The Harmonic Analysis of Polygons and Napoleon s Theorem

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1 Journal for Geometry and Graphics Volume 5 (00), No.,. The Harmonic nalysis of Polygons and Napoleon s Theorem Pavel Pech Pedagogical Faculty, University of South ohemia Jeronýmova 0, 7 5 České udějovice, zech Republic pech@pf.jcu.cz bstract. Plane closed polygons are harmonically analysed, i.e., they are expressed in the form of the sum of fundamental k-regular polygons. From this point of view Napoleon s theorem and its generalization, the so-called theorem of Petr, are studied. y means of Petr s theorem the fundamental polygons of an arbitrary polygon have been found geometrically. Key Words: finite Fourier series, polygon transformation MS 000: 5M0. Harmonic analysis of polygons plane n-gon Π is an ordered n-tuple of points 0,,..., n, which can be represented by complex numbers in the complex plane. The points 0,,..., n are called vertices of the n-gon Π. We shall write Π = ( 0,,..., n ). enote the n-roots of unity by ω j ν = eiνj π n = cos νj π n + i sin νj π n, where ν, j = 0,,..., n. Then the system of linear equations n ν = ϑ k ων, k ν = 0,,..., n () k=0 with unknown complex numbers ϑ k has a unique solution. The determinant of the system () is a determinant of the Fourier matrix Ω = (ωj k ) n k,j=0, which is not vanishing (Vandermonde). Multiplying each equation from () by ων j and adding these equations for ν = 0,,..., n we get n n n n n ν ων j = ϑ k ων k j = ϑ k ωk j ν = nϑ j. ν=0 k=0 ν=0 ISSN 4-857/$.50 c 00 Heldermann Verlag k=0 ν=0

2 4 P. Pech: The Harmonic nalysis of Polygons and Napoleon s Theorem The Fourier coefficients ϑ 0, ϑ,..., ϑ n of the polygon Π are then ϑ j = n n ν ων j. ν=0 We can see that ϑ 0 = n n ν=0 ν, i.e., the coefficient ϑ 0 coincides with the centroid of the n-gon Π. If we place the origin of the coordinate system into the centroid of the n-gon Π, we get ϑ 0 = 0. The system of linear equations () may be written in the form Π = ϑ 0 Π 0 + ϑ Π ϑ n Π n or Π = Φ 0 + Φ Φ n () with Φ k := ϑ k Π k. The right-hand side of () is expressed as a linear combination (with complex Fourier coefficients) of basic k-regular polygons Π j, where Π j = (ωj 0, ωj,..., ω n j ). From () it is seen that every closed plane n-gon Π has been uniquely represented as a sum of fundamental regular n-gons Φ 0, Φ,..., Φ n, i.e., it has been analysed harmonically, see Schoenberg [7]. The expansion () may also be expressed in the following form: Π = ϑ 0 Π 0 + (ϑ Π + ϑ n Π n ) + (ϑ Π + ϑ n Π n ) () It is easily seen that the sum ϑ k Π k + ϑ n k Π n k is an affine regular n-gon, i.e., the affine image of a k-regular n-gon. From () we get the fact, that every n-gon Π can be uniquely expressed as a sum of fundamental affine regular n-gons ϑ k Π k + ϑ n k Π n k. It is a question how to find the fundamental n-gons of an arbitrary plane n-gon geometrically? For the sake of it, we shall concern with Napoleon s theorem and its generalization, the theorem of Petr, see Naas and Schmid [4]. These theorems are based on polygon to polygon transformation.. Polygon transformation Let Π = ( 0,,..., n ) be an arbitrary plane n-gon. Let us suppose that P is a such polygon to polygon transformation that assigns to the n-gon Π an n-gon Π = ( 0,,..., n ), whose vertices are a linear combination of two consecutive vertices of the n-gon Π, i.e., j = a j + b j+, j = 0,,..., n, (4) where coefficients a, b are complex. We shall concern with a special type of (4) whose coefficients a, b fulfil a + b =. This condition assures, that all the triangles with vertices k erected on sides k k+ of the original triangle, are similar. If isocseles triangles with the angle j π are constructed on sides of an arbitrary n-gon Π, n we get the relation ( k k)ω j = k+ k. (5) From (5) we obtain that for a, b in (4) holds (Fig. ). a = ω j and b = ω j ω j

3 P. Pech: The Harmonic nalysis of Polygons and Napoleon s Theorem 5 π/n k k k+ Figure : Petr s construction If we denote a transformation, which maps a j-regular polygon Π j into the origin, by P j instead of P, we may write the image of Π in the form of the expansion where P j (Π) = ϑ 0 P j (Π 0 ) + ϑ P j (Π ) ϑ n P j (Π n ), (6) P j (Π ν ) = ω ν ω j ω j Π ν, j =,,..., n. (7) From (6) and (7) we see that P j (Π 0 ) = Π 0 and P j (Π j ) = 0. Let P k be another polygon transformation with k j. The Fourier expansion of the image of P j (Π) in the transformation P k is n ω ν ω j ω ν ω k P k P j (Π) = ϑ ν Π ν. (8) ω j ω k ν=0 From (8) it is seen that two of the regular polygons Π j, Π k of the Fourier expansion of the polygon P k P j (Π) vanished. ontinuing this process, after n steps of using the polygon transformation P ν for all values ν =,,..., n, we arrive at the point ϑ 0 Π 0 the common centroid of all polygons, which occur in this process. We have proved the theorem that was established by the zech mathematician K. Petr in 905 [6]. Petr s Theorem: On the sides of any closed plane n-gon Π construct isosceles triangles π with vertex angle j, where j n {,,..., n }. The resulting vertices form a new polygon π Π j. On the sides of this polygon construct isosceles triangles with vertex angle j, where n j {,,..., n }\{j }. We get a polygon Π j,j. ontinue this construction for all values,,..., n. Then the final polygon Π j,j,...,j n is a point the common centroid of all polygons Π, Π j, Π j,j,..., Π j,j,...,j n and the polygon Π j,j,...,j n is j n -regular. Petr s theorem has been rediscovered several times. This theorem has often been attributed to J. ouglas [] and.h. Neumann [5], but it seems the priority belongs to K. Petr. We shall call a polygon to polygon transformation, described in Petr s theorem, the Petr s construction P. Now we will investigate polygons, which we obtain from the original n-gon Π using successively n polygon transformations P ν for different values ν {,,..., n }. enote

4 6 P. Pech: The Harmonic nalysis of Polygons and Napoleon s Theorem such an n-gon by P j (Π), where the upper index j denotes the value, which has been omitted in this process. ccording to (8) we have and after a short calculation we get P j (Π) = ϑ 0 Π 0 + ϑ j n k=,k j ω j ω k ω k Π j, (9) P j (Π) = ϑ 0 Π 0 ϑ j ω j Π j. (0) From (8) and (9) we see that the n-gon P j (Π) doesn t depend on the order of polygon transformations P ν and therefore to a given n-gon Π there exist the only n-gon P j (Π) for every j =,,..., n. In this way we can assign to an arbitrary plane n-gon Π n firmly determined n-gons P j (Π). If we place the origin of the coordinate system into the centroid of Π, then ϑ 0 = 0 and instead of (0) we may write P j (Π) = ϑ j ω j Π j. The polygon P j (Π) is a j-regular n-gon, which differs from the fundamental n-gon ϑ j Π j in the Fourier expansion () of the n-gon Π only by the multiply ω j. This means that P j (Π) is a centrally symmetric image of ω j ϑ j Π j, which has the same vertices as ϑ j Π j. With respect to this fact, we can state: Theorem: Let Π be an arbitrary plane closed n-gon, whose centroid is placed at the origin of the artesian coordinate system. y the construction described above, we can assign to an arbitrary plane n-gon Π n firmly determined regular n-gons P j (Π) for every j =,,..., n. centrally symmetric image of P j (Π) shifted by ω j gives the fundamental n-gon Φ j and Π can be expressed as the sum of n-gons Φ j, j =,,..., n... The case n = s an application of the above theory we will mention Napoleon s theorem: If equilateral triangles are erected externally (or internally) on the sides of an arbitrary triangle, their centers form an equilateral triangle. This well known theorem has been attributed to Napoleon onaparte, although there is some doubt whether its proof is due to him. In the recent survey by Martini [], more than hundred references to this theorem and its generalizations are collected. ccording to this theorem we can assign to an arbitrary triangle two regular triangles and, which are called outer and inner Napoleon triangles (Fig. ). Our theory says that images of these Napoleon triangles in a point reflection with center T are fundamental triangles Φ, Φ of the original triangle. In Fig. it is shown, that the original triangle Π = (,, ) is a sum of its two fundamental triangles Φ = (,, ) and Φ = (,, ), i.e., Π = Φ + Φ. For a better view a decomposition of a triangle is shown in Fig. 4. generalization of Napoleon s theorem is the following Theorem: If similar triangles are constructed on the sides of an arbitrary triangle, their vertices form a new triangle whose fundamental triangles arise from fundamental triangles of in a similar way (Fig. 5). For isosceles triangles with the angle of 0, we get the subcase of Napoleon s theorem. see J. Fischer: Napoleon und die Naturwissenschaften, Franz Steiner Verlag Wiesbaden, Stuttgart 988

5 P. Pech: The Harmonic nalysis of Polygons and Napoleon s Theorem 7 Figure : Outer Napoleon triangle Inner Napoleon triangle Figure : The triangle is a sum of two fundamental triangles and... The case n = 4 Similarly as in the case of triangles we can assign to an arbitrary quadrangle three fundamental regular quadrangles (squares), one of which is a segment (Fig. 6). decomposition of a quadrangle into three regular quadrangles is shown in Fig. 7. In Fig. 8 isosceles right angled triangles are erected on the sides of a quadrangle. We obtain a new quadrangle which is the sum of (square) and (segment) whereas the third fundamental quadrangle vanished. From this decomposition it is easy to = + Figure 4: decomposition of a triangle into two fundamental triangles

6 8 P. Pech: The Harmonic nalysis of Polygons and Napoleon s Theorem = + Figure 5: The construction of similar triangles on the sides of an arbitrary triangle is preserved = = Figure 6: The quadrangle as the sum of the three squares,,. see that diagonals of the quadrangle are perpendicular and have equal length. If isosceles triangles with vertex angle 80 are erected on the sides of a quadrangle, i.e., is formed by centers of sides of, then a quadrangle is a sum of two oppositely oriented squares and, whereas the third vanished. is clearly a parallelogram (Fig. 9). The outward (or inward) erection of right angled isosceles triangles on the sides of a parallelogram gives the fundamental square as it is shown in Fig. 0, whereas the other square from the decomposition described in Fig. 0 vanishes. This is a special case of a more general theorem, often called arlotti theorem (see arlotti []), but it has already been mentioned by J. ouglas []: = = + + = Figure 7: ecomposition of a quadrangle into three fundamental squares

7 P. Pech: The Harmonic nalysis of Polygons and Napoleon s Theorem 9 = = + = Figure 8: ecomposition of a quadrangle with equal and perpendicular diagonals = + Figure 9: parallelogram as a sum of two squares Erecting regular n-gons outwardly (or inwardly) on the sides of any affinely regular n-gon, their centers form a regular n-gon. = + Figure 0: enters of squares erected on the sides of a parallelogram form a square From the decomposition of a quadrangle into four fundamental quadrangles as displayed in Fig. 7 another expression of a quadrangle follows. Summing up two squares and we obtain a parallelogram. The quadrangle is the sum of a parallelogram and a four times calculated segment with endpoints = and = (Fig. ).

8 0 P. Pech: The Harmonic nalysis of Polygons and Napoleon s Theorem = = + = Figure : ecomposition of a quadrangle into a parallelogram and a segment.. The case n = 5 plane pentagon E could be expressed as a sum of its fundamental pentagons Φ, Φ, Φ, Φ 4 which consist of two convex regular pentagons with the opposite order of vertices and two nonconvex star regular pentagons (Fig. ). pentagon E may also be expessed as a sum of two affine regular pentagons Φ + Φ 4 and Φ + Φ (see Schoenberg [7] and Fig. ). E = Figure : ecomposition of a pentagon into four fundamental pentagons E E = + E Figure : ecomposition of a pentagon into two affine regular pentagons. Spatial case It is possible to transfer our constructions into -space:

9 .. The case n = 4 P. Pech: The Harmonic nalysis of Polygons and Napoleon s Theorem Let be an arbitrary skew and closed quadrangle. It is easy to show that it can be expressed as sum of three squares, two of which lie in the plane of a parallelogram which is formed by centers of sides of, the other fundamental square coincides with a segment S S, whose endpoints are centers of diagonals which join opposite vertices of. In Fig. 4 it is shown that a quadrangle is the sum of a segment S S and a parallelogram, whose centers of sides are vertices of. Two of the fundamental squares lie in the plane of and are formed by the centers of squares erected on the sides of outwardly and inwardly. S S Figure 4: ecomposition of a skew quadrangle into a segment and a parallelogram.. The case n = 5 Let E be an arbitrary skew and closed pentagon. We shall construct pentagons E and E in the following way: enote by M M M M 4 M 5 the centers of the sides of E. The vertices and lie on the line M (outwardly and inwardly M ) with M = 5 (M ) and M = 5 (M ). Then E and E are plane convex and star affine regular pentagons (see ouglas [] and Fig. 5). In order to arrive at fundamental pentagons of E, erect in the plane of E on its sides isosceles triangles with vertex angle π (outwardly and inwardly). 5 We obtain two regular pentagons. y a homothety with the center at the common centroid of all the pentagons and the ratio 5 we get two of the fundamental pentagons of E. We obtain from E the other two (lying in a different plane) in the same way. cknowledgements This research has been performed within the Grant Project MSM

10 P. Pech: The Harmonic nalysis of Polygons and Napoleon s Theorem E M Figure 5: onstruction of two plane affine regular pentagons to any pentagon E References []. arlotti: Una proprieta degli n-agoni che si ottengono transformando in una affinita un n-agono regolare. oll. Un. Mat. Ital. 0, (955). [] J. ouglas: Geometry of polygons in the complex plane. J. of. Math. and Phys. 9, 9 0 (940). [] H. Martini: On the theorem of Napoleon and related topics. Math. Semesterber. 4, (996). [4] J. Naas, H.L. Schmid: Mathematisches Wörterbuch. erlin-leipzig 967. [5].H. Neumann: Some remarks on polygons. J. London Math. Soc. 6, 0 45 (94). [6] K. Petr: O jedné větě pro mnohoúhelníky rovinné. Čas. pro pěst. mat. a fyz. 4, 66 7 (905). [7] I.J. Schoenberg: The finite Fourier series and elementary geometry. mer. Math. Monthly 57, (950). Received ugust, 000; final form ecember 0, 000

AN EXTENSION OF A THEOREM OF BARLOTTI TO MULTIPLE DIMENSIONS

AN EXTENSION OF A THEOREM OF BARLOTTI TO MULTIPLE DIMENSIONS B. D. S. DON MCCONNELL In [1], A. Barlotti shows that any affinely regular polygon in the plane can be expressed as the vertex sum of two regular