §5 compares the branch partition with the cylinder decomposition: the crossing number and the overlap theorem. §6 proves the rigidity theorems at a pole, reduces $p = 0$ to two statements, proves the first (no prong of an interior pole ends at a zero), and proves the second, (ZZ), on the Veech locus at odd $Q$. §7 proves three sufficient conditions for complete periodicity and, by one global edge count, $CP \Longleftrightarrow p = 0$ at every centre, and draws the consequences of $p = 0$ for the cylinders. §8 bounds the count. §9 describes the involution on the cylinders. §10 computes: the whole lattice locus, the numerator three, and the family $n(5/(5m + 1)) = 5$, $m \ge3$. §11 treats the centres where complete periodicity fails. §12 draws the consequences for billiard paths in mirror directions, among them the sweep law. §13 lists what is open. Appendix A describes the computation.

The necklace

Throughout, $T = T(\alpha)$ is the right triangle of §1.1, with legs $L_1$, $L_2$ and hypotenuse $H$, and $n_{\mathrm{fix}}$, $m$ are as in §1.2, so that $n_{\mathrm{fix}} + 2m = Q$ and $m = (Q - 1)/2$ at odd $Q$.