A THEORY OF HARMONIC VARIATIONS

TRISTRAM DE PIRO

Abstract.

Let n be an odd number, and C a circle of radius 1, centred about the origin (0, 0), of a real coordinate system (x, y). Suppose that a regular n-sided polygon is inscribed inside the circle, with vertices \{p_{0}, …, p_{j}, …, p_{n−1}\}, with coordinates e^{\frac{2πij}{n}} and \{l_{0}, …, l_{j}, …, l_{n−1}\} are the lines formed by the edges of the polygon, so that l_{j} passes through the pair of vertices {p_{j}, p_{j+1}}, mod(n). By construction, the n intersections (l_{j} ∩ l_{j+1}), for 0 ≤ j ≤ n − 1, mod(n), lie on the unit circle. We claim, more generally, that;

Lemma 0.1. If 1 ≤ k ≤ \frac{n−1}{2}, the n intersections (l_{j} ∩ l_{j+k}), mod(n),

lie on a circle, centred about (0, 0), of radius \frac{sin( \frac{π}{2} (1− \frac{2}{n}))}{sin( \frac{π}{2} (1− \frac{2k}{n}))}, with equal angles subtended by consecutive pairs to the origin (0, 0).

Proof. For convenience of notation, let O denote the origin (0, 0), and let C denote the intersection (l_{j} ∩ l_{j+k}). Let {α, β, γ} denote the angles {Op_{j}C, p_{j}OC, p_{j}CO} of the triangle with vertices {p_{j}, O, C}, let δ denote the angle p_{j}Op_{j+1} of the triangle with vertices {p_{j}, O, p_{j+1}}, let ϵ denote the angle between the lines l_{j} and l_{j+1} and r the length of the edge OC. We have that;

δ = ϵ = \frac{2π}{n}

α = \frac{π−ϵ}{2} = \frac{π}{2}(1 − \frac{2}{n})

β = \frac{δ(k+1)}{2} = \frac{π(k+1)}{n}

γ = π − (α + β) = \frac{π}{2}(1 − \frac{2k}{n})

By the sine rule, applied to the triangle p_{j}OC, we have that;

r = \frac{sin(α)}{sin(γ)} = \frac{sin( \frac{π}{2} (1− \frac{2}{n}))}{sin( \frac{π}{2} (1− \frac{2k}{n}))}

as required. The last claim follows easily from calculating the angle C_{1}OC_{2} = \frac{2π}{n}, for two consecutive intersections in the set l_{j} ∩ l_{j+k}.

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Remarks 0.2. It follows that all of the C_{2}^{n} intersections between the n lines {l_{0}, …, l_{j}, …, l_{n−1}} lie on concentric circles about the origin O. The pattern of lines and intersections forms an attractive radiating pattern, harmoniously arranged in the plane.

If n is an even number, we can perform the same construction, but obtain a slightly modified version of the previous lemma;

Lemma 0.3. Let hypotheses and notation be as above. If 1 ≤ k ≤ \frac{n−2}{2}, the n intersections (l_{j} ∩ l_{j+k}) are arranged as in Lemma 0.1. If k = \frac{n}{2}, there exist \frac{n}{2} intersections in the set l_{j} ∩ l_{j+k}, situated on the circle at ∞, in the real projective plane RP^{2}.

Proof. It is sufficient to observe that, when k = \frac{n}{2}, the lines l_{j} and l_{j+k}, mod(n), are parallel, in the plane with affine coordinates (x, y). Embedding the real affine plane in the projective plane RP^{2}, we obtain \frac{n}{2} intersections between the \frac{n}{2} pairs of parallel lines in the set {l_{0}, …, l_{j}, …, l_{n−1}}, as the pair gradients are distinct.

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Definition 0.4. We say that n lines in the real projective plane RP^{2} are in general position, if no three of the lines intersect in a point. We say that the lines are in bounded position, if all of the intersections lie in the affine plane R^{2}. To n lines {l_{0}, …, l_{n−1}} in bounded general position, we can associate a graph G, whose vertices consist of the C_{2}^{n} intersections of the lines in the affine plane. We say that two vertices {v_{1}, v_{2}} ⊂ G are connected by an edge if;

(i). There exists a line in the set {l_{0}, …, l_{n−1}} containing v_{1} and v_{2}.

(ii). There does not exist a third vertex v_{3}, lying between v_{1} and v_{2}, on the same line.

We define an edge to be the closed line segment, connecting two such vertices. We define an alcove of the graph G by the following properties;

(i). A compact convex connected subset V of R^{2}.

(ii). The boundary δV is a union of edges, belonging to distinct lines.

(iii). V does not contain a proper subset W, satisfying properties (i) and (ii).

We now show that;

Lemma 0.5. If {v_{0}, …, v_{j}, …, v_{n−1}} are vertices, connected by edges {e_{0}, …, e_{j}, …, e_{n−1}}, lying on distinct lines, forming a convex n-polygon V, then V is an alcove.

Proof. Suppose not, then clearly condition (iii) fails. We can, therefore, find a proper subset W ⊂ V, satisfying conditions (i) and (ii). Let e be one of the edges of W, belonging to a line l. Suppose the edges of V belong to lines {l_{0}, …, l_{j}, …, l_{n−1}} respectively. If, l does not coincide with one of these lines, then, by the definition of general position, it must intersect one of them in a vertex, distinct from {v_{1}, …, v_{j}, …, v_{n−1}}, on δV. This contradicts the fact that {e_{0}, …, e_{j}, …, e_{n−1}} are edges. It follows, that l must coincide with one of the lines {l_{0}, …, l_{j}, …, l_{n−1}}, say l_{0}. If e does not coincide with the edge e_{0}, then l_{0} must contain a point in the interior (V \ δV) of V. It is straightforward to show that this contradicts the assumption that V is convex, (^{1}). It follows that the boundary δW ⊂ δV. As δW is connected, δW = δV, hence, W must coincide with V, showing the result. □

We make the following definition;

Definition 0.6. If {v_{0}, …, v_{j}, …, v_{n−1}} are vertices, lying on distinct lines, forming a convex n-polygon V, we define the vertex number of V, to be the number of vertices of the lines in bounded general position, either interior to V or interior to the line segments forming the boundary δV of V.

Remarks 0.7. By the previous lemma, a convex n-polygon, with vertex number 0, is an alcove.

We then have that; Lemma 0.8. If an edge e forms one side of a convex n-polygon V, which is not an alcove, having vertex number m > 0, then e forms one side of a convex, at most n + 1-sided polygon W, having vertex number 0 ≤ r < m.

1Let x be the interior point, and let {v_{0}, v_{1}, v_{2}} be the vertices, connecting the edges e_{0} and e_{1}. By convexity, the triangles with vertices {x, v_{1}, v_{2}} and {v_{0}, v_{1}, v_{2}} both lie inside V. This implies that the interior of the edge e_{1} is interior to V, contradicting the fact that e_{1} is contained in δV.

Proof. Let {l_{0}, …, l_{j}, …, l_{n−1}} enumerate the line segments, forming the boundary of V, with e corresponding to l_{0}. As V is not an alcove, there exists a vertex v, interior to one of these lines, on the boundary δV of V. As e is an edge, it cannot be interior to l_{0}, say v is interior to l_{1}. As V is convex, there exists a new line l_{n}, passing through v. As the lines are in general position, and e is an edge, the line l_{n} must intersect the interior of one of the lines {l_{2}, …, l_{j}, …, l_{n−1}}, say l_{j}. It is easily checked that the polygon W, formed by the sides {e, l_{1}, l_{n}, l_{j}, …, l_{n−1}} is convex, with (n − j + 3) sides, having vertex number r < m. □

As a straightforward consequence, we have that;

Lemma 0.9. Every edge e lies on the boundary of at least one alcove.

Proof. Suppose that the edge e has vertices v_{0} and v_{1}, belonging to a line l_{0}, let l_{1} and l_{2} be further lines containing these vertices, respectively, intersecting in a vertex v_{2}. If the triangle with vertices {v_{0}, v_{1}, v_{2}} is an alcove, the result is shown. Otherwise, it satisfies the hypotheses of the previous lemma; one may then apply the result inductively, together with the fact that the number of vertices are finite, and the previous remark, to obtain the same result.

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The following results use a different argument;

Lemma 0.10. If {V_{1}, V_{2}} are two distinct alcoves, then (V_{1} ∩ V_{2}) ⊂ (δV_{1} ∪ δV_{2}) and, consists of an edge or a vertex.

Proof. We clearly have that (V_{1}∩V_{2}) satisfies condition (i) in the definition of an alcove, and the boundary δ(V_{1}∩V_{2}) is contained in (δV_{1}∪δV_{2}). Suppose that (V_{1} ∩ V_{2}) contains an open subset of R^{2}, (∗) then the boundary δ(V_{1} ∩ V_{2}) is connected, therefore, must consist either of a vertex, or a union of line segments. That the line segments form edges, follows from the fact that the lines forming the boundary δV_{1} of V_{1}, intersect the lines forming the boundary δV_{2} of V_{2}, in vertices. It follows that (V_{1} ∩ V_{2}) also satisfies condition (ii) in the definition of an alcove. As the intersection is a proper subset of both V_{1} and V_{2}, this contradicts the assumption that V_{1} and V_{2} are both alcoves. It follows that (∗) fails, that is (V_{1} ∩ V_{2}) ⊂ (δV_{1} ∪ δV_{2}). As (V_{1} ∩ V_{2}) is connected, being convex, the intersection consists of a vertex or an edge, as required. □

Lemma 0.11. Every edge e lies on the boundary of at most two alcoves.

Proof. Suppose that e lies on the boundary of three distinct alcoves {V_{1}, V_{2}, V_{3}}. It is easily checked, that, if x is an interior point of the edge e, then x is an interior point of the union of alcoves (V_{1} ∪ V_{2}). It follows that the intersection of V_{3} with either V_{1} or V_{2}, must contain an open subset of R^{2}. This contradicts the previous result. □

Lemma 0.12. If n ≥ 3, there exist \frac{(n−1)(n−2)}{2} alcoves, associated to the graph of Definition 0.4.

Proof. When n = 3, it is easily checked that there is 1 alcove, formed by the vertices {v_{1}, v_{2}, v_{3}} of a triangle, obtained from the intersection of three lines {l_{1}, l_{2}, l_{3}} in bounded general position. We assume, inductively, that the result is true for n lines in bounded general position. Let l_{n+1} be a new line, added to n lines {l_{1}, …, l_{j}, …, l_{n}} in bounded general position. This introduces n new vertices {v_{1}, …, v_{n}} and (n − 1) new edges, corresponding to line segments e_{j} between the vertices v_{j} and v_{j+1}. We claim that an edge e_{j} is on the boundary of two alcoves in the graph G_{n+1} of (n + 1) lines, (∗), if and only if it passes through the interior of an alcove in the graph G_{n} of n lines, (∗∗). For assume that (∗) holds, and e_{j} lies on the boundary of two alcoves V_{1} and V_{2}. By Lemma 0.10, e_{j} = (V_{1} ∩ V_{2}). Let {e_{j}, f_{1}, …, f_{r}} and {e_{j}, g_{1}, …, g_{s}} enumerate the consecutive edges of the alcoves V_{1} and V_{2} respectively. By the definition of lines in general position, the edges {f_{r}, g_{1}} and {f_{1}, g_{s}} belong to the same lines l_{1} and l_{2} respectively, in particular, either the polygon defining the boundary δV_{1} or the polygon defining the boundary δV_{2} is inscribed within the triangles, having either edges {e_{j}, f_{1}, f_{r}} or {e_{j}, g_{1}, g_{s}}. In either case, it follows that the union of alcoves (V_{1} ∪ V_{2}) is convex. We now relabel the boundary δ(V_{1} ∪ V_{2}), after removing the edge e_{j}, consecutively, as {h_{1}, f_{2}, …, f_{r−1}, h_{2}, g_{2}, …, g_{s−1}}, where h_{1} are h_{2} are the new edges in the graph G_{n}, obtained by joining the edges {f_{r}, g_{1}} and {f_{1}, g_{s}} in the graph G_{n+1}. As all the faces are edges, belonging to distinct lines, from the above, it follows that (V_{1} ∪ V_{2}) is an alcove, by Lemma 0.5, hence (∗∗) follows. Conversely, assume that (∗∗) holds, and e_{j} passes through the interior of an alcove V in G_{n}. By the definition of an alcove, the boundary δV consists of a union of edges {k_{1}, …, k_{t}}, arranged in a convex polygon, belonging to distinct lines. By the definition of lines in general position, the edge e_{j} passes through the interior of two of these edges, say k_{1} and k_{l}, where 1 < l ≤ t, and forms two new pairs of edges {k_{11}, k_{12}} and {k_{l1}, k_{l2}} in the graph G_{n+1}. We let V_{1} be the region, bounded consecutively by the edges {k_{11}, …, k_{l−1}, k_{l1}, e_{j}}, and V_{2} the region, bounded consecutively by the edges {k_{l2}, k_{l+1}, …, k_{t}, k_{12}, e_{j}}. One of the regions is bounded, by the triangle with edges {e_{j}, l_{1}, l_{l}}, where l_{1} and l_{l} are the lines containing the edges k_{1} and k_{l} respectively. It follows, that both regions are convex, and, therefore, alcoves, by Lemma 0.5. Hence, (∗) is shown. Now, if l_{n+1} is a new line, each of the (n − 1) new edges, either passes through the interior of an alcove in G_{n}, in which case, by the above (∗)(∗∗), and Lemma 0.11, one extra alcove is introduced into the graph G_{n+1}, or, does not pass, through an interior, in which case, by the above (∗)(∗∗), and Lemma 0.9, an extra alcove is also introduced into the graph G_{n+1}. In total, (n − 1) new alcoves are introduced, which implies that the total number of alcoves in G_{n+1} is;

\frac{(n−1)(n−2)}{2} + (n − 1) = \frac{n(n−1)}{2} = \frac{([n+1]−1)([n+1]−2)}{2}

This implies the result, by induction.

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Remarks 0.13. We return to the notation of Lemma 0.1, and the following remark. For n odd, we can apply the previous lemma, to obtain that there exist C_{2}^{n−1} alcoves associated to a regular bounded arrangement of lines. One may also extend the above definition of an alcove to regions in the real projective plane RP^{2}, by, for example, assuming that all the intersections are in finite position. This may always be achieved by an appropriate choice of the line at ∞, so as not to include any of the vertices.

With the convention that any two such regions intersecting in a vertex, on the line at ∞, are counted as a single alcove, the reader is invited to check that there are again C_{2}^{n−1} alcoves, associated to a set of lines in general position. The reason for this

We can give a convenient description of the alcoves associated to a regular bounded line arrangement;

Lemma 0.14. In the situation of Lemma 0.1, and Lemma 0.3, the alcoves are defined by;

(i). For n ≥ 3, the central alcove, with boundary defined by the npolygon, inscribed in the unit circle.

(ii). For n ≥ 5, n peripheral alcoves of the first kind, inscribed between the unit and first concentric circle, with boundaries defined by the triangles, formed by the lines {l_{i}, l_{i+1}, l_{i+2}}, mod (n).

(iii). For n ≥ 7, n peripheral alcoves of the second kind, inscribed between the (j − 1, j, j + 1) concentric circles, with boundaries defined by the quadrilaterals, formed by the lines

\{l_{i}, l_{i+1}, l_{i+2j}, l_{i+2j+1}\}, mod (n), 1 ≤ j ≤ (\frac{n−5}{2})

Proof. The proof is left to the reader, one should observe that the total number of alcoves is correct, as;

1 + n + n.\frac{(n−5)}{2} = 1 + n.\frac{(n−3)}{2} = \frac{(n−1)(n−2)}{2}

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Cheltenham

  E-mail address: depiro300@hotmail.co.uk, depiro@rocketmail.com