Classification of rational angles in plane lattices
By Roberto Dvornicich, Francesco Veneziano, and Umberto Zannier
Abstract
This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of $\pi$. We shall study how many such angles may appear in a given lattice and in which positions, allowing the lattice to vary arbitrarily.
This classification turns out to be much less simple than could be expected, leading even to parametrizations involving rational points on certain algebraic curves of positive genus.
1. Introduction
The present paper will be concerned with lattices in $\mathbb{R}^2$ and in fact with the angles $\widehat{ABC}$ determined by an ordered triple of distinct points $A$,$B$,$C$ varying through the lattice. Our leading issue will involve angles which are rational multiples of $\pi$, which we will call rational angles for brevity. These angles of course appear in regular polygons, in tessellations of the plane, and in other similar issues, and it seems to us interesting to study in which lattices these angles appear and how.
Suppose $P$,$Q$ are points in the lattice $\mathbb{Z}^2$, and let $\widehat{POQ}$ be the angle formed between the rays $OP$ and $OQ$ (where $O$ denotes the origin); one might wonder when this angle is a rational multiple of $\pi$. It turns out that if $\theta$ is a rational multiple of $\pi$, then $\theta$ is one of $\pm \frac{\pi }{4}$,$\pm \frac{\pi }{2}$,$\pm \frac{3\pi }{4}$,$\pm \pi$, as shown by J. S. Calcut in Reference Cal09 (see the Appendix to this Introduction for a proof simpler than Calcut’s). Analogous properties hold for the Eisenstein lattice generated by the vertices of an equilateral triangle in the plane: the rational angles which occur are precisely the integer multiples of $\pi /6$.
Let us recall a few historical precedents of similar problems.
Considering the simplest lattice $\mathbb{Z}^2$, E. Lucas Reference Luc78 in 1878 answered in the negative the rather natural question of whether points $A$,$B$,$C\in \mathbb{Z}^2$ can determine an equilateral triangle. In fact, it is not very difficult to show that no angle $\widehat{ABC}$, with $A$,$B$,$C\in \mathbb{Z}^2$, can be equal to $\pi /3$. (This will follow as an extremely special case of our analysis, but we anticipate it in the Appendix to the Introduction (§ 1.1) with a short and simple argument.)
In 1946, in a related direction, W. Scherrer Reference Sch46 considered all regular polygons with all vertices in a given arbitrary lattice and proved that the polygons which may occur are precisely those with $3$,$4$, or $6$ sides. (See the Appendix to the Introduction (§ 1.2) for an account of Scherrer’s nice proof with some comments.)
These results prompt the question: what happens when we consider other planeFootnote^{1} lattices?
^{1}
Of course one could consider analogous problems in higher dimensions, however we expect the complexity will increase greatly. There is some work in progress by Poonen et al. which appears to be not unrelated.
Namely, for an arbitrary lattice $\Lambda \subset \mathbb{R}^2$, how can we describe all rational angles determined by three points in $\Lambda$? The most ambitious goal would be to obtain a complete classification, in some sense. In particular, we can list the following issues in this direction:
1.
Which rational angles can occur in a given plane lattice $\Lambda$? Intuitively, we should not expect many such angles when the lattice is fixed.
2.
In which positions can they occur? Namely, which triples of points $A$,$B$,$C\in \Lambda$ can determine such a given angle? Note that it is clearly sufficient to consider the case when $B$ is the origin and that we may replace $A$,$C$ by any two points in the lattice on the lines $OA$,$OC$. So we shall consider an angle determined by the origin and two lines passing through the origin and another lattice point.
3.
Adopting a reciprocal point of view, how can we classify plane lattices according to the structure of all rational angles determined by their points?
We shall give below a more precise meaning to these questions; for instance, we shall study plane lattices according to how many rational angles (with vertex in the origin) may occur and in which geometric configurations; for instance it is relevant which pairs of angles have a side in common.
It is clear that for any arbitrarily prescribed angle we are able to find a plane lattice in which this angle appears as the angle determined by three points of the lattice. It is easily seen that even a second (rational) angle can be prescribed arbitrarily. On adding further conditions on the rationality of other angles and their relative positions, the arithmetical information deduced from these conditions increases and imposes severe restrictions on the lattice, which can lead to a classification. Let us observe that the variables in our problem are
(i)
the lattices,
(ii)
the (rational) angles,
(iii)
the lines (through lattice points) determining these angles.
We shall see that for three or more rational angles in a lattice we have roughly the following possibilities:
(A)
either the angles belong to a certain finite set which is described in Section 5,
(B)
or the lattice belongs to one of finitely many families of lattices (and corresponding angles) which are described in Section 6.4.
More precisely, with a notation that will be properly introduced later, we will prove in Sections 5 and 6 the following theorem.
Furthermore, when the angles lie in the mentioned finite set and are fixed, then either
(A1)
the lattice belongs to a certain well-described family (called CM in this paper); moreover, for a fixed lattice in this family the points determining the angles are parameterized by the rational points in finitely many rational curves (see Section 3.1 and equation Equation 3.2); or
(A2)
both the lattice and the vertices are parametrized by the rational points on finitely many suitable algebraic curves; or
(A3)
we have a finite set of lattices-vertices.
We shall analyze all of these situations, also looking at the maximal number of different configurations of rational angles which can exist in a lattice, in the sense of item 2 above. We shall prove that indeed this number is finite apart from well-described families.
As it is natural to expect, this study involves algebraic relations among roots of unity, a topic which falls into a well-established theory. However, even with these tools at our disposal, our problem turns out to be more delicate than can be expected at first sight, in that some surprising phenomena will appear on the way to a complete picture. For instance we shall see that some of the curves mentioned in (A2) have positive genus, in fact we have examples of genus up to $5$.
In this paper we shall give an “almost” complete classification, in the sense alluded to above. In particular, this shall be complete regarding the sets of angles which may appear. Concerning the classification of specific configurations of angles, we shall confine them to a certain explicit finite list. In the present paper we shall treat in full detail only a part of them, in particular when the rational angles considered share a side.
We also give a full discussion of a case in which configurations arise that correspond to rational points on a certain elliptic curve (of positive rank). (See § 8)
We postpone to a second paper the discussion of the remaining few cases, which need not be treated differently, but which are computationally rather complicated due to the combinatorics of the configurations. We note that in view of Faltings’ theorem, there are only finitely many rational points on the curves of genus $>1$ which appear; however no known method is available to calculate these points (only to estimate their number), so we shall not be able a priori to be fully explicit in these cases.
A concrete example
As an illustration of the kind of problems one is faced with, we show here an example of a curve of high genus which arises from the general treatment of lattices with three nonadjacent rational angles.
Fixing the amplitudes of the angles to be $\frac{3}{5}\pi$,$\frac{3}{10}\pi$,$-\frac{1}{10}\pi$, we are led to study the rational solutions to the system $f_1=f_2=0$, where
$$\begin{align*} f_1= &-a^2 b^2 - a^2 b c + a b^2 c - 2 a b c^2 + 2 a b^2 d - a^2 c d \\ &+10 a b c d - 5 b^2 c d - a c^2 d + b c^2 d - 4 a b d^2 + 2 b c d^2 - c^2 d^2,\\ f_2= &-a^2 b^2 - 2 a^2 b c + a b^2 c - 5 a b c^2 + 2 a^2 b d - a^2 c d \\ &+ 16 a b c d - 2 b^2 c d - 2 a c^2 d + b c^2 d - 8 a b d^2 + 2 a c d^2 - c^2 d^2. \end{align*}$$
These equations define a variety in $\mathbb{P}_3$ which consists of the four lines $ab=cd=0$ and of an irreducible curve $\mathcal{C}$ of genus 5.
It is worth noting that the curve $\mathcal{C}$ contains some trivial rational points such as, for instance, $(1:1:1:1)$, but also nontrivial rational points such as $(12:2:-8:-3)$, which corresponds to the lattice generated by 1 and $\tau =r\theta$, with
In addition to the angle spanned by 1 and $\tau$, there are two more rational angles to be found between lines through the origin passing through elements of the lattice. The angle spanned by $\tau +12$ and $\tau +2$ has an amplitude of $\frac{3}{10}\pi$, and the one spanned by $\tau -3$ and $\tau -8$ has an amplitude of $\frac{1}{10}\pi$. Figure 1 illustrates these angles.
As will be clear soon, these problems are better analyzed not in plane lattices but by viewing the real plane $\mathbb{R}^2$ as the complex field $\mathbb{C}$ and considering, in place of the lattices, the $\mathbb{Q}$-vector subspaces of $\mathbb{C}$ generated by the lattices.
Organization of the paper
The paper will be organized roughly as follows.
•
In § 2 we shall introduce in detail our issues, giving also some notation and terminology.
In the first two parts we shall find general equations corresponding to the configurations that we want to study (as described in the previous section).
The third part will be devoted to obtaining a linear relation in roots of unity, with rational coefficients, after elimination from the equations obtained formerly (depending on the configuration).
In the fourth part we will study more in depth the elimination carried out in the previous part, in order to prove some geometric results.
The fifth part will recall known results from the theory of linear relations in roots of unity, which shall be used to treat the mentioned equations.
•
In § 4 we shall study spaces with special symmetries (for instance those which correspond to imaginary quadratic fields).
•
In § 5 we shall prove the bound appearing in Theorem 1.1. For this we shall use among other things results of the geometry of numbers (not applied to the original lattice however, but to a certain region in dimension $3$).
•
In § 6 we classify the finite number of continuous families of lattices which escape the previous theorem.
•
In § 7 we study the configurations where there exist four nonproportional points $P_1$,$P_2$,$P_3$,$P_4$ of the lattice such that every angle $\widehat{P_iOP_j}$ is rational. In this analysis we shall meet two rather surprising geometric shapes (which we shall call dodecagonal).
•
Section 8 will contain the complete study of an elliptic curve such that its rational points correspond to lattices with three nonadjacent rational angles. (The group of rational points will be found to be isomorphic to $\mathbb{Z}/(2)\times \mathbb{Z}$.)
Appendix to the Introduction
In this short appendix we give a couple of simple proofs related to the known results cited in the Introduction. These will be largely superseded by the rest of the paper, but due to their simplicity we have decided to offer independent short arguments for them.
1.1. Rational angles in the Gaussian lattice
By Gaussian lattice we mean as usual the lattice $\Lambda =\mathbb{Z}+\mathbb{Z}i\subset \mathbb{C}$. For our issue of angles, it is equivalent to consider the $\mathbb{Q}$-vector space generated by the lattice, i.e., $V=\mathbb{Q}+\mathbb{Q}i$. This space has the special feature of being a field, i.e., the Gaussian field $\mathbb{Q}(i)$, which makes our problem quite a bit simpler. In fact, let $\alpha =2\pi a/b$ be a rational angle occurring in $\Lambda$ or $V$, where $a$,$b$ are coprime nonzero integers, and let $\zeta =\exp (2\pi ia/b)$. That $\alpha$ occurs as an angle in $\Lambda$ means that there are nonzero points $P$,$Q\in \Lambda \subset \mathbb{C}$ such that $Q=\zeta rP$ where $r\in \mathbb{R}^*$. Conjugating and dividing, we obtain $\zeta ^2=x/\bar{x}$, where $x= Q\bar{P}\in \mathbb{Q}(i)$. So the root of unity $\zeta ^2$ lies in $\mathbb{Q}(i)$, and it is well known that $\mathbb{Q}(i)$ contains only the fourth roots of unity, so $\zeta$ has order dividing $8$. A direct argument is to observe that $\zeta ^2$ is an algebraic integer so must be of the shape $r+is$ with integers $r$,$s$, and being a root of unity this forces $r^2+s^2=1$, so $\zeta ^2$ is a power of $i$ and $\zeta ^8=1$, as required.
In the converse direction, observe that indeed $\exp (\pi i/4)$ is determined by the three points $(1,1)$,$(0,0)$,$(1,0)$ of $\Lambda$, (i.e., the complex numbers $1+i$,$0$,$1 \in \mathbb{Q}(i)$).
A similar argument holds for the Eisenstein lattice, which again generates over $\mathbb{Q}$ a field, namely the field generated by the roots of unity of order $6$; hence $\zeta$ can be a twelfth root of unity.
These special lattices will be treated in greater generality in § 4.2.
1.2. On Sherrer’s proof
As mentioned above, Sherrer proved that the only regular polygons with vertices in some lattice are the $n$-gons for $n=3$,$4$,$6$. The idea of his proof is by descent: if $P_1$, …, $P_n$ are the vertices of an $n$-gon in a given lattice $\Lambda$, then $P_{i+i}-P_i$ are again the vertices of a regular $n$-gon in $\Lambda$; however, if $n>6$, the new sides are smaller whence iterating the procedure, we obtain a contradiction (and similarly with a little variation for $n=5$). In fact, this argument amounts to the following: we assume as before that the lattice is inside $\mathbb{C}$ and (after an affine map) that $\mathbb{Q}\Lambda$ contains all the $n$th roots of unity $\zeta ^m$ where $\zeta =\exp (2\pi i/n)$. By appealing to the fact, already known to Gauss, that the degree $[\mathbb{Q}(\zeta ):\mathbb{Q}]$ is equal to $\phi (n)$, we may already conclude that $\phi (n)\le 2$, i.e., $n=1$,$2$,$3$,$4$,$6$. But we may also avoid using such result, on observing that for $n>6$, the ring $\mathbb{Z}[\zeta ]$ contains nonzero elements of arbitrarily small complex absolute value, so $\mathbb{Z}[\zeta ]$ cannot be contained in a lattice, which is discrete. To justify the assertion, consider the elements $(\zeta -1)^m$. If $n>6$, they have absolute value which is decreasing to $0$ since $0<|\zeta -1|<1$. For $n=5$, we may instead consider $1+\zeta ^2$ in place of $\zeta -1$.
2. Terminology and notation
We identify the euclidean plane with $\mathbb{C}$. We call a rational angle in $\mathbb{C}$ an ordered couple of distinct lines through the origin such that the measure of the euclidean angle between them is a rational multiple of $\pi$. We say that two points $v_1$,$v_2\in \mathbb{C}\setminus \{0\}$ such that $v_2/v_1\not \in \mathbb{R}$ determine (or form) a rational angle if the lines $(\mathbb{R}v_1, \mathbb{R}v_2)$ do (i.e., if the argument $v_2/v_1$ is a rational multiple of $\pi$). With a slight abuse of notation we write the angle $(\mathbb{R}v_1, \mathbb{R}v_2)$ as $(v_1,v_2)$.
Let $\Lambda \subseteq \mathbb{C}$ be a lattice. Given a rational angle determined by elements of $\Lambda$, many more pairs in $\Lambda ^2$ can be trivially found (by multiplication by integers) that determine the same angle. Therefore we prefer to tensor the whole lattice by $\mathbb{Q}$ and study angles in the tensored space. With this point of view, we can say that when we draw rational angles in $\Lambda$ we extend the sides indefinitely and we are not concerned with which points of $\Lambda$ actually meet the sides.
In this setting, the objects that we will study are two-dimensional $\mathbb{Q}$-vector spaces $V\subset \mathbb{C}$ that contain two $\mathbb{R}$-linearly independent vectors. These are precisely the sets obtained after tensoring a plane lattice by $\mathbb{Q}$. From now on, unless otherwise stated, we will refer to these sets simply as spaces.
Any angle-preserving transformation of $\mathbb{C}$ that sends the origin to itself will clearly establish a bijection between the rational angles of a space $V$ and the rational angles of its image.
These transformations are generated by complex homotheties of the form $z\mapsto \lambda z$ for a fixed $\lambda \in \mathbb{C}^*$ and by complex conjugation. For this reason we will say that two spaces $V_1$,$V_2$ are homothetic (and we write $V_1\sim _h V_2$) if they are sent one to the other by a homothety, and we will say that they are equivalent (and we write $V_1\sim V_2$) if $V_1$ is homothetic to $V_2$ or to its complex conjugate $\overline{V_2}$.
For ease of notation, we will often write $V(\tau )\coloneq \left\langle 1,\tau _1 \right\rangle _\mathbb{Q}$.
In every space $V$, given a rational angle $(v_1,v_2)$, we can obtain other rational angles by swapping them. We call the angles thus obtained equivalent, and we are not concerned with them.
Given two adjacent rational angles $(v_1,v_2)$ and $(v_2,v_3)$, we see immediately that $(v_1,v_3)$ is also a rational angle. For this reason it is more convenient to consider sets of adjacent angles as a single geometrical configuration, rather than as independent angles. This point of view is also supported by the shape of the equations that describe these cases. Therefore we call a rational $n$-tuple a set of $n$ vectors $\{v_1,\dotsc ,v_n\}$ such that $(v_i,v_j)$ is a rational angle for all $i\neq j$.
According to this definition, a rational $n$-tuple can be identified with an $n$-element subset of $\mathbb{P}(V)\subseteq \mathbb{P}(\mathbb{C})$. This point of view however is not particularly useful when writing up the equations that describe the configuration.
From the shape of equations Equation 3.5 and Equation 3.11 below, which describe rational $n$-tuples, it is clear that angles which belong to a rational $n$-tuple containing 1 are qualitatively different from angles which do not belong to such an $n$-tuple. In fact, an angle in a rational $n$-tuple containing 1 leads to an equation of degree 1 in $\tau$, while angles in rational $n$-tuples not containing 1 lead to equations of degree 2 in $\tau$.
In light of these considerations, we will say that a space $V$ is of type $\enclose {circle}{n}$ if it contains a rational $n$-tuple; we extend this notation additively, by saying that $V$ is of type $\enclose {circle}{n}+\enclose {circle}{m}$ if it contains a rational $n$-tuple and a disjoint rational $m$-tuple, and so on. There is an obvious partial order on the possible types, and we say that $V$ has an exact type $a_1\enclose {circle}{n_1}+\dots +a_k\enclose {circle}{n_k}$, if this type is maximal for $V$. We will characterize in § 4.2 the spaces for which such a maximal type exists.
We remark that spaces of type $\enclose {circle}{3}$ correspond to triangles in which all angles are rational multiples of $\pi$.
We will denote by $U$ the set of all roots of unity.
3. Equations
3.1. The equation of a rational angle
Let $\tau \in \mathbb{C}\setminus \mathbb{R}$, and let $V=\left\langle 1,\tau \right\rangle _\mathbb{Q}$. Let $a_0$,$a_1$,$b_0$,$b_1\in \mathbb{Q}$ be such that $(a_0\tau +a_1,b_0\tau +b_1)$ is a rational angle, which is to say, there exists a root of unity $\mu \in U\setminus \{\pm 1\}$ such that the ratio $\frac{b_0\tau +b_1}{a_0\tau +a_1}\mu$ is real. Setting this ratio equal to its conjugate leads to the equation
This shows that $\mu ^2\in \mathbb{Q}(\tau ,\overline{\tau })$. Equation Equation 3.1 is bihomogeneous of degree 1 in both $a_0$,$a_1$ and $b_0$,$b_1$, so it defines a curve $\mathcal{C}\subseteq \mathbb{P}_1\times \mathbb{P}_1$. Setting
$$\begin{equation} a_0 b_0 A+a_0 b_1 B + a_1 b_0 C + a_1 b_1 =0. \cssId{texmlid1}{\tag{3.2}} \end{equation}$$
The curve $\mathcal{C}$ has genus 0 and it is irreducible. In fact, it has bidegree $(1,1)$ and, if it were not irreducible, it would have two components of bidegrees $(0,1)$ and $(1,0)$. This happens if and only if $A=BC$, and a small computation shows that this happens if and only if $\tau \in \mathbb{R}$.
There is a bijection between $\mathcal{C}(\mathbb{Q})$ and the set of rational angles $(v_1,v_2)$ in $V$ with $\arg (v_2/v_1)=\mu$. However, $\mathcal{C}$ is in general not defined over $\mathbb{Q}$, but only over the field $\mathbb{Q}(\tau ,\overline{\tau })\cap \mathbb{R}$. This fact plays a role in the characterization of spaces with infinitely many rational angles.
3.2. The equations of a rational $n$-tuple containing $1$
Let $V$ be a space with a rational $n$-tuple($n\geq 3$). Up to homothety of the space and equivalence of rational angles, we can assume that $V=\left\langle 1,\tau \right\rangle _\mathbb{Q}$, and that the $n$-tuple is given by $\{1,\tau ,\tau +a_1,\dotsc ,\tau +a_{n-2}\}$, where the $a_j$ are distinct rational numbers different from 0. We write $\tau =r\theta _0$, with $r=\left| \tau \right|$ and $\theta _0\in U$. Similarly, let $\tau +a_j=\left| \tau +a_j \right|\theta _j$ for $j=1$, …, $n-2$.
In particular we have that $(r\theta _0+a_j)/\theta _j\in \mathbb{R}$ for $j=1$, …, $n-2$. Equating these numbers and their conjugates, we can write
where we have set $x_j=\theta _j^2$ for $j=0$, …, $n-2$.
3.3. The equations of a rational $n$-tuple not containing $1$
Let $V=\left\langle 1,\tau \right\rangle _\mathbb{Q}$ and $\tau =r\theta _0$, with $r=\left| \tau \right|$ and $\theta _0\in U$. Let $\{v_0,\dotsc ,v_{n-1}\}$ be a rational $n$-tuple($n\geq 2$) which does not contain vectors proportional to 1 or $\tau$. By rescaling we can assume $v_j=\tau +b_j$ for $j=0$, …, $n-1$ and the $b_j$ distinct nonzero rational numbers. Let $\mu _j\in U$ such that $\mu _j(\tau +b_0)/(\tau +b_j)\in \mathbb{R}$ for $j=1$, …, $n-1$. Then we have
We begin by observing that, if $V$ is a space of type $\enclose {circle}{2}$, it is homothetic to $\left\langle 1,\theta r \right\rangle$ with $\theta \in U\setminus \{\pm 1\}$.
If a space $V$ has two nonequivalent rational angles, we have seen in Sections 3.2 and 3.3 that we can derive equations for $\tau$ with coefficients in cyclotomic fields; therefore $\tau \in \overline{\mathbb{Q}}$.
If we have two independent such equations, we can eliminate $\tau$ and obtain one equation in roots of unity with rational coefficients. We shall then apply the results of Section 3.5 with the aim of bounding the degree of the roots of unity intervening in the equation, outside of certain families which admit a parametrization.
When eliminating $\tau$, we could have two equations of shape Equation 3.5 (which amounts to having a rational 4-tuple), or one of shape Equation 3.5 and one of shape Equation 3.11 (which amounts to one rational triple and one more angle not adjacent to it) or two equations of shape Equation 3.11, for which we need three rational angles pairwise nonadjacent. These are the cases that we denote as type $\enclose {circle}{4}$, type $\enclose {circle}{3}+\enclose {circle}{2}$ and type