Tilings of Convex Polygons with Congruent Triangles

M. Laczkovich

doi:10.1007/s00454-012-9404-x

Discrete & Computational Geometry

September 2012, Volume 48, Issue 2, pp 330–372 | Cite as

Tilings of Convex Polygons with Congruent Triangles

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M. LaczkovichEmail author

Article

First Online: 23 February 2012

Received: 28 February 2011

Accepted: 30 January 2012

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Abstract

By the spectrum of a polygon A we mean the set of triples (α,β,γ) such that A can be dissected into congruent triangles of angles α,β,γ. We propose a technique for finding the spectrum of every convex polygon. Our method is based on the following classification. A tiling is called regular if there are two angles of the triangles, α and β such that at every vertex of the tiling the number of triangles having angle α equals the number of triangles having angle β. Otherwise the tiling is irregular. We list all pairs (A,T) such that A is a convex polygon and T is a triangle that tiles A regularly. The list of triangles tiling A irregularly is always finite, and can be obtained, at least in principle, by considering the system of equations satisfied by the angles, examining the conjugate tilings, and comparing the sides and the area of the triangles to those of A. Using this method we characterize the convex polygons with infinite spectrum, and determine the spectrum of the regular triangle, the square, all rectangles, and the regular N-gons with N large enough.

Keywords

Tilings with congruent triangles Regular and irregular tilings

1 Introduction

This paper is concerned with the following problem. Suppose we are given a convex polygon A. Decide, whether or not A can be dissected into congruent triangles, and if there is such a dissection, find all triangles T such that A has a dissection into congruent triangles similar to T. By a dissection (or tiling) we mean a decomposition of A into pairwise nonoverlapping polygons. No other conditions are imposed on the tilings. In particular, it is allowed that two pieces have a common boundary point, but do not have a common side. By the spectrum of the polygon A we mean the set of triples (α,β,γ) such that A can be dissected into congruent triangles of angles α,β,γ.

A complete solution to this problem would be an algorithm producing the spectrum of every given convex polygon. Although we cannot solve the problem in this algorithmic sense of the word, we present a method which seems to work in most (and possibly, for all) cases. Our method is based on the following classification of tilings introduced in [3]. Suppose that A is dissected into triangles similar to T (we do not assume that the triangles are congruent). We say that the tiling is regular if there are two angles of T, say α and β such that at each vertex V of the tiling, the number of triangles having V as a vertex and having angle α at V is the same as the number of triangles having angle β at V. If this condition is not satisfied, then the tiling is called irregular. Thus the problem of finding the spectrum is divided into two separate questions: given A, determine those triangles T for which A has a regular (resp. irregular) tiling with congruent triangles similar to T.

We give a complete solution to the question concerning regular tilings. In Theorem 2.1 we list all pairs (A,T) such that A is a convex polygon, T is a triangle, and there is a regular tiling of A with congruent triangles similar to T. In fact, this is the main result of the paper, and its proof occupies Sects. 4–9.

As for irregular tilings, our starting point is [3, Theorem 4] stating that for every polygon A, the number of distinct non-similar triangles T such that A has an irregular tiling with triangles similar to T is finite. More precisely, the number of these triangles is at most cN ⁶, where N is the number of vertices of A and c is an absolute constant. The proof of [3, Theorem 4] is effective, and gives a list of triples (α,β,γ) such that the angles of every triangle which tiles A irregularly are given by one of the triples of the list.

As we shall see in Sect. 3, in many cases this list can be reduced considerably by using the system of equations satisfied by the angles, and considering conjugate tilings as in [1]. Then, assuming that the tiling consists of congruent triangles, we may compare the sides and the area of the triangles to those of A in order to obtain further number theoretical restrictions on the triples (α,β,γ). Discarding all triples violating these conditions, we arrive at the list of all triangles that tile A irregularly.

There can be two problems with the application of this argument. The first problem is that we cannot guarantee that the reduced list obtained by considering the equations, conjugate tilings and number theoretical restrictions only contains triples corresponding to a tiling. (Proving this fact would result in an algorithmic solution of the problem.) However, as we shall see in Sect. 3, in all cases we consider, each triple contained by the reduced list actually tiles A. One can hope that this happens to every convex polygon A.

The other problem is that when we arrive at a list of triples from which we cannot eliminate any item, we have to produce tilings corresponding to these triangles. In principle, this could be the most difficult step in the procedure. However, in the actual cases we consider, the existence of these tilings is either trivial (as in the case of rectangles or regular N-gons with N large enough), or relatively easy (as for the regular triangle). One can hope that this shows a general tendency.

The applications of the method outlined above will be given in Sect. 3. As an application of Theorem 2.1 we characterize the convex polygons with infinite spectrum (Theorem 3.1). Then we determine the spectrum of the regular triangle, the square, all rectangles, and the regular N-gons with N large enough (Theorems 3.3, 3.6, Corollary 3.7 and Theorem 3.4).

2 Regular Tilings of Convex Polygons

We recall the definition of regular tilings. Let A be a polygon with vertices V ₁,…,V _N, and suppose that A is decomposed into nonoverlapping similar triangles Δ ₁,…,Δ _t of angles α,β,γ. Let V ₁,…,V _m (m≥N) be an enumeration of the vertices of the triangles Δ ₁,…,Δ _t. For every i=1,…,m we shall denote by p _i (resp. q _i and r _i) the number of those triangles Δ _j whose angle at the vertex V _i is α (resp. β and γ). If i≤N and the angle of A at the vertex V _i is δ _i, then

$$ p_i \alpha +q_i \beta +r_i\gamma=\delta_i .$$

(1)

If i>N then we have either

$$ p_i \alpha +q_i \beta +r_i\gamma= 2\pi $$

(2)

or

$$ p_i \alpha +q_i \beta +r_i\gamma= \pi.$$

(3)

Namely, (2) holds if V _i is in the interior of A and whenever V _i is on the boundary of a triangle Δ _j then necessarily V _i is a vertex of Δ _j. In the other cases (3) will hold. It is clear that the coefficients p _i,q _i,r _i must satisfy

$$ \sum_{i=1}^mp_i =\sum_{i=1}^mq_i =\sum _{i=1}^mr_i =t.$$

(4)

The tiling will be called regular, if one of the following statements is true:

p _i=q _i for every i=1,…,m;
p _i=r _i for every i=1,…,m;
q _i=r _i for every i=1,…,m.

Otherwise the tiling is called irregular. We call a polygon rational, if its sides are pairwise commensurable.

Theorem 2.1

Let A be a convex N-gon, and suppose that A has a regular tiling with congruent triangles of angles α,β,γ such that at each vertex of the tiling the number of angles α is the same as that of β. Then one of the following statements is true.

(i)

A is a parallelogram of angles γ and α+β, and the ratio of the sides of A is a rational multiple of sinα/sinβ.
(ii)

A is a rational polygon, N−2 of its angles equal γ, the other two angles are integer multiples of α+β, and sinα,sinβ,sinγ are pairwise commensurable.
(iii)

A is a rational polygon, N−2 of its angles equal α+β, the other two angles are integer multiples of γ, and sinα,sinβ,sinγ are pairwise commensurable.
(iv)

A is a regular triangle, γ=π/3 or γ=2π/3, and sinα,sinβ,sinγ are pairwise commensurable.
(v)

A is a centrally symmetric rational N-gon, N=2k≥8, each angle of A equals α+β, and the side lengths of A are a ₁,…,a _k,a ₁,…,a _k in this order, where a ₁,…,a _k−1 constitute a geometric progression of quotient q=sinα/sinβ, and a _k≥q ^k−1⋅a ₁.
(vi)

A is a centrally symmetric rational hexagon, each angle of A equals 2π/3, γ=π/3 or γ=2π/3, and sinα/sinβ is rational.
(vii)

A is a regular N-gon, α=β=(π/2)−(π/N) and γ=2π/N.
(viii)

A is a rational N-gon with N≤6, each angle of A equals π/3 or 2π/3, α=β=π/6 and γ=2π/3.
(ix)

A is a rational hexagon, each angle of A equals 2π/3, γ=π/3, and either α=π/6, β=π/2 or α=π/2, β=π/6.

We shall prove Theorem 2.1 in Sects. 4–9. In this section first we discuss the most immediate consequences of Theorem 2.1. Then we show that the cases listed in the theorem correspond to existing tilings.

Theorem 2.2

The set of those non-similar convex polygons which are not parallelograms and have a regular tiling with congruent triangles is countable.

Proof

It is enough to check that the set of those non-similar convex polygons which satisfy the conditions formulated in the cases (ii)–(ix) of Theorem 2.1 is countable.

First note that the set of non-similar rational triangles is countable, and thus the set of those triples (α,β,γ) for which sinα,sinβ,sinγ are pairwise commensurable is countable. From this observation it is clear that the set of those non-similar convex polygons which satisfy the conditions of cases (ii)–(iv) is countable.

Now consider the case (v). Then the angles of A are equal π−(2π/N). Since the sides of A are commensurable, it is clear that the number of these polygons is countable.

The statement concerning the cases (vi)–(ix) is trivial. □

Remark 2.3

The analogous statement about irregular tilings is false. In fact, for every N, there are continuum many non-similar convex N-gons that have irregular tilings with congruent triangles. To see this, let XYZ be a triangle such that $\overline {XY}=\overline {XZ}$ and the angle γ=YXZ∡ satisfies γ<π/(N−2). Let T _i denote the triangle obtained from XYZ by rotating it about the vertex X by the angle i⋅γ (i=0,…,N−3). Then $A_{\gamma}=\bigcup_{i=0}^{N-3} T_{i}$ is a convex N-gon tiled with the triangles T ₀,…,T _N−3. It is clear that different values of γ lead to non-similar polygons A _γ.

One can show that the set of non-similar convex N-gons having irregular tilings with congruent triangles can be decomposed into countable many families depending on some continuous parameters. Unfortunately, the description or enumeration of these families seems to be difficult.

Theorem 2.4

Let A be a convex N-gon, and suppose that A has a regular tiling with congruent triangles of sides a,b,c. Then at least one of the following statements is true.

(i)

N≤6.
(ii)

A is a regular N-gon.
(iii)

A is rational and centrally symmetric.
(iv)

The lengths a,b,c are pairwise commensurable.

Proof

This is clear from Theorem 2.1. □

We close this section by showing that each case listed in Theorem 2.1 occurs. More precisely, we show that in each case (with the possible exception of (ix)), whenever the polygon A and the angles α,β,γ satisfy the conditions, then A has a regular tiling with congruent triangles of angles α,β,γ.

(i) Let A be a parallelogram of sides d ₁,d ₂ and angles γ and α+β such that d ₁/d ₂=(p/q)⋅(sinα/sinβ), where p and q are positive integers. Then we put τ=d ₁/(psinα). If a=τsinα and b=τsinβ, then d ₁=p⋅a and d ₂=q⋅b. Thus A can be decomposed into pq congruent parallelograms of sides a and b. Each of these parallelograms can be decomposed into two congruent triangles of angles α,β,γ and of sides a,b and c=τsinγ. It is clear that the tiling obtained is regular.

Next suppose that A and α,β,γ satisfy the conditions formulated in (ii) or (iii). Then cosγ is rational, since

$$\cos\gamma=\frac{\sin^2 \alpha +\sin^2 \beta -\sin^2 \gamma }{2\sin \alpha \sin\beta }.$$

Therefore, putting δ=γ if A satisfies (ii) and δ=α+β if A satisfies (iii), we find that cosδ is rational, N−2 angles of A equal δ, and the other two angles are integer multiples of π−δ.

Now it can be shown by induction on N that under these conditions A can be decomposed into finitely many nonoverlapping rational symmetric trapezoids of angles δ and π−δ. (See also Lemma 8 of [3], where a more general statement is proved, except that the rationality of the polygon A is not supposed, and thus the trapezoids obtained are not necessarily rational either. However, one can check that if A is rational and cosδ is rational, then the construction of [3, Lemma 8] yields rational trapezoids.) It follows from Lemma 2.2 of [2] that each of these symmetric trapezoids can be tiled with congruent copies of a triangle of angles α,β,γ. Then, by Lemma 2.1 of [2], A itself can be tiled with congruent copies of a triangle of angles α,β,γ. It is easy to check that the tiling obtained this way is regular.

(iv) See Theorem 3.1 of [2].

(v) By assumption, q=a ₂/a ₁ is a positive rational number. The conditions imply that α+β=π−(2π/N), and γ=2π/N=π/k.

Let V ₁,…,V _N=V ₀ be the vertices of A listed counterclockwise such that $\overline {V_{i-1}V_{i}} =a_{i}$ for every i=1,…,N (see Fig. 1). For every i=1,…,N, let h _i be the halfline starting from V _i such that the angle between the side V _i−1 V _i and h _i equals β, and the angle between the side V _i V _i+1 and h _i equals α. Let E be the intersection of h _N and h ₁. Then $\overline {EV_{1}}/\overline {EV_{0}}=\sin\alpha /\sin\beta =q$, and thus $\overline {EV_{1}}=q\cdot \overline {EV_{0}}$. Since $\overline {V_{1}V_{2}} =q\cdot \overline {V_{0} V_{1}}$ and EV ₁ V ₂∡=α, it follows that the triangle EV ₁ V ₂ is similar to the triangle EV ₀ V ₁, and thus E is the intersection of h ₁ and h ₂. Repeating this argument we find that E is on the halflines h _i for every i=1,…,k−1.

Since a _k≥q ^k⋅a ₁, there is a point W on the side V _k−1 V _k such that $\overline {V_{k-1}W}=q^{k-1} \cdot a_{1} $. (The point W coincides with V _k if a _k=q ^k⋅a ₁.) Since $\overline {W V_{k-1}}=q\cdot \overline {V_{k-2}V_{k-1}}$ and EV _k−1 W∡=α, it follows that the triangle EV _k−1 W is also similar to the triangle EV ₀ V ₁. By kγ=π we find that the points V ₀,E and W are collinear. Thus the convex polygon V ₀,…,V _k−1 WV ₀ is tiled by k triangles of angles α,β,γ such that the similarity ratio of any two of these triangles is rational.

The central symmetry of A implies that there is a point W′ on the side V _N−1 V ₀ such that $\overline {V_{N-1}W'}=q^{k-1} \cdot a_{1} $. Also, the convex polygon V _k,…,V _N−1 W′V _k is tiled by k triangles of angles α,β,γ such that the similarity ratio of any two of these triangles is rational.

The angles of the parallelogram W′V ₀ WV _k are α+γ and β. Since $\overline {WV_{k}}$ is a rational multiple of $\overline {V_{0} V_{1}}$ and $\overline {V_{0} W}$ is rational multiple of $\overline {EV_{0}}$, it follows that the ratio $\overline {WV_{k}}/\overline {V_{0} W}$ is a rational multiple of

$$\overline {V_0 V_1}/\overline {EV_0} =\frac{\sin\gamma}{\sin\beta } =q\cdot \frac{\sin\gamma}{\sin \alpha } .$$

Therefore, by (i), the parallelogram W′V ₀ WV _k can be tiled by congruent triangles of angles α,β,γ. In this way we decomposed A into nonoverlapping triangles of angles α,β,γ. It is easy to check that the similarity ratio of any two of these triangles is rational.

Let Δ ₁,…,Δ _n be the triangles of this decomposition, where each Δ _i is similar to Δ ₁. Then there are positive rational numbers r _i such that Δ _i is obtained from Δ ₁ by a similarity transformation with ratio r _i. Let p _i and q be positive integers such that r _i=p _i/q (i=1,…,n). For every i, we can dissect Δ _i into $p_{i}^{2}$ congruent triangles similar to Δ ₁. In this way we obtain a dissection of A into $\sum_{i=1}^{n} p_{i}^{2}$ congruent triangles similar to Δ ₁. It is easy to see that the tiling obtained is regular.

(vi) Let a ₁,a ₂,a ₃,a ₁,a ₂,a ₃ denote the lengths of the sides of A. It is easy to see that A can be decomposed into three parallelograms of sides a ₁,a ₂; a ₃,a ₁; and a ₂,a ₃, respectively. Therefore, these parallelograms are rational, and their angles equal π/3 and 2π/3. By (i), each of them can be tiled with congruent triangles of angles α,β,γ. In this way we decomposed A into nonoverlapping triangles of angles α,β,γ such that the similarity ratio of any two of these triangles is rational. From this we obtain a tiling of A into congruent triangles of angles α,β,γ as in the previous case. Clearly, the tiling obtained is regular.

(vii) Connecting the center of A with the vertices we obtain a tiling of A into congruent triangles of angles α,β,γ. Then we can label the angles (π/2)−(π/N) of the triangles with α and β in such a way that we obtain a regular tiling.

(viii) Since A is rational and each angle of A equals π/3 or 2π/3, it follows that A can be decomposed into congruent regular triangles. We can decompose each of these triangles into three triangles of angles α,β,γ. It is clear that we can label the acute angles of these triangles with α and β in such a way that we obtain a regular tiling.

(ix) It is clear that a regular triangle can be tiled with two congruent triangles having angles α=π/2,β=π/6,γ=π/3. Since every rational polygon with angles π/3 and 2π/3 can be tiled with congruent regular triangles, it follows that these polygons can be tiled with congruent triangles having angles α,β,γ. However, we are looking for regular tilings, and this extra condition excludes the cases when A is a triangle or a trapezoid or a pentagon. (We shall prove this in Sect. 8.) It is not clear whether or not every rational hexagon with angles 2π/3 has a regular tiling with triangles having angles α=π/2,β=π/6,γ=π/3. For some hexagons there is such a tiling. For example, if A is centrally symmetric, then the existence of such a tiling follows from (vi), since sinα/sinβ=2.

3 Applications of Theorem 2.1

For every polygon A we shall denote by c(A) the cardinality of the spectrum of A.

Theorem 3.1

Let A be a convex N-gon. Then c(A)=∞ if and only if A satisfies one of the following conditions.

(i)

A is a regular triangle.
(ii)

A is a parallelogram.
(iii)

A is rational, and there is a δ such that cosδ is rational, N−2 angles of A equal δ, and the other two angles are integer multiples of π−δ.

Proof

Let A be a given convex polygon. By [3, Theorem 4], the number of triples (α,β,γ) such that A has an irregular tiling with congruent triangles of angles (α,β,γ) is finite. Therefore, if c(A)=∞, then there are infinitely many triples (α,β,γ) such that A has a regular tiling with congruent triangles of angles (α,β,γ). In particular, there are such triples different from those listed in cases (v), (vii), (viii) and (ix) of Theorem 2.1. (Note that in (v) of Theorem 2.1 the triple (α,β,γ) is uniquely determined. Indeed, γ must be equal to 2π/N, and then the condition sinα/sinβ=q determines α and β as well.) Therefore, A must satisfy one of the conditions of (i)–(iv) and (vi) of Theorem 2.1. It is clear that in each case, one of (i)–(iii) of Theorem 3.1 holds. This proves the “only if” part of the theorem.

Now we prove the “if” part. If Δ is a rational triangle and one of its angles equals π/3 or 2π/3 then, by Theorem 3.1 of [2], the regular triangle can be tiled with congruent triangles similar to Δ. In Lemma 3.2 of [2] it is shown that there are infinitely many such triangles Δ, and thus c(A)=∞ holds for the regular triangle.

If A is a parallelogram of sides a,b and of angles γ and π−γ, then A can be tiled with congruent triangles of angles α,β,γ whenever sinα/sinβ is a rational multiple of a/b. It is easy to see that there are infinitely many such triples (α,β,γ), and thus c(A)=∞.

Next suppose that A satisfies (iii). We claim that there are infinitely many triples (α,β,γ) such that α,β,γ are the angles of a triangle, γ=δ, and sinα,sinβ,sinγ are pairwise commensurable. For each of these triples, A and (α,β,γ) satisfy the conditions of (iii) of Theorem 2.1 and thus, as we saw in the previous section, there is a tiling of A with congruent triangles of angles α,β,γ. This will prove c(A)=∞.

Let 0<γ<π be given such that cosγ is rational. Then d=sin² γ=1−cos² γ is a positive rational number. It is well-known that there are infinitely many points on the ellipse x ²+d⋅y ²=1 having rational coordinates; moreover, the set of these points is everywhere dense in the ellipse. Indeed, for every s∈ℚ, the point with coordinates x=±(s ²−d)/(s ²+d) and y=2s/(s ²+d) satisfies the equation x ²+d⋅y ²=1, and the set of these points is everywhere dense in the ellipse.

Let (x,y) be such a point with a small positive y. We define α∈(0,π/2) by sinα=y⋅sinγ. Then sinα/sinγ=y is rational, and

$$\cos\alpha =\sqrt{1-\sin^2 \alpha } =\sqrt{1-y^2 \cdot \sin^2 \gamma} = \sqrt{1-y^2 \cdot d} =|x|.$$

If y is small enough, then α+γ<π, and we can define β by β=π−α−γ. Then we have

$$\frac{\sin\beta }{\sin\gamma}=\frac{\sin\alpha \cos\gamma +\cos\alpha \sin\gamma}{\sin \gamma} = y\cdot \cos\gamma+\cos\alpha =y\cdot \cos \gamma+|x|,$$

and thus sinα/sinγ and sinβ/sinγ are both rational. It is clear that this construction gives infinitely many triples (α,β,γ) with the required properties. □

In the following applications we consider the cases when A is a regular triangle, a square, a rectangle or a regular N-gon with N large enough, and determine the set of triples (α,β,γ) such that A can be tiled with congruent triangles of angles (α,β,γ). We shall need the following lemma.

Lemma 3.2

Suppose that the convex polygon A has an irregular tiling with congruent triangles of angles α,β,γ. Then α,β,γ are linear combinations of the angles of A with rational coefficients.

Proof

Consider (1), (2), and (3) at the vertices of the tiling. Since the tiling is irregular, it follows from [3, Lemma 10] that the determinant

$$D_{ij}=\left| \begin{array}{c@{\quad}c@{\quad}c}1&1&1\\p_i & q_i & r_i \\p_j & q_j &r_j\end{array} \right|$$

is nonzero for at least one pair of indices (i,j). Then the corresponding system of equations

Open image in new window

determines α,β,γ (here δ _i and δ _j are either angles of A or equal π or 2π). Applying Cramer’s rule, we find that α,β,γ are linear combinations of the angles of A and of π with rational coefficients. Since π equals the sum of the angles of A divided by N−2, we obtain the statement of the lemma. □

Theorem 3.3

The regular triangle can be tiled with congruent triangles of angles α,β,γ if and only if a permutation of (α,β,γ) satisfies one of the following conditions:

(i)

α=β=π/6 and γ=2π/3;
(ii)

α=π/6, β=π/2, γ=π/3;
(iii)

γ∈{π/3,2π/3} and sinα,sinβ,sinγ are pairwise commensurable.

Proof

We saw already that if (α,β,γ) is one of the triples listed in the theorem, then there exists a tiling with the required properties.

Next suppose that the regular triangle is tiled with congruent triangles of angles α,β,γ. If the tiling is regular then one of (ii)–(iv), (vii) and (viii) of Theorem 2.1 must hold. It is clear that in each of these cases the statement of Theorem 3.3 is true.

Therefore, we may assume that the tiling is irregular. Then, by Lemma 3.2, α,β,γ are rational multiples of π. Then we can apply Theorem 5.1 of [2], and find that either a permutation of (α,β,γ) satisfies (i), (ii) or (iii) of Theorem 3.3, or a permutation of (α,β,γ) equals one of the triples

$$ \biggl(\frac{\pi}{3},\frac{\pi}{12}, \frac{7\pi}{12}\biggr),\ \biggl(\frac{\pi}{3},\frac{\pi}{30}, \frac{19\pi}{30} \biggr),\ \biggl(\frac{\pi}{3},\frac{7\pi}{30}, \frac{13\pi}{30} \biggr).$$

(5)

We prove that none of the triples listed in (5) gives an irregular tiling of the regular triangle. Suppose first (α,β,γ)=(π/3,π/12,7π/12). Since the tiling is irregular, there is an equation pα+qβ+rγ=δ such that q<r and δ∈{π/3,π,2π}. Multiplying the equation by 12/π we find that 4p+q+7r is divisible by 4. Then r−q is also divisible by 4. However, we have 0<r−q≤r≤3, which is impossible.

Next suppose (α,β,γ)=(π/3,π/30,19π/30). Since the tiling is irregular, there is an equation pα+qβ+rγ=δ such that q<r. Multiplying the equations by 30/π we find that 10p+q+19r is divisible by 10. Then r−q is also divisible by 10. However, we have 0<r−q≤r≤3, which is impossible.

A similar argument works in the case when (α,β,γ)=(π/3,7π/30,13π/30). □

Let R _N denote the regular N-gon. Connecting the center of R _N with the vertices, we obtain a tiling of R _N with congruent triangles of angles α=β=(π/2)−(π/N) and γ=2π/N. Another tiling is obtained by decomposing each of these isosceles triangles into two right triangles. The angles of the triangles of the new tiling are (π/2)−(π/N),π/2,π/N. This shows that for every N the regular N-gon can be tiled with congruent triangles with angles given by any of the triples

$$ \biggl( \frac{\pi}{2} -\frac{\pi}{N}, \frac{\pi}{2} -\frac{\pi}{N}, \frac{2\pi}{N} \biggr),\ \biggl( \frac{\pi}{2} -\frac{\pi}{N}, \frac{\pi}{2} , \frac{\pi}{N} \biggr) .$$

(6)

For N=3,4,6 there are other tilings of R _N. Moreover, in these cases there are infinitely many other triangles tiling R _N since, by Theorem 3.1, we have c(R ₃)=c(R ₄)=c(R ₆)=∞. Next we show that this behavior is exceptional among the regular polygons.

Theorem 3.4

(i)

If N≠3,4,6, then c(R _N)<∞.
(ii)

If N≠3,4,6 and R _N has a regular tiling with congruent triangles, then the angles of the triangles are α=β=(π/2)−(π/N), γ=2π/N.
(iii)

If N>420, then c(R _N)=2.

Lemma 3.5

If N≠3,4,6, then R _N cannot be tiled with congruent triangles of angles α=β=π/N, γ=π−(2π/N).

Proof

Suppose there is such a tiling. We may assume that the sides of the triangles are a,a,c, where a=sinα and c=sin(π−(2π/N))=sin2α=2sinαcosα. Then the side of R _N equals xa+yc, where x,y are nonnegative integers. If the number of tiles is t then, comparing the areas we obtain

$$N\cdot \frac{(xa+yc)^2}{4}\cdot \cot\frac{\pi}{N} =\frac {t}{2}\cdot \sin^2 \frac{\pi}{N} \cdot \sin\frac{2\pi}{N} .$$

Using a=sinα, c=2sinαcosα and cot(π/N)=cosα/sinα, we obtain

$$\frac{N}{4} \cdot (x+2y \cos\alpha )^2 =t\cdot \sin^2 \alpha =t\cdot \bigl(1-\cos^2 \alpha \bigr)$$

and

$$ \bigl(Ny^2 +t\bigr) \cos^2 \alpha +Nxy\cos \alpha + \bigl( \bigl(Nx^2 /4\bigr)-t \bigr) =0.$$

(7)

Thus the degree of cosα is at most 2. However, by Theorem 3.9 of [5], the degree of cosα=cos(π/N) is ϕ(2N)/2, and thus ϕ(2N)=2 or ϕ(2N)=4. Since N≠3,4,6, the only possibility is N=5. Then $\cos\alpha =\cos\pi/5=(\sqrt{5} +1)/4$, and thus the minimal polynomial of cosα is 4X ²−2X−1. Then (7) gives Ny ²+t=−4⋅((Nx ²/4)−t) and Nxy=2((Nx ²/4)−t). From the first equation we obtain t=N(x ²+y ²)/3>Nx ²/4. Thus the second equation gives Nxy<0, which is impossible. This contradiction completes the proof. □

Proof of Theorem 3.4

(i) Suppose c(R _N)=∞. The angles of R _N equal δ=π−(2π/N). Then, by Theorem 3.1, cosδ is rational, and then so is cos2π/N=−cos(π−δ). Thus cos2π/N=0,±1/2,±1 by [5, Corollary 3.12], and hence N∈{3,4,6}.

(ii) Let α,β,γ be the angles of the triangles of a regular tiling of R _N. If N≠3,4,6, then R _N has to satisfy one of (ii), (iii), (v) and (vii) of Theorem 2.1. If R _N satisfies (vii), then there is nothing to prove.

Suppose R _N satisfies (ii). Then γ=π−(2π/N) and sinα,sinβ,sinγ are commensurable. Since 2cosγ=(sin² α+sin² β−sin² γ)/(sinαsinβ), it follows that cosγ is rational. Then, as we saw above, we have N∈{3,4,6} which is impossible.

If R _N satisfies (iii), then γ=2π/N and the same argument works.

Finally, suppose that R _N satisfies (v). Then α+β=π−(2π/N) and γ=2π/N. Since the sides of R _N are equal, we have sinα/sinβ=1, α=β, and α=β=(π/2)−(π/N).

(iii) We have to prove that if N>420 and R _N has a tiling with congruent triangles, then the angles of the triangles are one of given by (6). By (ii), this is true if the tiling is regular, so we may assume that it is irregular. By Lemma 3.2, this implies that α,β,γ are rational multiples of π. Now we shall apply the following result proved in [4]: Let N>420, and suppose that R _N has a tiling with similar triangles of angles α,β,γ. If α,β,γ are rational multiples of π, then (α,β,γ) is one of the triples of ( 6 ), or equals (π−(2π/N),π/N,π/N). Since, by Lemma 3.5, the latter case is impossible, this completes the proof. □

It is very likely that c(R _N)=2 for every N≠3,4,6. In order to prove this, we have to show that if N≤420 and N≠3,4,6, then there is no irregular tiling of R _N. This amounts to checking a given finite set of triples. Unfortunately, the number of cases to consider is enormous, and it seems to be hopeless to exclude these triples without the use of computer. We plan to return to this computation in the forthcoming paper [4].

Our next aim is to determine those triangles that tile a rectangle.

Theorem 3.6

If the congruent copies of a triangle T tile a rectangle, then T is a right triangle, and the ratio of the sides of the rectangle is a rational multiple of the ratio of the perpendicular sides of T. Therefore, a rectangle A can be tiled with congruent triangles of angles α,β,γ if and only if a permutation of (α,β,γ) satisfies the following condition: γ=π/2 and sinα/sinβ is a rational multiple of the ratio of the sides of A.

We note the following consequence of Theorem 3.6.

Corollary 3.7

The square can be tiled with congruent triangles of angles α,β,γ if and only if a permutation of (α,β,γ) satisfies the following condition: γ=π/2 and sinα/sinβ is rational. Consequently, in every tiling of the square with congruent triangles, the pieces must be right triangles with commensurable perpendicular sides.

Lemma 3.8

Suppose that a rectangle A is tiled with congruent copies of a right triangle. Then the ratio of the perpendicular sides of the triangle is a rational multiple of the ratio of the sides of A.

Proof

Let α,β,γ be the angles of the triangle, where γ=π/2 and α≥β. If α=β, then the triangles are isosceles right triangles, and thus the sides of A are commensurable by [1, Theorem 2]. In this case the statement of the lemma is true. Therefore, we may assume that α>β, and thus π/4<α<π/2. If the tiling is regular, then one of (i), (ii), (iii) and (vii) of Theorem 2.1 must hold. It is clear that in each of these cases the statement of the lemma is true.

Therefore, we may assume that the tiling is irregular. Then there is an equation pα+qβ+rγ=δ with p>q and δ∈{π/2,π,2π}. This implies (p−q)α+q(α+β)+rγ=δ, and thus α+β=γ=π/2 gives mα=sπ/2, where m=p−q and s are both positive integers. Note that s≤2δ/π≤4. Since π/4<α<π/2, we have 1/2<s/m<1, and thus s/m is one of the fractions

$$ \frac{2}{3}, \, \frac{3}{4}, \, \frac{3}{5}, \,\frac{4}{5}, \, \frac{4}{7}.$$

(8)

Suppose s/m=2/3. Then α=π/3 and β=π/6. We may assume that the sides of the triangles are $1,2,\sqrt{3} $. Then the sides of A are $x\sqrt{3} +y$ and $z\sqrt{3} +u$, where x,y,z,u are nonnegative integers. If the tiling contains t triangles, then comparing the areas we obtain

$$(x\sqrt{3} +y)\cdot (z\sqrt{3} +u)= \frac{t}{2} \cdot \sqrt{3} .$$

This implies x=u=0 or y=z=0. In both cases, one side of A is an integer multiple of $\sin\alpha =\sqrt{3} /2$, and the other side is an integer multiple of cosα=1/2; that is, the statement of the lemma is true.

Next suppose s/m=3/4. Then α=3π/8 and β=π/8. We may assume that the sides of the triangles are sinα,cosα and 1. Then the sides of A are xcosα+ysinα+z and ucosα+vsinα+w, where x,y,z,u,v,w are nonnegative integers. If the tiling contains t triangles, then, comparing the areas we obtain

$$ (x\cos\alpha +y\sin\alpha +z)\cdot (u\cos\alpha +v\sin\alpha +w)=\frac{t}{2} \cdot \cos\alpha \cdot \sin\alpha .$$

(9)

By Theorem 3.9 of [5], the degree of the algebraic numbers cosα and sinα equals four. On the other hand, $\tan\alpha =\sqrt{2} +1$, and thus each of the numbers cos² α,sin² α,sinαcosα,tanα belongs to the field $\mathbb{Q}(\sqrt{2} )$. Thus the left hand side of (9) equals I+J, where I=(xw+uz)cosα+(yw+zv)sinα and $J=xu \cos^{2} \alpha +(xv+yu)\cos\alpha \sin\alpha +yv\sin^{2} \alpha +zw\in\mathbb {Q}(\sqrt{2} )$. Since the right hand side of (9) belongs to $\mathbb{Q}(\sqrt{2} )$, we find that $I\in\mathbb{Q}(\sqrt{2} )$, and thus

$$\cos\alpha \cdot \bigl[ (xw+uz)+(yw+zv)\tan\alpha \bigr]\in\mathbb {Q}(\sqrt{2} ).$$

By $\cos\alpha \notin\mathbb{Q}(\sqrt{2} )$ we obtain I=0. Now each term of I is nonnegative, and thus I=0 implies xw=uz=yw=zv=0. If z≠0, then we get u=v=0. Since ucosα+vsinα+w=w equals a side of A, we have w≠0, and thus x=y=0. Then the left-hand side of (9) is zw, while the right-hand side of (9) is irrational. This is a contradiction, and thus z=0. The same argument shows w=0. Then, dividing both sides of (9) by cos² α we obtain

$$(x+y\tan\alpha )\cdot (u+v\tan\alpha )=\frac{t}{2} \cdot \tan \alpha $$

or yvtan² α+(yu+xv−(t/2))tanα+xu=0. Since the minimal polynomial of $\tan\alpha =\sqrt{2} +1$ is X ²−2X−1, it follows that xu=−yv, and thus xu=yv=0. Then we have either x=v=0 or u=y=0. In both cases, one side of A is an integer multiple of sinα, and the other side is an integer multiple of cosα; that is, the statement of the lemma is true.

Finally, suppose that s/m≠2/3, 3/4. Then s/m is one of the fractions 3/5, 4/5, 4/7. We can see, applying Theorem 3.9 of [5], that in each case the degree of sinα is greater than the degree of cosα. Consequently, sinα is not an element of the field ℚ(cosα). Note also that cosα is irrational.

We may assume that the sides of the triangles are sinα,cosα and 1. Then the sides of A are xcosα+ysinα+z and ucosα+vsinα+w, where x,y,z,u,v,w are nonnegative integers. If the tiling contains t triangles, then, comparing the areas we obtain (9). Since sin² α∈ℚ(cosα) but sinα∉ℚ(cosα), (9) implies

$$ xu\cos^2 \alpha +xw\cos\alpha +yv\sin^2\alpha +uz\cos\alpha +zw=0$$

(10)

and

$$ \bigl[(xv+yu)\cos\alpha +(yw+zv)\bigr]\cdot \sin\alpha =(t/2)\cdot \cos \alpha \cdot \sin\alpha .$$

(11)

By (10) we have xu=xw=yv=uz=zw=0. Also, (11) gives (xv+yu)cosα+(yw+zv)=(t/2)⋅cosα. Since cosα is irrational, this implies yw=zv=0. Since max(x,y,z)>0 and max(u,v,w)>0, these equations imply either x=z=v=w=0 or y=z=u=w=0. In both cases, one side of A is an integer multiple of sinα, and the other side is an integer multiple of cosα, which completes the proof. □

Proof of Theorem 3.6

Suppose that the rectangle A is tiled with congruent copies of a triangle T. By [1, Theorem 23], one of the following must hold: (i) T is a right triangle; (ii) the angles of T are (π/6,π/6,2π/3); or (iii) the angles of T are given by one of the following triples:

$$ \biggl(\frac{\pi}{8},\frac{\pi}{4}, \frac{5\pi}{8}\biggr),\ \biggl(\frac{\pi}{4},\frac{\pi}{3}, \frac{5\pi}{12} \biggr),\ \biggl(\frac{\pi}{12},\frac{\pi}{4}, \frac{2\pi}{3} \biggr) .$$

(12)

If T is a right triangle, then the statement of the theorem is true by Lemma 3.8.

Suppose that the angles of T are (π/6,π/6,2π/3). We may assume that the sides of the triangles are $1,1,\sqrt{3} $. Then the sides of A are $x\sqrt{3} +y$ and $z\sqrt{3} +u$, where x,y,z,u are nonnegative integers. If the tiling contains t triangles, then, comparing the areas we obtain

$$(x\sqrt{3} +y)\cdot (z\sqrt{3} +u)= \frac{t}{2} \cdot \sqrt{3} .$$

This implies x=u=0 or y=z=0. In both cases, there is a side XY of A such that $\overline {XY}$ is an integer. Let X=U ₀,U ₁,…,U _k=Y be a division of XY such that each subinterval U _i−1 U _i (i=1,…,k) is a side of a triangle T _i of the tiling. Since $\overline {U_{i-1} U_{i}}=1$ for every i, it follows that the angle of T _i at one of the vertices U _i−1 and U _i equals 2π/3. However, the angle of T ₁ at U ₀=X and the angle of T _k at U _k=Y must be π/6, and thus there is a 0<i<k such that the angle of both T _i−1 and T _i at U _i equals 2π/3. Then the triangles T _i−1 and T _i overlap, which is impossible.

In order to complete the proof of Theorem 3.6, we have to prove the following.

Lemma 3.9

Suppose that the angles of T are given by one of the triples of (12). Then no rectangle can be tiled with congruent copies of T.

Proof

First we suppose that a rectangle A of vertices V ₁,V ₂,V ₃,V ₄ is tiled with congruent triangles with angles α=π/8, β=5π/8, γ=π/4. We may assume that V ₁ is the origin and V ₂ is the point (1,0). Since each of the numbers $\cot\pi/8 = \sqrt{2} +1 ,\ \cot5\pi/8 = \sqrt{2} -1$ and cotπ/4=1 belongs to $\mathbb {Q}(\sqrt{2} )$, it follows from [1, Theorem 2] that the coordinates of the vertices of the triangles belong to $\mathbb{Q}(\sqrt{2} )$.

Let a,b,c denote the sides of the triangles. Then a/c=(sinπ/8)/(sinπ/4) and b/c=(sin5π/8)/(sinπ/4). By Theorem 3.9 of [5], the degree of sinπ/8 and of sin5π/8 equals 4, and thus the ratios a/c and b/c do not belong to $\mathbb{Q}(\sqrt{2} )$.

There is a division V ₁=U ₀,…,U _k=V ₂ of the side V ₁ V ₂ such that U _i−1 U _i is the side of a triangle T _i of the tiling for every i=1,…,k. Let x _i denote the first coordinate of U _i. Since $x_{i} -x_{i-1} \in\mathbb{Q}(\sqrt{2})$ for every i=1,…,k and $a/c,b/c\notin\mathbb{Q}(\sqrt{2})$, it follows that either $x_{i} -x_{i-1} =\overline {U_{i-1}U_{i}} \in\{a,b\}$ for every i, or $x_{i} -x_{i-1} =\overline {U_{i-1}U_{i}} =c$ for every i.

Suppose $\overline {U_{i-1}U_{i}}=c$ for every i. Then the angles of T _i at the points U _i−1 and U _i are π/8 and 5π/8 in some order. Since 5π/8>π/2, it follows that the angle of T ₁ at U ₀=V ₁ and the angle of T _k at U _k=V ₂ must be π/8, and thus there is a 0<i<k such that the angle of both T _i−1 and T _i at U _i equals 5π/8. Then the triangles T _i−1 and T _i overlap, which is impossible. This proves that $\overline {U_{i-1}U_{i}}\in\{a,b\}$ for every i.

Next we prove that either $\overline {U_{i-1}U_{i}} =a$ for every i, or $\overline {U_{i-1}U_{i}}=b$ for every i. In order to prove this we shall need to consider a conjugate tiling as described in [1].

Let ϕ denote the automorphism of the field $\mathbb{Q}(\sqrt{2} )$ defined by $\phi(x+y\sqrt{2} )=x-y\sqrt{2}$ (x,y∈ℚ). Then Φ(x ₁,x ₂)=(ϕ(x ₁),ϕ(x ₂)) defines a collineation on the set of vertices of the tiling. Let X′ denote the image of X under Φ. Then $V'_{1} =V_{1}$ and $V'_{2} =V_{2} $. The points $V'_{1} ,V'_{2} ,V'_{3} ,V'_{4}$ are the vertices of a rectangle A′ and, according to [1], the images of the triangles are nonoverlapping and constitute a tiling of A′.

Since the images of the triangles T _i are nonoverlapping, it follows that the points $U'_{0} ,\ldots,U'_{k}$ constitute a division of the segment $V'_{1} ,V'_{2}$ in this order. Since $U'_{i} =(\phi(x_{i} ),0)$ for every i, it follows that the sequence $(\phi(x_{i} ))_{i=0}^{k}$ is strictly increasing.

Suppose that there are indices 1≤i,j≤k such that x _i−x _i−1=a and x _j−x _j−1=b. Then ϕ(x _i)=ϕ(x _i−1)+ϕ(a) and ϕ(x _j)=ϕ(x _j−1)+ϕ(b). Therefore, the numbers ϕ(a),ϕ(b) are positive. On the other hand, $a/b=(\sin\pi/8 )/(\sin5\pi/8 )=\sqrt{2} -1$, and thus

$$\phi(a)/\phi(b)=\phi(a/b)=-\sqrt{2} -1<0,$$

which is a contradiction. This proves that either $\overline {U_{i-1}U_{i}} =a$ for every i, or $\overline {U_{i-1}U_{i}}=b$ for every i.

Thus $\overline {V_{1} V_{2}}$ is an integer multiple of either a or b. The same is true for the side V ₂ V ₃, and thus the area of A is an integer multiple of one of the numbers a ²,b ²,ab. On the other hand, the area of any of the triangles is $ab(\cos\gamma)/2=ab \sqrt{2} /4$, and thus the area of A is an integer multiple of $ab \sqrt{2} /4$. Therefore, one of the numbers $(ab\sqrt{2} )/a^{2} ,(ab\sqrt{2} )/b^{2} ,(ab\sqrt{2} )/(ab)$ is rational. However, $a/b=\sqrt{2} -1 $, and thus each of these numbers is irrational, a contradiction. This proves that no rectangle can be tiled with congruent triangles with angles α=π/8, β=5π/8, γ=π/4.

Next suppose that a rectangle A of vertices V ₁=(0,0),V ₂=(1,0) and V ₃,V ₄ is tiled with congruent triangles with angles α=π/4, β=5π/12, γ=π/3. Since each of the numbers cotπ/4=1, $\cot5\pi/12 = 2-\sqrt{3} , \ \cot\pi/3 = \sqrt{3} /3$ belongs to $\mathbb{Q}(\sqrt{3} )$, it follows from [1, Theorem 2] that the coordinates of each vertex of A and of each triangle belong to $\mathbb{Q}(\sqrt{3} )$.

Let a,b,c denote the sides of the triangles. Then

$$a/c=(\sin\pi/4) / (\sin\pi/3)=\sqrt{2} /\sqrt{3} \notin\mathbb{Q}(\sqrt{3} ).$$

Since $\sin5\pi/12 =(\sqrt{3} +1)/(2\sqrt{2} )$, we have

$$b/c=(\sin5\pi/12) /(\sin\pi/3)=(\sqrt{3} +1)/(\sqrt{6} ),$$

and thus b/c does not belong to $\mathbb{Q}(\sqrt{3} )$ either.

There is a division V ₁=U ₀,…,U _k=V ₂ of the side V ₁ V ₂ such that U _i−1 U _i is the side of a triangle T _i of the tiling for every i=1,…,k. Since $a/c,b/c\notin\mathbb{Q}(\sqrt{3} )$, it follows that either $\overline {U_{i-1}U_{i}} \in\{a,b\}$ for every i, or $\overline {U_{i-1}U_{i}} =c$ for every i.

Suppose $\overline {U_{i-1}U_{i}}=c$ for every i. Then the angles of T _i at the points U _i−1 and U _i are π/4 and 5π/12. It is clear that the angle of T ₁ at U ₀=V ₁ must be π/4 and, similarly, the angle of T _k at U _k=V ₂ must be π/4. Therefore, there exists an index 0<i<k such that the angle of T _i−1 at U _i is 5π/12 and the angle of T _i at U _i is also 5π/12. Since π−2⋅(5π/12)=π/6<min(α,β,γ), this is clearly impossible. Thus $\overline {U_{i-1}U_{i}} \in\{a,b\}$ for every i.

Next we prove that either $\overline {U_{i-1}U_{i}} =a$ for every i, or $\overline {U_{i-1}U_{i}}=b$ for every i. Let ψ denote the automorphism of the field $\mathbb{Q}(\sqrt{3} )$ defined by $\psi(x+y\sqrt{3} )=x-y\sqrt{3}$ (x,y∈ℚ). Then Ψ(x ₁,x ₂)=(ψ(x ₁),ψ(x ₂)) defines a collineation on the set of vertices of the tiling. Let X′ denote the image of X under Ψ. Then $V'_{1} =V_{1}$ and $V'_{2} =V_{2} $. The points $V'_{1} ,V'_{2} ,V'_{3} ,V'_{4}$ are the vertices of a rectangle A′ and the images of the triangles are nonoverlapping.

Therefore, the points $U'_{0} ,\ldots,U'_{k}$ constitute a division of the segment $V'_{1} ,V'_{2}$ in this order. Let x _i denote the first coordinate of U _i. Then $U'_{i} =(\psi(x_{i} ),0)$ for every i, and hence the sequence $(\psi(x_{i} ))_{i=0}^{k}$ is strictly increasing.

Suppose that there are indices 1≤i,j≤k such that x _i−x _i−1=a and x _j−x _j−1=b. Then ψ(x _i)=ψ(x _i−1)+ψ(a) and ψ(x _j)=ψ(x _j−1)+ψ(b). Therefore, the numbers ψ(a),ψ(b) are positive. On the other hand, $a/b=(\sin\pi/4 )/(\sin5\pi/12 )=\sqrt{3} -1$, and thus

$$\psi(a)/\psi(b)=\psi(a/b)=-\sqrt{3} -1<0,$$

which is a contradiction. This proves that either $\overline {U_{i-1}U_{i}} =a$ for every i, or $\overline {U_{i-1}U_{i}}=b$ for every i.

Thus $\overline {V_{1} V_{2}}$ is an integer multiple of either a or b. The same is true for the side V ₂ V ₃, and thus the area of A is an integer multiple of one of the numbers a ²,b ²,ab. On the other hand, the area of any of the triangles is $ab(\cos\gamma)/2=ab \sqrt{3} /4$, and thus the area of A is an integer multiple of $ab \sqrt{3} /4$. Therefore, one of the numbers $(ab\sqrt{3} )/a^{2} ,(ab\sqrt{3} )/b^{2} ,(ab\sqrt{3} )/(ab)$ is rational. However, $a/b=\sqrt{3} -1 $, and thus each of these numbers is irrational, a contradiction. This proves that no rectangle can be tiled with congruent triangles with angles α=π/4, β=5π/12, γ=π/3.

Finally, suppose that a rectangle is tiled with congruent triangles with angles α=π/4, β=π/12, γ=2π/3. It is easy to check that the conjugate tiling corresponding to the automorphism of the field $\mathbb{Q}(\sqrt{3} )$ defined by $\psi(x+y\sqrt{3} )=x-y\sqrt{3}$ (x,y∈ℚ) is a tiling of a rectangle with congruent triangles with angles α=π/4, β=5π/12, γ=π/3 (see [1], p. 291). As we proved above, this is impossible. □

4 Proof of Theorem 2.1: Some Preliminary Results

We start with the following simple observation.

Lemma 4.1

Let A be a parallelogram of angles γ and π−γ. Suppose that A is tiled with congruent copies of a triangle of angles α,β,γ and of sides a,b,c. If one of the sides of A is an integer multiple of a, then the ratio of the sides of A is a rational multiple of sinα/sinβ, and thus (i) of Theorem 2.1 holds.

Proof

If the other side of A is of length d, then the area of A equals kad⋅sinγ with an integer k. Since the area of each triangle of the tiling equals (absinγ)/2, we obtain 2kadsinγ=t⋅absinγ and d=(t/2k)⋅b, where t is the number of triangles of the tiling. Thus the ratio of the sides of A equals ka/d=(2k ²/t)⋅(a/b)=(2k ²/t)⋅(sinα/sinβ). □

In the sequel we fix a regular tiling of the convex polygon A with the congruent triangles Δ ₁,…,Δ _t of angles α,β,γ such that equations (1)–(3) are of the form p _i(α+β)+r _i γ=δ _i, p _i(α+β)+r _i γ=2π and p _i(α+β)+r _i γ=π, respectively. The vertex V _i of the tiling is called normal, if p _i=r _i. If p _i≠r _i and V _i is not a vertex of A, then we say that the vertex V _i is exceptional. Whenever

$$ p(\alpha +\beta )+r\gamma=v\pi $$

(13)

is an equation satisfied by α,β,γ, then we shall call (p−s)(α+β)+(r−s)γ=(v−s)π the reduced form of (13), where s=min(p,r). Thus the reduced form of the equation at a normal vertex is 0⋅(α+β)+0⋅γ=0⋅π.

Let a,b,c denote the sides of the triangles Δ _i opposite to the angles α,β,γ. Then a/b=sinα/sinβ and b/c=sinβ/sinγ.

We shall say that a triangle Δ is supported by a segment UV, if one of the sides of Δ is a subset of UV.

First we assume that a is not a linear combination of b and c with nonnegative rational coefficients. We define a directed graph Γ _a on the set of vertices of the tiling as follows. Let XY be a maximal segment belonging to the union of the boundaries of the triangles Δ _i (i=1,…,t), and suppose that the segment XY lies in the interior of A, except perhaps the endpoints X and Y. Then there are divisions X=U ₀,U ₁,…,U _k=Y and X=V ₀,V ₁,…,V _ℓ=Y of the segment XY such that each subinterval U _i−1 U _i (i=1,…,k) is a side of a triangle T _i of the tiling supported by XY and lying on the same side of the segment XY, and each subinterval V _j−1 V _j (j=1,…,ℓ) is a side of a triangle $T'_{j}$ of the tiling supported by XY and lying on the other side of XY. Suppose that exactly one of the lengths $\overline {U_{0} U_{1}}$ and $\overline {V_{0} V_{1}}$ equals a. By symmetry, we may assume that $\overline {U_{0} U_{1}} =a\ne \overline {V_{0} V_{1}} $. Since a is not a linear combination of b and c with nonnegative rational coefficients, it follows that there is a unique index 1≤i ₀<k such that $\overline {U_{i-1} U_{i}} =a$ for every i≤i ₀ and $\overline {U_{i_{0}} U_{i_{0} +1}} \ne a$. If all these conditions are satisfied, then we connect the vertices X and $U_{i_{0}}$ by a directed edge $\overrightarrow{XU_{i_{0}}} $. Note that $U_{i_{0}}$ belongs to the interior of A and is different from each of the points V _j (j=0,…,ℓ). Thus $U_{i_{0}}$ is in the interior of one of the sides of the triangle $T'_{j}$ for a suitable j.

Let Γ _a denote the set of all directed edges defined as above. It is clear that the in-degree of any vertex is zero or one. As we saw above, a vertex V can have an incoming edge only if V belongs to the interior of A and if V is in the interior of one of the sides of a triangle of the tiling.

Lemma 4.2

Suppose that a is not a linear combination of b and c with nonnegative rational coefficients. If V is a normal vertex and V is the endpoint of an edge of Γ _a, then at least one edge of Γ _a starts from V.

Proof

Let $\overrightarrow{XV}$ be an edge, and let $V=U_{i_{0}}$, where XY is a maximal segment belonging to the union of the boundaries of the triangles Δ _i, and X=U ₀,U ₁,…,U _k=Y and T ₁,…,T _k are as in the definition of the graph. Put $T=T_{i_{0}}$ and $T'=T_{i_{0} +1}$. Then the side of T lying on the segment XY equals a, and the side of T′ lying on the segment XY is different from a, and thus the angle of T off the line XY is α, and the angle of T′ off the line XY is different from α. Since V is normal, the equation at V must be α+β+γ=π. Therefore, V is the common vertex of three triangles.

Considering all possible positions of the angle α in these triangles, we can see that in each case at least one edge starts from V (see Fig. 2). Note that a≠b and a≠c.

□

Lemma 4.3

Suppose that a is not a linear combination of b and c with nonnegative rational coefficients. Let XY be a side of A, and suppose that every vertex V lying in the interior of the segment XY is normal and is such that no edge of the graph Γ _a starts from V. Then each of the following statements is true.

(i)

At least one of the angles of A at X and Y is such that the corresponding equation is of the form p(α+β)=δ with a positive integer p.
(ii)

If there is a triangle Δ supported by XY such that its angles on the side XY are β and γ, then every triangle supported by XY is a translated copy of Δ. In particular, the length of XY equals k⋅a, where k is a positive integer.
(iii)

If no edge of the graph Γ _a starts from X or from Y then at least one of the angles of A at X and Y is such that the corresponding equation is different from α+β=δ.

Proof

There is a division X=U ₀,U ₁,…,U _k=Y of the segment XY such that each subinterval U _i−1 U _i (i=1,…,k) is a side of a triangle T _i of the tiling supported by XY. By assumption, U _i is normal, and no edge starts from U _i for every 1≤i≤k−1. It is easy to check, by inspecting the possible cases, that either T _i+1 is a translated copy of T _i, or one of the four cases presented in Fig. 3 holds.

It follows from the convexity of A that the equation at X is of the form p(α+β)=δ or rγ=δ. Suppose it is rγ=δ. Then the angle of T ₁ at the vertex U ₀=X is γ. Then, considering the possible cases according to Fig. 3, we can see that each triangle T _i is either a translated copy of T ₁, or its angle at the vertex U _i is α. This implies that the angle of T _k at Y is α or β, and thus the equation at Y is p(α+β)=δ. This proves (i).

If T _i has angles β and γ on XY then, considering again the possible cases according to Fig. 3, we can see that each triangle T _j is a translated copy of T _i. This proves (ii).

Suppose that the equation at the vertex X is of the form α+β=δ. If no edge starts from X, then either the angle of T ₁ at X is β and at U ₁ is γ, or its angle at the vertex X is α. In the first case each triangle T _i must be a translated copy of T ₁. Then the angle of T _k at Y equals γ, and thus the equation at Y cannot be of the form α+β=δ. Suppose that the angle of T ₁ at X is α. Then, as Fig. 3 shows, the angle of T _i at U _i−1 is α for every i. In particular, the angle of T _k at U _k−1 is α. If the angle of T _k at Y equals γ, then the equation at Y cannot be of the form α+β=δ. On the other hand, if the angle of T _k at Y equals β and the equation at Y is α+β=δ, then there is an edge starting from the vertex Y. This proves (iii). □

Next we consider the case when c is not a linear combination of a and b with nonnegative rational coefficients. We define the directed graph Γ _c the same way as we defined Γ _a, except that we replace a by c in the definition.

Lemma 4.4

Suppose that c is not a linear combination of a and b with nonnegative rational coefficients. Let XY be a side of A, and suppose that every vertex V lying in the interior of the segment XY is normal and is such that no edge of the graph Γ _c starts from V. Then

(i)

at least one of the angles of A at X and Y is such that the corresponding equation is of the from p(α+β)=δ with a positive integer p; and
(ii)

if there is a triangle Δ supported by XY such its angles on the side XY are α and β, then every triangle supported by XY is a translated copy of Δ. In particular, the length of XY equals k⋅c, where k is a positive integer.

Proof

There is a division X=U ₀,U ₁,…,U _k=Y of the segment XY such that each subinterval U _i−1 U _i (i=1,…,k) is a side of a triangle T _i of the tiling supported by XY. It is easy to check, by inspecting the possible cases, that either T _i+1 is a translated copy of T _i, or one of the presented in Fig. 4 four cases holds. Then we can repeat the argument of the proof of Lemma 4.3.

□

Lemma 4.5

Suppose that c is not a linear combination of a and b with nonnegative rational coefficients. Let the segment XY belong to the union of the boundaries of the triangles, and let T and T′ be two triangles supported by XY and lying on the same side of XY. Suppose that T and T′ have a common vertex V on the segment XY and that V is either on the boundary of A, or is an inner point of a side of a triangle. If the side of T lying on XY equals c and the side of T′ lying on XY is different from c, then there is an edge of Γ _c starting from V.

Proof

If V is normal, then we can easily check that an edge starts from V (see Fig. 2 with α replaced by γ).

Suppose the vertex V is exceptional. Then the equation at V is of the form rγ=π or p(α+β)=π. If the equation is rγ=π, then neither of the triangles T and T′ can have a side of length c lying on XY, which is impossible. Thus the equation is p(α+β)=π. Let T ₁,…,T _2p denote the triangles having V as a vertex listed clockwise. We may assume, by symmetry, that T ₁=T and T _2p=T′. There are halflines h _i (i=0,…,2p) starting from V such that T _i lies in the angular domain bounded by h _i−1 and h _i for every i=1,…,2p. Let α _i and β _i denote the angles of T _i at its vertices lying on the halflines h _i−1 and h _i, respectively. Then α ₁≠γ and β _2p=γ by the conditions on T and T′. Since the angle of T _i at V is different from γ for every i, it follows that there exists an index i<2p such that α _i≠γ and β _i+1=γ or α _i=γ and β _i+1≠γ. Then an edge of Γ _c starts from V along the halfline h _i, which completes the proof of the lemma. □

Lemma 4.6

If a and b are commensurable and c is not a rational multiple of a, then the graph Γ _c is empty: it has no edge.

Proof

By Lemma 4.5, the in-degree is not greater than the out-degree at each vertex. Therefore, the in-degree equals the out-degree at each vertex. Since no edge arrives at the boundary of A, it follows that no edge of Γ _c starts from the boundary of A.

Suppose that Γ _c is not empty, and let G denote the set of vertices of nonzero out-degree. Then each vertex V∈G is the starting point of an edge and also the endpoint of another edge. Let V be a vertex of the convex hull of G. Then V∈G, and thus V is the endpoint of an edge $\overrightarrow{XV}$. Let $V=U_{i_{0}}$, where XY is a maximal segment belonging to the union of the boundaries of the triangles, and let X=U ₀,U ₁,…,U _k=Y and T ₁,…,T _k be as in the definition of the graph. Since X∈G, it follows that the segment VY is outside the convex hull of G, and thus no edge starts or arrives at any point of VY except at V.

Suppose that an inner point W of the segment XY is the endpoint of an edge $\overrightarrow{ZW}$. Then W is an inner point of a side of a triangle which must be supported by XY. Thus the starting point Z of the edge must be either X or Y. It cannot be Y, as Y is outside the convex hull of G, and thus Y∉G. Thus Z=X, and then W=V. Therefore, among the vertices that are inner points of XY only V belongs to G.

Let ℓ be the smallest positive index for which U _ℓ is a common vertex of triangles lying on different sides of XY. Then i ₀<ℓ≤k. It follows from Lemma 4.5 that for every i ₀<i≤ℓ, the length of the segment U _i−1 U _i is either a or b. Therefore, $\overline {XU_{\ell}}=i_{0} c+ra$ with a positive rational r.

Let d ₁,…,d _ν be the side lengths of the triangles supported by XU _ℓ and lying on the side opposite to the triangles T _i. Then either each d _i equals c or each is different from c. Indeed, otherwise an edge would start from an inner point of XY different from V which is impossible. Thus the length $\overline {XU_{\ell}}$ is either k′c with a positive integer k′ or sa with a positive rational s. Since both of the equation i ₀ c+ra=k′c and i ₀ c+ra=sa implies that c is a rational multiple of a, both cases are impossible. □

Lemma 4.7

If a and b are commensurable and c is not a rational multiple of a, then the sides of A are pairwise commensurable.

Proof

Since Γ _c is empty, the situation described in Lemma 4.5 cannot happen. In particular, if XY is a side of A then either all triangles supported by XY have sides of length c on XY, or all have sides of length different from c on XY. Thus the length of each side of A is either an integer multiple of c or a rational multiple of a. We prove that only one of these cases can occur. It is enough to show that if X,Y,Z are consecutive vertices of A, then side lengths $\overline {XY}$ and $\overline {YZ}$ are commensurable.

Let T ₁,…,T _s denote the triangles having Y as a vertex listed counterclockwise. The equation at Y is of the form rγ=δ or p(α+β)=δ. If the equation is rγ=δ, then s=r, and the sides of T ₁ and T _r lying on the boundary of A are of length a or b. In this case the lengths of XY and of YZ are rational multiples of a, hence commensurable.

Suppose that equation at Y is p(α+β)=δ. Then s=2p, and there are halflines h _i (i=0,…,2p) starting from Y such that T _i lies in the angular domain bounded by h _i−1 and h _i for every i=1,…,2p. Let α _i and β _i denote the angles of T _i at its vertices lying on the halflines h _i−1 and h _i, respectively. Since no edge starts from Y, for every i=1,…,2p−1 we have either β _i=α _i+1=γ or α _i=β _i+1=γ. This implies that we have either α ₁=β _2p=γ or α ₁≠γ, β _2p≠γ. In the first case the lengths of XY and of YZ are rational multiples of a, and in the second case they are integer multiples of c. In both cases, they are commensurable. □

Lemma 4.8

Suppose that a and b are commensurable and c is not a rational multiple of a. Then the triangles that have their side of length c in the interior of A come in pairs. The triangles of each pair have a common side of length c, and thus they form a quadrilateral which is either a parallelogram of sides a,b and of angles γ,α+β, or a kite of sides a,b and of angles 2β,γ,2α,γ.

Proof

Let Δ be a triangle of the tiling such that its side UV of length c lies in the interior of A (except perhaps the endpoints). The side UV is contained in a segment XY with the following properties: XY belongs to the union of the boundaries of the triangles, X and Y are common vertices of triangles lying on different sides of XY, and no point in the interior of XY has this property. Then it follows from Lemma 4.5 and from the fact that Γ _c is empty that every triangle supported by XY and lying on the same side of XY as Δ has a side of length c on XY. Thus the length of XY is an integer multiple of c. Since c is not a linear combination of a and b with integer coefficients, it follows that the triangles supported by XY and lying on the other side of XY also have their sides of length c on XY. Therefore, there exists a triangle Δ′ such that the sides of length c of Δ and of Δ′ coincide. □

Let G be an additive subgroup of the reals such that π∈G. We shall denote by $\mathcal{P}_{G}$ the family of all simple, closed polygons satisfying the condition that for every side XY of P, the angle between the line going through the side XY and the x-axis belongs to G.

Suppose that χ:G→ℂ is a multiplicative function; that is, χ satisfies the functional equation χ(θ ₁+θ ₂)=χ(θ ₁)⋅χ(θ ₂) (θ ₁,θ ₂∈G). Also, we assume that χ(π)=−1. Then χ(θ+π)=−χ(θ) and χ(θ+2π)=χ(θ) hold for every θ∈G.

For a polygon $P\in\mathcal{P}_{G} $, let X ₀,…,X _k−1,X _k=X ₀ be the vertices of P listed counterclockwise, let h _i denote the halfline starting from X _i−1 and going through X _i, and let θ _i denote the directed angle between the positive x-axis and h _i (i=1,…,k). Then we define

$$ \varPhi(P)=\sum_{i=1}^k \chi(\theta_i )\cdot \overline {X_{i-1} X_i} .$$

(14)

The definition makes sense, as the function χ is periodic mod 2π. Using the fact that χ(θ+π)=−χ(θ) for every θ∈G it is easy to see that the function Φ is additive in the following sense: if P is decomposed into the nonoverlapping polygons $P_{1} ,\ldots,P_{t} \in\mathcal{P}_{G} $, then $\varPhi(P)=\sum_{i=1}^{t} \varPhi(P_{i} )$. It is clear that Φ is invariant under translations. Also, it follows from the multiplicative property of χ that if $P\in \mathcal{P}_{G}$ and P′ is obtained from P by a rotation of angle θ∈G, then Φ(P′)=χ(θ)⋅Φ(P).

Let α _i be the angle of P at the vertex X _i (i=1,…,k), and put $\alpha'_{i} =\pi-\alpha_{i} $. Since the halfline h _i+1 is obtained from h _i by a rotation of angle $\alpha'_{i} $, it follows that

$$\theta_i \equiv\theta_1 +\alpha'_1+ \cdots+\alpha'_{i-1}\quad(\mathrm{mod}\ 2\pi)$$

for every 1≤i≤k. Thus $\chi(\theta_{i} )=\chi(\theta_{1} )\cdot \chi(\alpha'_{1} )\cdots\chi(\alpha'_{i-1})$ for every 1≤i≤k, and

$$ \varPhi(P)= \chi( \theta_1 )\cdot \bigl(\overline {X_0 X_1 } +\chi\bigl(\alpha'_1\bigr) \cdot \overline {X_1 X_2 } +\cdots+ \chi\bigl(\alpha'_1 \bigr) \cdots\chi\bigl(\alpha'_{k-1}\bigr) \cdot \overline {X_{k-1} X_k } \bigr) . $$

(15)

Now we turn to the proof of Theorem 2.1. We shall consider the following five cases separately: γ>π/2 and γ≠2π/3; γ<π/2 and γ≠π/3; γ=2π/3; γ=π/3; γ=π/2.

5 Case I: γ>π/2, γ≠2π/3

In this case, in each of equations (1), if r _i>0, then the equation must be of the form γ=δ _i. We claim that in each of equations (2) and (3) we have r _i≤p _i. Indeed, otherwise the reduced form of the equation in question would be rγ=vπ, where r is a positive integer and v=1 or 2. However, as π/2<γ<π and γ≠2π/3, no such equation is possible.

Let P(α+β)+Rγ=(N−2)π be the sum of equations (1). Then, by (4), we have P≤R. If N=3 then P(α+β)+Rγ=π, and thus P≤R gives P=0 or P=R=1. If P=0 then Rγ=π which is impossible. If P=R=1, then at one of the angles of A the equation is α=δ which contradicts the regularity of the tiling. Thus N=3 is impossible, and we have N≥4.

By P(α+β)+Rγ=(N−2)π and P≤R we have R≥N−2, and thus the possible values of R are N−2, N−1 and N. If R=N−2, then necessarily P=N−2, and in each of the equations (2) and (3) we have p _i=r _i.

If R=N−1, then N−1 of the angles of A equal γ. The equation P(α+β)+(N−1)γ=(N−2)π gives P(α+β)+(N−1)(π−α−β)=(N−2)π and π=(N−1−P)(α+β). There must exist an equation p _i(α+β)+r _i γ=vπ with r _i<p _i and v=1,2. The reduced form of this equation is p(α+β)=vπ, where we have p=(N−1−P)v. Since ∑p _i=∑r _i, it follows that p+P≤R; that is, (N−1−P)v+P≤N−1. Thus v=1, and there is no other vertex with r _j<p _j and j>N. Therefore, apart from the vertices of A, there is only one vertex of the tiling with p _i≠r _i, and the reduced equation at this vertex equals (N−1−P)(α+β)=π.

Finally, if R=N, then each angle of A equals γ. Thus α+β=π−γ=2π/N. There must exist an equation p _i(α+β)+r _i γ=vπ with r _i<p _i and v=1,2. The reduced form of this equation is p(α+β)=vπ, where we have 2p=Nv. Since ∑p _i=∑r _i, it follows that p+P≤R; that is, N⋅(v/2)≤N. If v=2, then there is no other vertex with r _j<p _j and j>N. If v=1, then there is one more such vertex, where the reduced equation is (N/2)⋅(α+β)=π. Therefore, apart from the vertices of A, there are at most two vertices with p _i≠r _i. If there is one such vertex, then the equation at this vertex is N(α+β)=2π, and if there are two such vertices, then N is even and the corresponding reduced equations are (N/2)⋅(α+β)=π.

Summing up: there are three cases. In the first case P=R=N−2, and p _i=r _i for every i>N; that is, every vertex other than the vertices of A is normal. In this case N−2 of the vertices of A equals γ, and the other two vertices equal p(α+β) and p′(α+β), where p+p′=N−2.

In the second case R=N−1, every vertex other than the vertices of A is normal with one exception. In the exceptional vertex the reduced equation is (N−1−P)(α+β)=π. In this case N−1 of the vertices of A equals γ, and one equals P(α+β).

In the third case R=N, every vertex other than the vertices of A is normal with at most two exceptions. If there is only one exceptional vertex, then the corresponding equation is N(α+β)=2π; and if there are two exceptional vertices, then their reduced equations are (N/2)⋅(α+β)=π. In this case each angle of A equals γ.

5.1 Subcase Ia

First we assume that a is not a linear combination of b and c with nonnegative rational coefficients. Then we may consider the directed graph Γ _a.

Suppose that P=R=N−2. Then there is no exceptional point other than the vertices of A and thus, by Lemma 4.2, the out-degree is not smaller than the in-degree at each vertex. Therefore, they are equal everywhere, and thus the graph Γ _a is the union of disjoint directed cycles. Since the in-degree is zero at each vertex belonging to the boundary of A, it follows that no edge starts from the boundary of A. Therefore, by (i) of Lemma 4.3, A does not have two consecutive vertices with angle γ. However, we know that A has N−2 vertices where its angle is γ. Since N−2≥2, it follows that N=4. Then P=2, and thus two equations at the vertices of A are of the form α+β=δ. In other words, A is a quadrilateral with angles γ,α+β,γ,α+β in this order. Therefore, A is a parallelogram.

Let X be a vertex of A with angle γ, and let T be the unique triangle of the tiling having X as a vertex. Let U be the vertex of T with angle β, and suppose that U is on the side XY. Then, by (ii) of Lemma 4.3, the length of the side XY equals ka, where k is a positive integer. Therefore, by Lemma 4.1, (i) of Theorem 2.1 holds in this case.

Next suppose that R=N−1. Let V denote the exceptional vertex where the reduced equation is (N−1−P)(α+β)=π. If, at the vertex V, the out-degree is not smaller than the in-degree then, as this is also true at every other vertex, it follows that the out-degree is equal to the in-degree everywhere, and Γ _a is the union of disjoint directed cycles. Since the in-degree is zero at each vertex belonging to the boundary of A, it follows that no edge starts from the boundary of A. We know that N−1≥3 vertices of A equals γ, and thus there are at least N−2≥2 sides of A such that A has angle γ at each endpoint. However, the exceptional point V can be an inner point of at most one of these sides, so there is a side without exceptional point and outgoing edge such that A has angle γ at each endpoint. By (i) of Lemma 4.3, this is impossible.

Therefore, the out-degree is smaller than the in-degree at the vertex V. This means that there is an edge arriving at V, but no edge starts from V. Then V is an inner point of A, and thus there is no exceptional point on the boundary of A. Since the out-degree is not smaller than the in-degree at the vertices different from V, it follows that Γ _a is the union of disjoint directed cycles and one path arriving at V. Therefore, on the boundary of A there is at most one vertex with positive out-degree. Then we can find again a side of A without exceptional points and without outgoing edge such that A has angle γ at each endpoint, which is impossible.

Next we consider the case R=N. Then all angles of A equal γ, and then there are N≥4 sides such that the angles of A at the endpoints equal γ. We know that there are at most two exceptional points, where the out-degree can be smaller than the in-degree. The deficit can be at most two, and thus Γ _a is the union of disjoint directed cycles and at most two paths arriving at the exceptional point(s). Considering the possible cases according to the number of exceptional points on the boundary of A, we can check that in each case there is a side of A without exceptional points and without outgoing edge such that A has angle γ at each endpoint. But this is a contradiction again. This completes the proof in the subcase when a is not a linear combination of b and c with nonnegative rational coefficients.

5.2 Subcase Ib

Suppose that b is not a linear combination of a and c with nonnegative rational coefficients. Since the roles of a and b are symmetric in the conditions as well as in the statements of the theorem, this subcase can be treated similarly to Subcase Ia.

Therefore, we may assume that a is a linear combination of b and c with nonnegative rational coefficients, and b is a linear combination of a and c with nonnegative rational coefficients. This implies that either a,b,c are pairwise commensurable, or a and b are commensurable and c is not a rational multiple of a.

If a,b,c are pairwise commensurable, then sinα,sinβ,sinγ are pairwise commensurable, and so are the sides of A. We show that this implies P=R=N−2, and thus in this case statement (ii) of the theorem holds. Indeed, if R=N−1 or R=N, then α and β satisfy an equation of the form p(α+β)=vπ, and thus γ is a rational multiple of π. Since cosγ=(c ²−a ²−b ²)/(2ab) is rational as well, it follows that γ=π/2,π/3 or 2π/3 (see [5, Corollary 3.12]), which is not the case.

5.3 Subcase Ic

Thus we are left with the case when a and b are commensurable and c is not a rational multiple of a. Then the graph Γ _c is empty by Lemma 4.6.

If P=R=N−2, then using (i) of Lemma 4.4 and the corresponding argument above, we can see that A is a parallelogram. Since A is rational by Lemma 4.7, we find that the case (i) of Theorem 2.1 holds.

The cases R=N−1 and R=N are impossible. Indeed, in these cases there is a side XY of A such that the angle of A at the vertices X and Y equals γ, and XY does not contain exceptional points. This, together with the fact that Γ _c is empty, contradicts (i) of Lemma 4.4.

6 Case II: γ<π/2, γ≠π/3

In this case, in each of equations (1), if p _i>0, then the equation must be of the form α+β=δ _i. We claim that in each of equations (2) and (3) we have r _i≥p _i. Indeed, otherwise the reduced form of the equation in question would be p(α+β)=vπ, where p is a positive integer and v=1 or 2. However, as π/2<α+β<π and α+β≠2π/3, no such equation is possible.

Let P(α+β)+Rγ=(N−2)π be the sum of equations (1). Then, by (4), we have P≥R. If N=3 then P(α+β)+Rγ=π, and thus P≥R gives R=0 or P=R=1. If R=0 then P(α+β)=π which is impossible. If P=R=1, then at one of the angles of A the equation is α=δ _i which contradicts the regularity of the tiling. Thus N=3 is impossible, and we have N≥4.

By P(α+β)+Rγ=(N−2)π and P≥R we have P≥N−2, and thus the possible values of P are N−2, N−1 and N. If P=N−2, then necessarily R=N−2, and in each of the equations (2) and (3) we have p _i=r _i.

If P=N−1, then N−1 of the angles of A equal α+β. The equation (N−1)(α+β)+Rγ=(N−2)π gives (N−1−R)γ=π. There must exist an equation p _i(α+β)+r _i γ=vπ with r _i>p _i and v=1,2. The reduced form of this equation is rγ=vπ, where we have r=(N−1−R)v. Since ∑p _i=∑r _i, it follows that, apart from the vertices of A, there is only one vertex of the tiling with p _i≠r _i, and that the reduced equation at this vertex is (N−1−R)γ=π.

Finally, if P=N, then each angle of A equals α+β. Thus γ=π−(α+β)=2π/N. There must exist an equation p _i(α+β)+r _i γ=vπ with r _i>p _i and v=1,2. The reduced form of this equation is rγ=vπ, where we have 2r=Nv. Since ∑p _i=∑r _i, it follows that, apart from the vertices of A, there are at most two vertices with p _i≠r _i. If there is one such vertex, then the reduced equation at this vertex is Nγ=2π, and if there are two such vertices, then N is even and the corresponding reduced equations are (N/2)γ=π.

Summing up: there are three cases. In the first case P=R=N−2, and p _i=r _i for every i>N. In this case N−2 of the vertices of A equals α+β, and the other two vertices equal rγ and r′γ, where r+r′=N−2.

In the second case P=N−1, and p _i=r _i for every i>N with exactly one exception, where the reduced equation is (N−1−R)γ=π. In this case N−1 of the vertices of A equals α+β, and one equals Rγ.

In the third case P=N, and p _i=r _i for every i>N with at most two exceptions. If there is only one exceptional vertex, then the corresponding equation is Nγ=2π, and if there are two exceptional vertices, then their reduced equations are (N/2)γ=π. In this case each angle of A equals α+β.

6.1 Subcase IIa

First we assume that a is not a linear combination of b and c with nonnegative rational coefficients, and consider the graph Γ _a. If R=N−2, then there are no exceptional points apart from the vertices of A, and the in-degree equals the out-degree at every vertex. Thus no vertex starts from the boundary of A. By (iii) of Lemma 4.3 this implies that there are no adjacent vertices of A with angle α+β. As in case I, we can infer that A is a parallelogram, and using an analogous argument, we can check that in this case the statement (i) of the theorem holds. Repeating the analogous argument of case I, we can also see that the cases R=N−1 and R=N are impossible.

6.2 Subcase IIb

Since the roles of a and b are symmetric, we have the same conclusion if b is not a linear combination of a and c with nonnegative rational coefficients.

Therefore, we may assume that a is a linear combination of b and c with nonnegative rational coefficients, and b is a linear combination of a and c with nonnegative rational coefficients. This implies that either a,b,c are pairwise commensurable, or a and b are commensurable and c is not a rational multiple of a. If a,b,c are pairwise commensurable, then (iii) of the theorem is true. Indeed, if R=N−1 or R=N, then γ is a rational multiple of π, which is impossible (see the analogous argument in case I).

6.3 Subcase IIc

In this subcase we assume that a and b are commensurable and c is not a rational multiple of a. Then the graph Γ _c is empty by Lemma 4.6.

Lemma 6.1

If a and b are commensurable and c is not a rational multiple of a, then either (i) of Theorem 2.1 holds, or P=N; that is, each angle of A equals α+β.

Proof

Let X,Y,Z be consecutive vertices of A, and suppose that the equation at X is rγ=δ. We show that if there are no exceptional points on the sides XY and YZ, then the equation at Z is also of the form rγ=δ. Indeed, by (i) of Lemma 4.4, the equation at Y must be α+β=δ. The proof of Lemma 4.4 also shows that there is a triangle T _k supported by the side XY, and there is a point U _k−1 in the interior of XY such that U _k−1 Y is a side of T _k, and the angle of T _k at U _k−1 is γ.

Let $T'_{1} ,\ldots,T'_{m}$ be the triangles supported by the side YZ, and let Y=V ₀,…,V _m=Z be a division of YZ such that V _i−1 V _i is a side of $T'_{i}$ for every i=1,…,m. Then the angle of $T'_{1}$ at the vertex Y equals α or β. Since no edge of Γ _c starts from Y and the angle of T _k at U _k−1 equals γ, it follows that the angle of $T'_{1}$ at V ₁ must be γ. Now the vertices V _i are normal by assumption, and no edge of Γ _c starts from any of them. Therefore, each $T'_{i}$ has angle γ at the vertex V _i (see Fig. 4). In particular, $T'_{m}$ has angle γ at V _m=Z, and thus the equation at Z is of the form rγ=δ.

Suppose R=N−2. Then there are no exceptional points, and thus it follows from what we proved above that the equation at every second vertex of A is of the form rγ=δ. Then N=4 and A is a parallelogram. By Lemma 4.7, the sides of A are commensurable. Since a/b is rational, it follows that (i) of Theorem 2.1 holds.

Next suppose P=N−1. Then there is one single exceptional point. Let Z ₁,Y ₁,X,Y ₂,Z ₂ be consecutive vertices of A such that the equation at the vertex X is of the form rγ=δ. (The vertices Z ₁ and Z ₂ may coincide.) Since only one of the sides Z ₁ Y ₁, Y ₁ X, XY ₂, Y ₂ Z ₂ can contain the exceptional point, it follows that the equation at either Z ₁ or Z ₂ is of the form rγ=δ. This, however, contradicts the assumption P=N−1. Therefore, the only remaining possibility is P=N, which completes the proof. □

Therefore, we are left with the case when P=N. If N=4, then A is a rectangle. Since α+β=π/2, sinα/sinβ=a/b is rational and A is rational by Lemma 4.7, it follows that (i) of Theorem 2.1 holds. Thus we may assume that N≥5. Our next aim is to show that in this case (v), (vi) or (vii) of Theorem 2.1 holds. The rest of the section is devoted to the proof of this statement.

In Lemma 4.8 we proved that those triangles of the tiling that have their side of length c in the interior of A come in pairs, and each pair forms a quadrilateral which is either a parallelogram of sides a,b and of angles γ,α+β, or a kite of sides a,b and of angles 2β,γ,2α,γ.

Lemma 6.2

Suppose that a and b are commensurable, c is not a rational multiple of a, and P=N. Then among the pairs described above only parallelograms can occur. More precisely, there are no kites, unless α=β when each kite is a parallelogram.

Proof

Suppose this is not true; that is, α≠β and there exists at least one kite. We shall denote by Γ _kite the set of those directed segments $\overrightarrow{XY}$ for which the segment XY is the common side of two triangles having angles α at X and β at Y. Then the line going through the edge $\overrightarrow{XY}$ of Γ _kite is the axis of symmetry of a kite described in Lemma 4.8.

If $\overrightarrow{XY}$ is an edge of Γ _kite, then Y is in the interior of A. Indeed, the left hand side of the equation at Y is of the form p(α+β)+rγ with p≥2. Since r≥p≥2, we have p=r=2, and the equation at Y is 2α+2β+2γ=2π.

By Lemma 4.8, the triangles having Y as a vertex and having angle α or β at Y come in pairs. Each pair forms a quadrilateral which is either a parallelogram having angle α+β or γ at Y, or a kite having angle 2α or 2β at Y.

Since p=r=2, it follows that the arrangement of the triangles around Y must be one of two cases presented in Fig. 5.

Since there is exactly one kite having angle 2α at Y, it follows that there is exactly one edge of Γ _kite starting from Y. Therefore, Γ _kite is the union of disjoint cycles, and every vertex of Γ _kite is in the interior of A. Note also that each cycle of Γ _kite is a simple polygon of angles α+β or α+2γ+β.

We shall need another graph on the set of vertices. Let XZ be a maximal segment contained by the union of the boundaries of the triangles Δ _i and such that XZ belongs to the interior of A except perhaps the endpoints X and Z. There are divisions X=U ₀,U ₁,…,U _k=Z and X=V ₀,V ₁,…,V _ℓ=Z of the segment XZ such that each subinterval U _i−1 U _i (i=1,…,k) is a side of a triangle T _i of the tiling supported by XZ and lying on the same side of the segment XZ, and each subinterval V _j−1 V _j (j=1,…,ℓ) is a side of a triangle $T'_{j}$ of the tiling supported by XZ and lying on the other side of XZ. Suppose that the angle of T ₁ at the vertex U ₁ and the angle of $T'_{1}$ at the vertex V ₁ both equal γ. Then there is a maximal index 1≤i ₀≤k such that the angle of T _i at the vertex U _i equals γ for every i=1,…,i ₀. Similarly, there is a maximal index 1≤j ₀≤ℓ such that the angle of $T'_{j}$ at the vertex V _j equals γ for every j=1,…,j ₀. Then we connect the vertices X and Y by a directed edge $\overrightarrow{XY}$ where $Y=U_{i_{0}}$ if $U_{i_{0}}$ is closer to X than $V_{j_{0}}$, $Y=V_{j_{0}}$ if $V_{j_{0}}$ is closer to X than $U_{i_{0}} $, and $Y=U_{i_{0}} =V_{j_{0}}$ otherwise. We denote by Γ _except the set of these edges. (The notation will be justified later, when we show that each edge of Γ _except starts from a vertex of A and arrives at an exceptional point.)

Let $\overrightarrow{XY}$ be an edge of Γ _except. We show that (i) no inner point of the segment XY can be a vertex of Γ _kite , and (ii) Y is a vertex of Γ _kite or Y is an exceptional point.

In the proof of these statements we shall use the notation of the definition of the graph Γ _except. By symmetry, we may assume that $Y=U_{i_{0}} $. Let V be an inner point of XY, and suppose that V is a vertex of Γ _kite. Then the arrangement of the triangles around V cannot be as shown by (B) of Fig. 5 since 2α+γ≠π and 2β+γ≠π. But it cannot be as shown by (A) of Fig. 5 either. Indeed, if i and j are such that $T_{i} ,T'_{j}$ have V as a vertex, then they are supported by the segment XZ and have angles γ at V. However, those two triangles in (A) of Fig. 5 which have angle γ at the given point are not supported by a common segment. This proves (i).

Next suppose that Y is not a vertex of Γ _kite and that Y is normal. Since $Y=U_{i_{0}}$, we have either $U_{0}=V_{j_{0}}$ or U ₀ is closer to X than $V_{j_{0}}$. This implies that either Y is in the interior of a side of one of the triangles $T'_{j} $, or Y is the vertex of a triangle $T'_{j}$ such that the angle of $T'_{j}$ at Y equals γ. In the first case the equation at Y is α+β+γ=π, and the angle of $T_{i_{0}}$ at Y equals γ. By Lemma 4.6, no edge of Γ _c starts from Y. This implies that the angle of the triangle $T_{i_{0}+1}$ at the point $U_{i_{0} +1}$ must be γ. (Indeed, by considering the possible arrangements of the angles around Y we can see that otherwise an edge of Γ _c would start from Y.) This, however, contradicts the choice of i ₀, since i ₀ was the largest index such that T _i has angle γ at U _i for every i≤i ₀.

In the second case there are two triangles supported by the continuation of the segment XY and having angle γ at Y. Since Y is normal, there are four other triangles having Y as a vertex, and their angles at Y are α,α,β,β. The two triangles with angle α at V cannot be adjacent, because in that case, depending on the location of their angle γ, either there would be an edge of Γ _c starting from Y, or Y would be a vertex of Γ _kite. Thus the angles of the four triangles at V are either α,β,α,β or β,α,β,α in this order. Thus Y is an inner point of the segment XZ. Then we can check, the same way as in the previous case, that the angle of $T_{i_{0} +1}$ at the point $U_{i_{0} +1}$ must be γ. This, again, contradicts the maximality of i ₀, which completes the proof of (ii).

If P is a simple polygon, then the bounded component of ℝ²∖P will be denoted by P°.

Let C be a cycle of Γ _kite. We shall also denote by C the polygon formed by the edges belonging to C. Let X be a vertex of C such that the angle of C at X is convex; that is, equals α+β. (For short, in this case we shall say that X is a convex vertex of C.) Let h denote the halfline starting from X as in (B) of Fig. 5. Then there is a point Y on h such that $\overrightarrow{XY}$ is an edge of Γ _except. Since h intersects C°, and no interior point of XY can be a vertex of Γ _kite, it follows that one of the following must hold: (i) Y is a vertex of C; (ii) Y∈C° and Y is a vertex of Γ _kite; or (iii) Y∈C° and Y is exceptional.

Note that in case (i) the angle of C at Y is concave; that is, equals α+2γ+β. Indeed, Y must be a vertex of C as in (B) of Fig. 5. Since the halfline h arrives at Y from C°, it follows that the angle of C at Y equals α+2γ+β.

Our next aim is to show that for every cycle C of Γ _kite , C° contains all exceptional points. In order to prove this we may assume that C° does not contain any vertex of Γ _kite. Indeed, if C ^∘∩Γ _kite≠∅, then we take a cycle C ₁ of Γ _kite belonging to C° and having minimal area. Then $C^{\circ}_{1}$ does not contain any vertex of Γ _kite, and if $C^{\circ}_{1}$ contains all exceptional points then so does C°.

As we proved above, if V is a convex vertex of C, then there is an edge $\overrightarrow{VW}$ of Γ _except such that either W is a concave vertex of C or W∈C° and W is exceptional. (Here we used the fact that C° does not contain any vertex of Γ _kite.) If u and v denote the number of convex and concave vertices of C, respectively, then we have

$$u(\alpha +\beta )+v(\alpha +\beta +2\gamma)=(u+v-2)\pi.$$

Since γ=2π/N and α+β=π(1−(2/N)), this implies u=v+N. Let V and V′ be distinct convex vertices of C, and let $\overrightarrow{VW}$, $\overrightarrow{V'W'}$ be the corresponding edges of Γ _except. It is clear that if W and W′ are concave vertices of C then they must be distinct (see (B) of Fig. 5). Since u=v+N, it follows that there are at least N convex vertices of C such that the corresponding edge a$\overrightarrow{VW}$ arrives at an exceptional point belonging to C°. This proves the claim if there is only one exceptional point. Suppose there are two exceptional points: E and F. Then N is even, and the equation at E is either (N/2)γ=π or α+β+((N/2)+1)γ=2π. In both cases, the number of edges of Γ _except ending at E is at most N/2; this follows from the fact that if an edge arrives at E then there are two triangles having E as a vertex and such that they are supported by the same segment, and their angles at E equal γ. The same is true for F. Since there are at least N edges arriving at exceptional points belonging to C°, it follows that both E and F must be in C°. This proves our claim.

Let C ₁ and C ₂ be disjoint cycles of Γ _kite. Since both of $C^{\circ}_{1}$ and $C^{\circ}_{2}$ contain the exceptional points, we have $C_{1}^{\circ}\cap C_{2}^{\circ}\ne\emptyset $, and thus we have either $C_{2} \subset C^{\circ}_{1}$ or $C_{1} \subset C^{\circ}_{2}$. Therefore, we may list the cycles of Γ _kite as C ₁,…,C _s, where $C_{i+1} \subset C^{\circ}_{i}$ for every i=1,…,s−1, and $C^{\circ}_{s}$ contains all exceptional points. In particular, there are no exceptional points in $A\setminus C^{\circ}_{1}$.

We prove that C ₁ is convex. Suppose this is not true, and let V be a concave vertex of C ₁. Then there is an edge $\overrightarrow{VW}$ of Γ _except starting from V (see (B) of Fig. 5). Since the edge starts from $A\setminus C_{1}^{\circ}$ and $A\setminus C_{1}^{\circ}$ contains no vertex of Γ _kite and no exceptional points, it follows that W belongs to the boundary of $A\setminus C_{1}^{\circ}$. That is, either W is in the boundary of A, or W is a vertex of C ₁. But W cannot be a boundary point of A, since W is either a vertex of Γ _kite or is exceptional. Hence W is a vertex of C ₁. Clearly, the segment VW is in the exterior of C ₁ (except the points V and W). The argument above proves that for every concave vertex V of C ₁ there is another vertex W of C ₁ such that the segment VW is in the exterior of C ₁. These segments VW are pairwise disjoint, since any intersection would be an exceptional point.

For every concave vertex V, the vertices V and W divide C ₁ into the subarcs $\sigma^{V}_{1}$ and $\sigma^{V}_{2} $. Let $P^{V}_{i}$ denote the simple polygon $(VW) \cup\sigma^{V}_{i} \ (i=1,2)$. We may choose the indices in such a way that $(P^{V}_{1} )^{\circ}$ is disjoint from $C_{1}^{\circ}$, and $(P^{V}_{2} )^{\circ}$ contains $C_{1}^{\circ}$. Let V be a concave vertex of C ₁ for which the area of $(P^{V}_{1} )^{\circ}$ is minimal. Since $(P^{V}_{1} )^{\circ}\cap C_{1}^{\circ}=\emptyset $, it follows that there exists a concave vertex V′ in the subarc $\sigma^{V}_{1} $. If $\overrightarrow{V'W'}$ is the edge of Γ _except starting from V′ then, as the segments VW and V′W′ disjoint, W′ belongs to $\sigma^{V}_{1} $. Then the area of $(P^{V'}_{1})^{\circ}$ is smaller than that of $(P^{V}_{1} )^{\circ}$, contradicting the choice of V. This contradiction proves that C ₁ is convex. Then every angle of C ₁ equals α+β=(1−(2/N))π, and hence C ₁ has N vertices.

Let X be a vertex of A. Since no edge of Γ _c starts from X, the triangles having X as a vertex must be arranged as in Fig. 6.

Suppose that they are arranged as in (B) of Fig. 6. Let XY be a side of A. Let X=U ₀,…,U _k=Y be a division of XY such that each U _i−1 U _i is the side of a triangle T _i of the tiling. Since the angle of T ₁ at the vertex U ₁ is γ and every vertex on XY is normal, it follows that the angle of T _i at the vertex U _i is γ for every i (see Fig. 4). In particular, the angle of T _k at the vertex U _k=Y is γ, which is impossible.

Therefore, the triangles T and T′ having X as a vertex must be arranged as in (A) of Fig. 6. Then the angles of T and of T′ opposite to the sides on the boundary of A equal γ, which means that there exists an edge $\overrightarrow{XW}$ of Γ _except starting from X. Since $A\setminus C_{1}^{\circ}$ contains no vertex of Γ _kite and no exceptional points, it follows that W belongs to the boundary of $A\setminus C_{1}^{\circ}$. We can check, using a previous argument, that W is a vertex of C ₁. Clearly, the segment XW is in the exterior of C ₁ (except the point W).

Let V ₁,…,V _N=V ₀ denote the vertices of A such that V _i−1 and V _i are adjacent for every i=1,…,N. As we saw above, there are vertices W _i of C ₁ such that $\overrightarrow{V_{i} W_{i}}$ is a vertex of Γ _except for every i=1,…,N. The segments V _i W _i are pairwise disjoint, since an intersection of two of them would be an exceptional point.

Since C ₁ is an N-gon, it has no vertices other than W ₁,…,W _N. Put W ₀=W _N. Then W _i−1 and W _i are adjacent vertices of C ₁ for every i=1,…,N. Indeed, suppose that W ₁,W _i,W _j,…,W _k is a list of the consecutive vertices of C ₁, where 2<i<N. Then P={V ₁ V ₂…V _i W _i W ₁ V ₁} and Q={V _i V _i+1…V _N,V ₁,W ₁,W _i,V _i} are simple polygons. If W _j∈P°, then all vertices of C ₁ other than W ₁ and W _i belong to P°, since the sides of C ₁, with the exception of W _k W ₁,W ₁ W _i and W _i W _j, are disjoint from the boundary of P. Since W _N∉P°, this is impossible. If W _j∈Q°, then we obtain a contradiction by W ₂∈P°. This shows that W ₁ and W _i can be adjacent vertices of C ₁ only if i=0 or i=2. We find, in the same way, that W _i−1 and W _i are adjacent for every i=1,…,N. Therefore, the quadrilaterals R _i={V _i−1,V _i,W _i,W _i−1} (i=1,…,N) form a ring along the boundary of A as shown in Fig. 7.

Now we prove that this is impossible (assuming α≠β). Since the equations at the vertices of A are of the form α+β=δ, the angles of R _i at the vertices V _i−1 and R _i are equal to either α or β. We shall assume that the angle of R ₁ at the vertex V ₀ equals α. (The same argument applies if it equals β.) Let T denote the triangle having V ₀ as a vertex and lying in R ₁. Since $\overrightarrow{V_{0} W_{0}}$ is an edge of Γ _except, the angle of T opposite to the side supported by V ₀ V ₁ equals γ. By (ii) of Lemma 4.4, each triangle supported by V ₀ V ₁ is a translated copy of T. This implies that the angle of R ₁ at the vertex V ₁ equals β. Then the angle of R ₂ at the vertex V ₁ equals α, and (ii) of Lemma 4.4 gives that the angle of R ₁ at the vertex V ₂ equals β. Continuing the argument we find that for every i, the angle of R _i at the vertex V _i−1 equals α, and its angle at the vertex V _i equals β.

The vertex W ₀ is a vertex of Γ _kite, and it is the endpoint of the edge $\overrightarrow{V_{0} W_{0}}$ of Γ _except. Therefore, the angle of R ₁ at the vertex W ₀ equals either γ+α or γ+β (see (B) of Fig. 5). Suppose it is γ+α. Then the angle of R ₁ at the vertex W ₁ equals γ+β, since 2π−α−β−(γ+α)=γ+β. Thus the angle of R ₂ at the vertex W ₁ equals γ+α, and its angle at W ₂ equals γ+β. Continuing the argument we find that for every i, the angles of R _i are as in (A) of Fig. 8. If the angle of R ₁ at the vertex W ₀ equals γ+β, then we find that for every i, the angles of R _i are as in (B) of Fig. 8.

Now we prove that both cases are impossible if α≠β. First suppose that each R _i looks like (A) of Fig. 8. There is a point Y on the segment V _i−1 V _i such that V _i−1 W _i−1 Y∡=γ and W _i W _i−1 Y∡=α. Then the segments YW _i−1 and V _i W _i are parallel to each other. Thus the quadrilateral YV _i W _i W _i−1 is a trapezoid. If α>β, then (γ+β)+β<π, and hence $\overline {V_{i} W_{i}}>\overline {YW_{i-1}}$. The angles of the triangle V _i−1 YW _i−1 are α,β,γ, and thus the condition α>β implies $\overline {Y W_{i-1}}>\overline {V_{i-1} W_{i-1}}$. Therefore, α>β implies $\overline {V_{i} W_{i}} >\overline {V_{i-1} W_{i-1}}$ for every i. Thus

$$ \overline {V_1 W_1} < \overline {V_2 W_2} <\cdots< \overline {V_NW_N} < \overline {V_1 W_1},$$

(16)

which is impossible. If α<β, then a similar argument gives

$$ \overline {V_1 W_1} > \overline {V_2 W_2} >\cdots> \overline {V_NW_N} > \overline {V_1 W_1},$$

(17)

also impossible. Next suppose that the quadrilaterals R _i look like (B) of Fig. 8. Then the sides W _i−1 W _i and V _i−1 V _i are parallel to each other. There is a point Y on the segment V _i−1 V _i such that V _i−1 W _i−1 Y∡=γ and W _i W _i−1 Y∡=β. Then YV _i W _i W _i−1 is a parallelogram, and thus $\overline {V_{i} W_{i}} =\overline {YW_{i-1}}$. Suppose α>β. Then, as the angles of the triangle V _i−1 YW _i−1 are α,β,γ, we have $\overline {Y W_{i-1}} >\overline {V_{i-1} W_{i-1}}$. Therefore, α>β implies $\overline {V_{i} W_{i}} >\overline {V_{i-1} W_{i-1}}$ for every i, which implies (16). If α<β, then a similar argument gives (17). Since both are impossible, the proof is complete. □

Lemma 6.3

Suppose that a and b are commensurable, c is not a rational multiple of a, and P=N; that is, each angle of A equals α+β. Then one of (v), (vi) or (vii) of Theorem 2.1 holds.

Proof

We shall use the notation introduced in the proof of Lemma 6.2. We proved there that if $\overrightarrow{XY}$ is an edge of Γ _except, then Y is either a vertex of Γ _kite, or is an exceptional point. (Note that the argument proving this statement did not use the condition α≠β.) If α≠β, then Γ _kite is empty, and thus Y must be an exceptional point. The same conclusion holds if α=β. Indeed, suppose that α=β and the endpoint Y is normal. Then Y is a vertex of Γ _kite, and the triangles around Y are arranged as in Fig. 5. It is clear that the case of (A) is impossible. In the case of (B), the halfline h is a continuation of the segment XY, since α+α+γ=α+β+γ=π. Now the two triangles supported by h have angle γ at their vertices lying on h, which contradicts the fact that Y is an endpoint of an edge. Indeed, in the definition of Γ _except Y is defined as $U_{i_{0}}$, where i ₀ is the maximal index such that the triangle T _i has angle γ at U _i for every i≤i ₀, while, in (B) of Fig. 5, i ₀+1 also has this property. This proves that Y must be exceptional.

In the proof of Lemma 6.2 it was also shown that each vertex of A is a starting point of an edge of Γ _except. (This argument was also independent of the condition α≠β.) Let $\overrightarrow{V_{i} W_{i}}$ be an edge of Γ _except for every i=0,…,N, where W ₁,…,W _N=W ₀ are exceptional points.

As we saw earlier, we have either W _i−1 V _i−1 V _i∡=α and V _i−1 V _i W _i∡=β for every i=1,…,N, or W _i−1 V _i−1 V _i∡=β and V _i−1 V _i W _i∡=α for every i=1,…,N. Since the roles of α and β are symmetric, we may assume the former.

Suppose that there is one exceptional point, E. Then W _i=E for every i, and the edges $\overrightarrow{V_{i} E}$ (i=1,…,N) decompose A into the nonoverlapping triangles V _i−1 V _i E. Since EV _i−1 V _i∡=α and V _i−1 V _i E∡=β for every i=1,…,N, the triangles V _i−1 V _i E are similar to each other. If $\overline {V_{0} E} > \overline {V_{1} E} $, then $\overline {V_{i-1} E} > \overline {V_{i} E}$ for every i, and we obtain

$$\overline {V_0 E} > \cdots>\overline {V_{N-1} E} >\overline {V_0 E},$$

which is impossible. We get a similar contradiction if $\overline {V_{0} E} < \overline {V_{1} E} $. Thus $\overline {V_{0} E} = \overline {V_{1} E} $, α=β, A is a regular N-gon, and we obtain (vii) of Theorem 2.1.

Next suppose that there are two exceptional points, E and F. Then N is even, N=2k, and the equation at E is either kγ=π or α+β+(k+1)γ=2π. It is clear that in the first case the number of edges of Γ _except with endpoint E is at most k−1, and in the second case this number is at most k. The same is true for F. Since there are (at least) N=2k edges arriving at E and F, it follows that the equations at E and F must be α+β+(k+1)γ=2π. In particular, E and F are in the interior of A, and both of them are the endpoints of k edges. By shifting the indices we may assume that there is an index 1≤j<N such that W _i=E for every i=0,…,j−1, and W _N−1=W _j=F.

The edges starting from the vertices of A are pairwise disjoint, except the endpoints. This implies that E is in the interior of the simple polygon {V _N−1,…,V _j,F}, and thus W _i=F for every i=j,…,N−1. Therefore, we have j=k, W ₀=⋯=W _k−1=E and W _k,…,W _N−1=F. (See Fig. 9.)

The polygons P={V ₀,…,V _k−1,E} and Q={V _k,…,V _N−1,F} are convex, since their angle at E and F, respectively, equal (k−1)γ=(1−(1/k))π. Put q=sinα/sinβ=a/b. Then, as a and b are commensurable, q is rational. The triangles EV _i−1 V _i (i=1,…,k−1) and FV _i−1 V _i (i=k+1,…,N) are similar, and we have

$$ \overline {V_{i}V_{i+1}} /\overline {V_{i-1}V_i} =q \quad(i\ne0, k).$$

(18)

This implies that P and Q are similar polygons.

Let D denote the middle point of the segment V ₀ V _k. We show that A is centrally symmetric with center D.

Let e _i denote the line going through the vertices V _i−1 and V _i. Since each angle of A equals α+β=(1−(1/k))π, it follows that the lines e ₀ and e _k are parallel to each other.

Let ρ denote the reflection about the point D. Then ρ(V _k)=V ₀ and ρ(e ₀)=e _k. Since V _N−1∈e ₀, it follows that ρ(V _N−1)∈e _k. Now we have V _N−1 V ₀ E∡=V _k−1 V _k F∡=β, and thus ρ(F) is on the halfline starting from V ₀ and going through E. Since the triangles {V ₀,V _k−1,E} and {V _k,V _N−1,F} are similar to each other and ρ(V _N−1) is on the line e _k, it follows that ρ({V _k,V _N−1,F}) coincides with ρ({V ₀,V _k−1,E}). Thus ρ(Q)=P and ρ(A)=A; that is, A is centrally symmetric.

By Lemma 4.7, A is a rational polygon. Then, it follows from (18) and q=sinα/sinβ that (vi) of Theorem 2.1 holds if N=6, and (v) of Theorem 2.1 holds if N≥8. This completes the proof of the lemma. □

7 Case III: γ=2π/3

In this case each angle of A equals π/3 or 2π/3. Thus A can be a regular triangle, a parallelogram, a trapezoid, a pentagon or a hexagon. If A is parallelogram or a trapezoid then two of its angles equal π/3 and the other two angles equal 2π/3; if A is a pentagon then one of its angles equals π/3 and the other angles equal 2π/3; if A is a hexagon, then all its angles equal 2π/3.

7.1 Subcase IIIa

First we assume that a is not a linear combination of b and c with nonnegative rational coefficients. Then we consider the directed graph Γ _a.

Lemma 7.1

Suppose that a is not a linear combination of b and c with nonnegative rational coefficients. Then

(i)

the out-degree of Γ _a equals the in-degree at each vertex;
(ii)

no edge of Γ _a starts from the boundary of A; and
(iii)

every vertex lying on the boundary of A but different from the vertices of A is normal.

Proof

It is easy to check that the equation at an exceptional point must be one of 3γ=2π, 3α+3β=π, 4α+4β+γ=2π, 6α+6β=2π.

In order to prove (i) it is enough to show that if the in-degree of a vertex X is positive, then so is the out-degree of X. If X is normal, then the out-degree at X is positive by Lemma 4.2. If X is exceptional then the equation at X must be 3α+3β=π, since X is an interior point of a side of a triangle. Let T ₁,…,T ₆ denote the triangles having X as a vertex listed counterclockwise, and let h _i denote the halfline starting from X and supporting the triangles T _i and T _i+1 (i=1,…,5). Three of the triangles T _i have angle β at X, and at least two of them are of the same orientation in the sense that a rotation about the point X brings one of them onto the other. Let T _i and T _j be such triangles. If j=i+1, then an edge of Γ _a starts from X along the halfline h _i. If j>i+1, then an edge of Γ _a starts from X along one of the halflines h _i and h _j−1, depending on the location of the angle γ in T _i and T _j. This proves (i).

The argument above shows that if the equation at a vertex X is 3α+3β=π, then an edge of Γ _a starts from X. Since no edge arrives at any boundary point, we obtain (ii) from (i). Therefore, if X is a boundary point and not a vertex of A, then the equation at X cannot be 3α+3β=π. Since the right hand side of all other equations at exceptional points equals 2π, it follows that X must be normal, which proves (iii). □

It is easy to check that if the equation at a vertex is X is 3γ=2π, then the in-degree of X is zero and the out-degree of X is positive. Since this contradicts (i) of Lemma 7.1, it follows that the equation cannot be 3γ=2π at any vertex. Therefore, we have r _i≤p _i for every i>N. Let P(α+β)+Rγ=(N−2)π be the sum of equations at the vertices of A. Then we have R≥P. In particular, we obtain R>0, and thus N=3 is impossible.

Suppose N=4. By (i) of Lemma 4.3, there are no adjacent vertices of A at which the equation is γ=δ. Thus R≤2, and then P≤R implies that P=R=2, and that the angles of A must be α+β, γ, α+β, γ in this order. Thus A is a parallelogram. Since R>0, there is a side XY of A such that the equation at X is γ=δ, and the triangle having X as a vertex has angles γ at X and has angle β at its vertex lying on the side XY. By (ii) of Lemma 4.3, the length of XY is k⋅a with a positive integer k. Therefore, by Lemma 4.1, (i) of Theorem 2.1 holds.

Next suppose N=5. Since there are no adjacent vertices of A at which the equation is γ=δ, we have R≤2. Then P≥3, which contradicts P≤R.

Finally, if N=6, then R≤3, and thus there are at least three vertices of A where the equation is 2(α+β)=δ. Thus P≥6, which contradicts P≤R again. This completes the proof in the subcase when a is not a linear combination of b and c with nonnegative rational coefficients.

7.2 Subcase IIIb

Suppose that b is not a linear combination of a and c with nonnegative rational coefficients. Since the roles of a and b are symmetric in the conditions as well as in the statements of the theorem, this subcase can be treated similarly to Subcase IIIa.

Therefore, we may assume that a is a linear combination of b and c with nonnegative rational coefficients, and b is a linear combination of a and c with nonnegative rational coefficients. This implies that either a,b,c are pairwise commensurable, or a and b are commensurable and c is not a rational multiple of a.

If a,b,c are pairwise commensurable, then so are the sides of A. We know that A is an N-gon with 3≤N≤6. If 4≤N≤6, then A has N−2 vertices with angle 2π/3, and the other two angles of A are integer multiples of π/3. Thus, in these cases, (ii) of Theorem 2.1 holds. If N=3, then (iv) of Theorem 2.1 holds.

7.3 Subcase IIIc

Thus we are left with the case when a and b are commensurable and c is not a rational multiple of a.

If A is a parallelogram, then (i) of Theorem 2.1 holds, since A is rational by Lemma 4.7, and sinα/sinβ=a/b∈ℚ. Therefore, we may assume that A is not a parallelogram; that is, A is a triangle, a trapezoid, a pentagon or a hexagon.

We show that α is not a rational multiple of π except when α=π/6. Indeed, we have

$$\frac{b}{a}=\frac{\sin\beta }{\sin\alpha } =\frac{\sin((\pi /3)-\alpha )}{\sin \alpha } =\frac{\sqrt{3}}{2}\cdot \cot\alpha -\frac{1}{2},$$

and thus $\sqrt{3} \cdot \cot\alpha $ is rational. Thus tan² α is rational, and then so are cos² α=1/(1+tan² α) and cos2α. By [5, Corollary 3.12]), this implies cos2α=0,±1,±1/2. Since α<π/3, we have 0<2α<2π/3, and thus 2α=π/3 or π/2, and α=π/6 or π/4. If α=π/4, then $\sqrt{3} \cdot \cot\alpha =\sqrt{3}$ is irrational, so the only possibility is α=π/6.

If α=π/6, then β=π/6 and, taking into consideration that A is rational by Lemma 4.7, we find that (viii) of Theorem 2.1 holds. Therefore, we may assume that α/π is irrational.

Let G denote the set of real numbers n⋅(π/3)+m⋅α, where n,m∈ℤ. Then G is an additive subgroup of the reals such that π∈G. Recall that we denote by $\mathcal{P}_{G}$ the family of all simple, closed polygons such that, for every side XY of P, the angle between the line going through the side XY and the x-axis belongs to G.

We may assume that the x-axis contains one of the sides of A. Then $A\in\mathcal{P}_{G}$ and $\varDelta_{i} \in\mathcal{P}_{G}$ for every i=1,…,t. Indeed, let T be any of the triangles Δ _i, and let e be a line containing one of the sides of T. Then there is a sequence of triangles T ₀,…,T _k and there is a sequence of lines e ₀,e ₁,…,e _k such that T _k=T, e ₀ is the x-axis, e _k=e, and e _i contains a side of both T _i−1 and T _i for every i=1,…,k. Then the angle θ _i of e _i−1 and e _i is one of α, β=(π/3)−α and 2π/3 for every i=1,…,k, and thus the angle between e _k and e ₀ equals $\sum_{i=1}^{k}\theta_{i} \in G$.

If θ=n⋅(π/3)+m⋅α where n,m∈ℤ, then we put χ(θ)=(−1)ⁿ. Then χ is well-defined on G. Indeed, if n⋅(π/3)+m⋅α=n′⋅(π/3)+m′⋅α where n,n′,m,m′∈ℤ, then n=n′ by α/π∉ℚ. Clearly, χ:G→{1,−1} is a multiplicative function; that is, χ satisfies the functional equation χ(θ ₁+θ ₂)=χ(θ ₁)⋅χ(θ ₂) (θ ₁,θ ₂∈G). Also, we have χ(π)=−1, and thus χ(θ+π)=−χ(θ) and χ(θ+2π)=χ(θ) for every θ∈G. We also have χ(−θ)=χ(θ) for every θ∈G. Let $\varPhi:\mathcal{P}_{G} \to\mathbb{C}$ be defined by (14).

Let V ₁,…,V _N=V ₀ be the vertices of A listed counterclockwise. We may assume that the side X ₀ X ₁ lies on the x-axis. Moreover, we shall assume that if N=4, then X ₀ X ₁ and X ₂ X ₃ are parallel sides of A, and if N=5, then the only acute angle of A is at the vertex X ₁. We put $d_{i} =\overline {X_{i-1}X_{i}}$ for every i=1,…,N.

Lemma 7.2

If N=3, then Φ(A)=3d ₁. If 4≤N≤5, then Φ(A)=3d ₄. If N=6, then

$$ \varPhi(A)=d_1 -d_2 +d_3-d_4 +d_5 -d_6 .$$

(19)

Proof

Using (15) and χ(±π/3)=−1, χ(±2π/3)=1 we find that if N=3, then Φ(A)=d ₁+d ₂+d ₃=3d ₁.

If N=4, then A is a trapezoid, and thus d ₂=d ₄ and d ₁=d ₃+d ₄. Therefore, we get Φ(A)=d ₁+d ₂−d ₃+d ₄=3d ₄.

If N=5, then A is a pentagon, d ₁=d ₃+d ₄ and d ₂=d ₄+d ₅. Therefore, Φ(A)=d ₁+d ₂−d ₃+d ₄−d ₅=3d ₄.

It is clear that if N=6, then (19) holds. □

In Lemma 4.8 we proved that those triangles of the tiling that have their side of length c in the interior of A come in pairs, and each pair forms a quadrilateral which is either a parallelogram of sides a,b and of angles γ,α+β, or a kite of sides a,b and of angles 2β,γ,2α,γ. Let Q ₁,…,Q _u be a list of these quadrilaterals. We prove that Φ(Q _i)=0 for every i=1,…,u. This is clear if Q _i is a parallelogram. Suppose Q _i is a kite. Since χ(2α)=1, χ(2β)=χ((2π/3)−2β)=1 and χ(2π/3)=1, (15) gives Φ(Q _i)=χ(θ ₁)(a−b+b−a)=0. Therefore, if the quadrilaterals Q ₁,…,Q _u tile A, then Φ(A)=0. By Lemma 7.2, this implies N=6 and d ₁−d ₂+d ₃−d ₄+d ₅−d ₆=0.

It is easy to check, using the fact that each angle of A equals 2π/3, that d ₁+d ₂=d ₄+d ₅ and d ₂+d ₃=d ₅+d ₆. Thus d ₁−d ₂+d ₃−d ₄+d ₅−d ₆=0 implies d ₁=d ₄, d ₂=d ₅ and d ₃=d ₆. Therefore, A is centrally symmetric. Since A is rational by Lemma 4.7 and sinα/sinβ=a/b∈ℚ, it follows that (vi) of Theorem 2.1 holds.

Next we suppose that Q ₁,…,Q _u do not tile A. Then, by Lemma 4.8, there is a triangle of the tiling that has its side of length c on the boundary of A. Let T ₁,…,T _v be a list of all these triangles. Then we have

$$ \varPhi(A)=\sum_{i=1}^u\varPhi(Q_i )+ \sum_{j=1}^v\varPhi(T_j )= \sum_{j=1}^v\varPhi(T_j ) .$$

(20)

Since the graph Γ _c is empty by Lemma 4.6, it follows from Lemma 4.5 that there is a side XY of A which is covered by the sides of length c of the triangles T _j supported by the side XY. Since A is rational by Lemma 4.7, we find that each side of A has this property; that is, the boundary of A is covered by the sides of length c of the triangles T ₁,…,T _v. Thus d _i=k _i⋅c (i=1,…,N), where k ₁,…,k _N are positive integers.

It is easy to check, using χ(α)=1, χ(β)=−1 and χ(γ)=1, that if a triangle T _j is supported by X ₀ X ₁, then Φ(T _j)=c+a−b. Then we find by checking each case that if N=3, then

$$\sum_{j=1}^v \varPhi(T_j )=3k_1 (c+a-b)=3d_1 +3k_1 (a-b),$$

and if 4≤N≤5, then

$$\sum_{j=1}^v \varPhi(T_j )=3k_4 (c+a-b)= 3d_4 +3k_4 (a-b).$$

Since a≠b by α/π∉ℚ, it follows from (20) and Lemma 7.2 that these cases are impossible. Thus we have N=6, when

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where k=k ₁−k ₂+k ₃−k ₄+k ₅−k ₆. By (20) and Lemma 7.2 we have k=0, and thus d ₁−d ₂+d ₃−d ₄+d ₅−d ₆=k⋅c=0. As we proved above, this implies that A is centrally symmetric, and that (vi) of Theorem 2.1 holds.

8 Case IV: γ=π/3

If γ=π/3 then, similarly to the previous case, each angle of A equals π/3 or 2π/3. Thus A can be a regular triangle, a parallelogram, a trapezoid, a pentagon or a hexagon. If A is parallelogram or a trapezoid then two of its angles equals π/3 and the other two angles equal 2π/3; if A is a pentagon then one of its angles equals π/3 and the other angles equal 2π/3; if A is a hexagon, then all its angles equal 2π/3.

8.1 Subcase IVa

First we assume that a is not a linear combination of b and c with nonnegative rational coefficients. Then we consider the directed graph Γ _a. Unfortunately, it can happen that the in-degree is different from the out-degree at certain points (unlike in the previous case). Therefore, we define another directed graph as well. We shall denote by Γ _k the set of those directed segments $\overrightarrow{XY}$ for which the segment XY is the common side of two triangles having angles γ at X and β at Y. We shall consider the set Γ _a∪Γ _k of all directed edges belonging to either Γ _a or Γ _k.

We show that the in-degree of Γ _a∪Γ _k is zero or one at each vertex. Indeed, if the in-degree of Γ _a is positive at a vertex V, then V is an inner point of a side of a triangle, and then either V is normal or the equation at V is 3γ=π. The in-degree of Γ _k at the vertex V is zero in both cases. On the other hand, it is easy to see that the in-degree of Γ _k is at most one at every vertex.

Lemma 8.1

Suppose that a is not a linear combination of b and c with nonnegative rational coefficients. Then

(i)

the out-degree of Γ _a∪Γ _k equals the in-degree at each vertex;
(ii)

no edge of Γ _a∪Γ _k starts from the boundary of A; and
(iii)

every vertex lying on the boundary of A but different from the vertices of A is normal.

Proof

It is easy to check that the equation at an exceptional point must be one of 3γ=π, 6γ=2π, 3α+3β=2π, α+β+4γ=2π.

In order to prove (i) it is enough to show that if the in-degree of a vertex X is positive, then so is the out-degree of X.

First assume that X is the endpoint of an edge of Γ _a. If X is normal, then the out-degree at X is positive by Lemma 4.2. If X is exceptional then the equation at X must be 3γ=π, since X is an interior point of a side of a triangle. It is easy to see that in this case either an edge of Γ _a or an edge of Γ _k starts from X.

Next assume that X is the endpoint of an edge $\overrightarrow{ZX}$ of Γ _k. Then the equation at X is 2α+2β+2γ=2π or 3α+3β=2π. Let T ₁,…,T ₆ denote the triangles having X as a vertex listed counterclockwise, and such that the common side of T ₁ and T ₆ is the segment ZX. Then the angles of T ₁ and T ₆ at Z and X equal γ and β, respectively. Let h _i denote the halfline starting from X and supporting the triangles T _i and T _i+1 (i=0,…,5), where we put T ₀=T ₆.

Suppose that the equation at X is 3α+3β=2π. Then the angle between h _i and h _j is different from π for every i≠j. Indeed, otherwise pα+qβ=π would hold for some integers 0≤p,q≤3. Since α+β=2π/3, this implies either α=β=π/3 or {α,β}={π/6,π/2}. However, in each case we would have a/b∈ℚ, which contradicts the assumption that a is not a linear combination of b and c with nonnegative rational coefficients. Therefore, no two of the halflines h ₀,…,h ₅ can form a line.

Since three of the triangles T _i have angle β at X, at least two of them are of the same orientation in the sense that a rotation about the point X brings one of them onto the other. Let T _i and T _j be such triangles. If j=i+1, then an edge of Γ _a starts from X along the halfline h _i. If j>i+1, then an edge of Γ _a starts from X along one of the halflines h _i and h _j−1, depending on the location of the angle γ in T _i and T _j.

Next suppose that the equation at X is 2α+2β+2γ=2π. One can check, by considering the possible cases, that if the angle between h _i and h _j is different from π for every i≠j then an edge of Γ _a∪Γ _k must start from X. If there are i≠j such that the angle between h _i and h _j equals π, then pα+qβ+rγ=π holds for some integers 0≤p,q,r≤2. It is easy to see that the only possibility is p=q=r=1. In this case the triangles T ₁ and T ₃ are supported by a line which is the union of the halflines h ₀ and h ₃. It is easy to check that in this case either an edge of Γ _a starts from X, or the common side of T ₃ and T ₄ is an edge of Γ _k starting from X. This proves (i).

Since no edge arrives at any boundary point, we obtain (ii) from (i). Therefore, if X is a boundary point and not a vertex of A, then the equation at X cannot be 3γ=π. Since the right hand side of all other equations at exceptional points equals 2π, it follows that X must be normal, which proves (iii). □

The equations at the vertices of A must be of the form α+β=δ, γ=δ or 2γ=δ.

Suppose that the angle of A at the vertex V ₁ equals π/3. Then the equation at V ₁ is γ=δ, and there is a triangle T ₁ of the tiling such that V ₁ is a vertex of T ₁ and the angle of T ₁ at V ₁ is γ. We may assume that T ₁ has angle β at its vertex lying on the side V ₁ V ₂. By (ii) of Lemma 4.3, every triangle supported by V ₁ V ₂ is a translated copy of T, and thus the length of V ₁ V ₂ is an integer multiple of a.

There is a triangle T _k such that T _k is supported by V ₁ V ₂, V ₂ is a vertex of T _k, and T _k is a translated copy of T ₁. Then the angle of T _k at V ₂ is β, and thus the equation at V ₂ is α+β=δ ₂. Then there is a triangle $T'_{1}$ supported by V ₂ V ₃ such that V ₂ is a vertex of $T'_{1}$ and the angle of $T'_{1}$ at V ₂ equals α. Let V ₂=U ₀,…,U _ℓ=V ₃ be a division of V ₂ V ₃ such that each U _i−1 U _i is the side of a triangle $T'_{i}$ of the tiling. Then every triangle $T'_{i}$ has angle α at the vertex U _i−1 (see Fig. 3). In particular, $T'_{\ell}$ has angle α at U _ℓ−1, and then the angle of $T'_{\ell}$ at V ₃ is different from α. If it is β, then the equation at V ₃ is α+β=δ ₂. However, in this case an edge of Γ _a would start from V ₃ which is impossible. Thus the angle of $T'_{\ell}$ at V ₃ is γ. Then the equation at V ₃ is γ=δ or 2γ=δ. The latter is impossible, because in that case an edge of Γ _a∪Γ _k would start from V ₃. Thus the angle of A at V ₃ equals π/3. Since the angle of A at V ₁ is also π/3, it follows that A is a parallelogram such that the length of its side V ₁ V ₂ is an integer multiple of a. Then, by Lemma 4.1, (i) of Theorem 2.1 holds.

Next suppose that no angle of A equals π/3. Then each angle of A equals 2π/3, and the equations at the vertices of A are 2γ=δ or α+β=δ.

Suppose that there is a vertex of A where the equation is 2γ=δ. Let V ₁ be such a vertex. Since no edge of Γ _k starts from V ₁, one of the triangles having V ₁ as a vertex must have angle β at its vertex lying on the boundary of A. Let T ₁ be such a triangle; we may assume that T ₁ is supported by V ₁ V ₂. By (ii) of Lemma 4.3, each triangle supported by V ₁ V ₂ is a translated copy of T ₁. Then the equation at V ₂ is α+β=δ ₂. Then there is a triangle $T'_{1}$ supported by V ₂ V ₃ such that V ₂ is a vertex of $T'_{1}$ and the angle of $T'_{1}$ at V ₂ equals α. Following the argument above we can see that in this case the angle of A at V ₃ is π/3 which is impossible.

Finally, suppose that the equation at every vertex of A is α+β=δ. Then, since no edge of Γ _a starts from V ₁, one of the triangles having V ₁ as a vertex must have angle β at V ₁ and angle γ at its vertex lying on the boundary of A. Let T ₁ be such a triangle; we may assume that T ₁ is supported by V ₁ V ₂. By (ii) of Lemma 4.3, each triangle supported by V ₁ V ₂ is a translated copy of T ₁. Then the equation at V ₂ is 2γ=δ ₂ which is impossible. This completes the proof in the subcase when a is not a linear combination of b and c with nonnegative rational coefficients.

8.2 Subcase IVb

Suppose that b is not a linear combination of a and c with nonnegative rational coefficients. Since the roles of a and b are symmetric in the conditions as well as in the statements of the theorem, this subcase can be treated similarly to Subcase IVa.

Therefore, we may assume that a is a linear combination of b and c with nonnegative rational coefficients, and b is a linear combination of a and c with nonnegative rational coefficients. This implies that either a,b,c are pairwise commensurable, or a and b are commensurable and c is not a rational multiple of a.

If a,b,c are pairwise commensurable, then so are the sides of A. We know that A is an N-gon with 3≤N≤6. If 4≤N≤6, then A has N−2 vertices with angle 2π/3, and the other two angles of A are integer multiples of π/3. Thus, in these cases, (iii) of Theorem 2.1 holds. If N=3, then (iv) of Theorem 2.1 holds.

8.3 Subcase IVc

Thus we are left with the case when a and b are commensurable and c is not a rational multiple of a.

If A is a parallelogram, then (i) of Theorem 2.1 holds, since A is rational by Lemma 4.7, and sinα/sinβ=a/b∈ℚ. Therefore, we may assume that A is not a parallelogram.

Suppose that N≤4; that is, A is a triangle or a trapezoid. Then the angle of A equals π/3 at two consecutive vertices, say, X and Y, and the corresponding equations at X and Y must be γ=δ. Since Γ _c is empty, it follows from (i) of Lemma 4.4 that there is an exceptional vertex in the interior of XY. Let X=U ₀,…,U _k=Y be a division of XY such that each U _i−1 U _i is the side of a triangle T _i of the tiling. Let i be the smallest positive index such that U _i is exceptional. Since T ₁ has angle γ at U ₀, it follows that T _j has angle γ at U _j−1 for every j=1,…,i (see Fig. 4). In particular, T _i has angle γ at U _i−1, and thus the angle of T _i at U _i is different from γ. However, as U _i is exceptional, the equation at U _i must be 3γ=π, which is a contradiction. Therefore, we have N≥5, and thus A is a pentagon or a hexagon.

We prove that A must be a centrally symmetric hexagon, and thus (vi) of Theorem 2.1 holds.

First we assume that α/π is irrational. Then we consider the group G, the multiplicative function χ:G→{−1,1} and the additive function $\varPhi:\mathcal{P}_{G} \to\mathbb{R}$ as in the Subcase IIIc. Repeating the argument of IIIc we find that A must be a centrally symmetric hexagon.

Finally, we consider the case when α/π is rational. We prove that this happens only if α=π/6 or α=π/2. Indeed, we have

$$\frac{b}{a}=\frac{\sin\beta }{\sin\alpha } =\frac{\sin((2\pi /3)-\alpha )}{\sin \alpha } =\frac{\sqrt{3}}{2}\cdot \cot\alpha +\frac{1}{2},$$

and thus $\sqrt{3} \cdot \cot\alpha $ is rational. Thus tan² α is rational, and then so are cos² α=1/(1+tan² α) and cos2α. By [5, Corollary 3.12]), this implies cos2α=0,±1,±1/2. Since α<2π/3, we have 0<2α<4π/3, and thus 2α∈{π/3,π/2,2π/3,π} and α∈{π/6,π/4,π/3,π/2}. If α=π/4 then $\sqrt{3} \cdot \cot\alpha =\sqrt{3}$ is irrational, which is impossible. If α=π/3, then α=β=γ and a=b=c, which contradicts the condition that c/a is irrational. Therefore, we have either α=π/6 or is α=π/2. If α=π/6 then β=π/2. Since the roles of α and β are symmetric, we may assume that α=π/2 and β=π/6. Then we have a=2b.

Now we prove that A has to be a hexagon.

By Lemma 4.8, the triangles having their side of length c in the interior of A come in pairs. The triangles of each pair have a common side of length c, and thus they form a quadrilateral which is either a parallelogram of sides a,b and of angles γ=π/3,α+β=2π/3, or a kite of sides a,b and of angles 2β=π/3,γ=π/3,2α=π,γ=π/3. Note that the kites are, in fact, regular triangles of side 2b. For every kite Q we denote by V(Q) the vertex of Q with angle labeled with 2β, and by W(Q) the middle point of the side opposite to V(Q). Since the tiling is regular, it has the property that for every kite Q, we have V(Q)=W(Q′) and W(Q)=V(Q″) for some kites Q′ and Q″.

We divide each parallelogram into two rhombuses of side b and of angles π/3,2π/3. Thus A is tiled with regular triangles of side 2b and rhombuses of side b and of angles π/3,2π/3. In addition, each triangle Q of the tiling has a selected vertex V(Q) and a point W(Q) opposite to V(Q) with the property described above. We show that the existence of such a tiling implies that A is either a parallelogram or a hexagon.

Suppose this is not true, and let A be a counterexample with a minimal number of pieces of the tiling. Since A is not a hexagon, it has a vertex X where the angle of A is π/3. Then X cannot be a vertex of a triangle Q, because in that case either V(Q) or W(Q) would be in the boundary of A, which is impossible. Thus X is the vertex of a rhombus R ₁. Let Y be a vertex of A adjacent to X. Then R ₁ has a vertex U ₁ on the side XY. If U ₁≠Y, then U ₁ is also the vertex of a triangle Q ₁ or a rhombus R ₂. But U ₁ cannot be the vertex of a triangle Q ₁, because in that case either V(Q ₁) or W(Q ₁) would be a point of XY, which is impossible. Thus U ₁ is the vertex of a rhombus R ₂. Let U ₂ be a vertex of R ₂ lying on XY and different from U ₁. If U ₂≠Y then, by repeating the argument we find that U ₂ is the vertex of a rhombus R ₃ etc. In this way we obtain a division X=U ₀,U ₁,…,U _k=Y of the segment XY and a sequence of rhombuses R ₁,…,R _k such that U _i−1 U _i is a side of R _i for every i=1,…,k.

Then P=R ₁∪⋯∪R _k is a parallelogram. Since, by assumption, A is not a parallelogram, we have P≠A. Then A∖P is a convex polygon tiled with triangles and rhombuses satisfying the condition described above. Now, the number of pieces in the tiling of A∖P is smaller than that of the tiling of A, and thus A∖P is either a parallelogram or a hexagon. However, A∖P has an angle π/3, and thus it cannot be a hexagon. Therefore, A∖P is a parallelogram, and then so is A, which is a contradiction. This proves that A is a hexagon, and thus (ix) of Theorem 2.1 holds.

9 Case V: γ=π/2

Under this assumption each angle of A equals π/2, and thus A is a rectangle. Our aim is to prove that in this case (i) of Theorem 2.1 holds.

9.1 Subcase Va

First we assume that a is not a linear combination of b and c with nonnegative rational coefficients. Then we consider the directed graph Γ _a.

Lemma 9.1

Suppose that γ=π/2 and a is not a linear combination of b and c with nonnegative rational coefficients. Then

(i)

the out-degree of Γ _a equals the in-degree at each vertex;
(ii)

no edge of Γ _a starts from the boundary of A; and
(iii)

the length of one of the sides of A is an integer multiple of a.

Proof

In order to prove (i) it is enough to show that if the in-degree of a vertex Y is positive, then so is the out-degree of Y. If Y is normal, then the out-degree at Y is positive by Lemma 4.2. If Y is exceptional then the equation at Y must be 2α+2β=π or 2γ=π, since Y is an interior point of a side of a triangle. An inspection of all possible cases shows that at least one edge of Γ _a starts from Y.

Since no edge arrives at any boundary point, (ii) follows from (i).

Let X be a vertex of A. In order to prove (iii), first we show that there is a triangle T such that X is a vertex of T and the side of T of length a is on the boundary of A. This is clear if the equation at X is γ=δ. Therefore, we may assume that the equation at X is α+β=δ. Let T be the triangle having X as a vertex and having angle β at X. Since, by (ii), no edge of Γ _a starts from X, it follows that the vertex of T with angle α is not on the boundary of A. Then, the side of T of length a is on the boundary of A.

We proved that there is a side XY of A and there is a division X=U ₀,…,U _k=Y of XY such that each U _i−1 U _i is the side of a triangle T _i of the tiling, and $\overline {U_{0}U_{1}} =a$. We prove that $\overline {U_{i-1}U_{i}} =a$ for every i. Suppose this is not true. Then there is an 1≤i<k such that $\overline {U_{i-1}U_{i}} =a$ and $\overline {U_{i}U_{i+1}}\ne a$. Then an edge of Γ _a must start from U _i; this can be shown by considering the same cases as in the proof of (i). This, however, contradicts (ii). Thus $\overline {U_{i-1}U_{i}} =a$ for every i, which proves (iii). □

Now Lemma 4.1 and (iii) of Lemma 9.1 imply that (i) of Theorem 2.1 holds, which completes the proof in the subcase when a is not a linear combination of b and c with nonnegative rational coefficients. The same argument applies if b is not a linear combination of a and c with nonnegative rational coefficients.

Therefore, we may assume that a is a linear combination of b and c with nonnegative rational coefficients, and b is a linear combination of a and c with nonnegative rational coefficients. This implies that either a,b,c are pairwise commensurable, or a and b are commensurable and c is not a rational multiple of a.

If a,b,c are pairwise commensurable, then so are the sides of A, and we find that (i) of Theorem 2.1 holds again.

Finally, if a and b are commensurable and c is not a rational multiple of a, then the sides of A are commensurable by Lemma 4.7, and we have the same conclusion. This completes the proof of Theorem 2.1.

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Authors and Affiliations

M. Laczkovich
- 1
- 2
Email author

1.Department of AnalysisEötvös Loránd UniversityBudapestHungary
2.Department of MathematicsUniversity College LondonLondonUK