User:Tomruen/Polygon

From Are Your Polyhedra the Same as My Polyhedra? Branko Grunbaum

Polygons[edit]

Since we consider polyhedra as families of points, segments and polygons (subject to appropriate conditions), it is convenient to discuss polygons first. Like “polyhedron”, the word “polygon” has been (and still is) interpreted in various ways. A polygon (specifically, an n-gon for some n≥3) is a cyclically ordered sequence of arbitrarily chosen points V₁, V₂, ..., V_n, (the vertices of the polygon), together with the segments E_i determined by pairs of vertices V_i,V_i+1 adjacent in the cyclic order (the edges of the polygon). Each vertex V_i is said to be incident with edges E_i−1 and E_i, and these edges only. Here and in the sequel subscripts should be understood mod n. Polygons were first considered in close to this generality by Meister [41] nearly 250 years ago. (The assertion by Gunther [30, p. 25] and Steinitz [52, p. 4] that already Girard [20] had this perception of “polygon” seems unjustified.) As explicitly stressed by Meister, this definition implies that distinct vertices of a polygon may be represented by the same point, without losing their individuality, and without becoming incident with additional edges even if the point representing a vertex is situated on another edge. Hence the definition admits various unexpected possibilities: edges of length 0; collinear edges –adjacent or not; edges overlapping or coinciding in pairs or larger sets; the concurrence of three or more edges. In order to simplify the language, the locution “vertices coincide” is to be interpreted as “the points representing the distinct vertices coincide”; similarly for edges. Most of the important writings on polygons after Meister (such as Poinsot [45], Cauchy [7], A.F.Mobius [42], Wiener [55], Steinitz [52], Coxeter [9]) formulate the definition in the same way or equivalent ones, even though in some cases certain restrictions are added; for example, A.F.Mobius insists that the polygon not be contained in a line. However, all these writers tacitly assume that no two vertices fall on coinciding points. It is unfortunate that Gunther [30, pp. 44ff] misunderstands Meister and imputes to him the same restriction. Other authors (for example, Bruckner [3, p. 1], Edmund Hess [32, p. 611]) insist explicitly in their definitions that no two vertices of a polygon are at the same point; but later Bruckner [3, p. 2] gives another definition, that coincides with ours, apparently written under the impression that it has the same meaning as his earlier one. However, disallowing representation of distinct vertices by one point is a crippling restriction which, I believe, is one of the causes for the absence of an internally consistent general theory of polyhedra. (The present definition coincides with what were called unicursal polygons in [23], where more general objects were admitted as “polygons”.) It should be mentioned that all these authors define a polygon as being a single circuit; A.F. Mobius explicitly states that it would be contrary to the customary meaning of the word if one were to call “hexagon” the figure formed by two triangles (hexagram 2{3}). This seems to have had little influence on later writers. For example, without any formalities or explanations, Bruckner [3, p. 6] introduces such figures as “discontinuous polygons”, in contradiction to his own earlier definition of “polygon”. Hess [32] has a more vague definition of polygons, and explicitly allows “discontinuous” ones. Since the number of “essentially different” shapes possible for n-gons in-creases very rapidly with increasing n, it is reasonable and useful to consider various special classes which can be surveyed more readily. The historically and practically most important classes are defined by symmetries, that is, by isometric transformations of the plane of the polygon that map the polygon onto itself. In case some of the vertices coincide, symmetries should be considered as consisting of an isometry paired with a permutation of the vertices. Thus, the quadrangle in Figure 2 does not admit a 120◦ rotational symmetry, but it admits a reflection in a vertical mirror paired with the permutation(12)(34). Clearly, all symmetries of any polygon form a group, its symmetry group. A polygon is called isogonal [isotoxal, regular] provided its vertices [edges, flags] form a single orbit under its symmetry group. (A flag is a pair consisting of a vertex and one of the edges incident with it.) It is easily proved that a polygon is regular if and only if it is both isogonal and isotoxal. Moreover, if n≥3 is odd, every isogonal n-gon is regular, as is every isotoxal one. The more interesting situation of even n is illustrated for n=6 in Figure 3. Similar illustrations of the possibilities for other values of n appear in [23], [24], [25]. Two consequences of the above definition of polygons deserve to be specifically mentioned; both are evident in Figure 3, and become even more pronounced for larger n. First, all isogonal n-gons fit into a small number of continua, and so do all isotoxal n-gons. If polygons having some coinciding vertices were excluded, the continua would be artificially split into several components, and the continuity would largely disappear. Second, the number of regular polygons would be considerably decreased. Under our definitions, for every pair of integers n and d, with 0≤d≤n/2, there exists a regular n-gon, denoted by its Schlafli symbol {n/d} . The construction of star polygons {n/d} inscribed in a unit circle can be described as follows (this was first formulated by Meister [41], and was also stated by Louis Poinsot [45]): From a point on the circle, taken as the first vertex, advance to the next vertex by turning through an angle of 2πd/n, and repeat this procedure from each resulting point till the starting point is reached at the n^th step. Naturally, depending on the values of n and d, some of the intermediate vertices may coincide, but their identities are determined by the number of steps that led to them. Thus, for example, {6/0} has six coinciding vertices, {6/2} has three pairs of coinciding vertices, and {6/3} has two triplets of coinciding vertices; in contrast, all vertices of {6/1} are distinct. It takes no effort to realize that all vertices of {n/d} will be distinct if and only if n and d>0 are relatively prime. This connection between geometry and number theory was stressed by Poinsot and all writers following his example, and seems to have been one of the reasons why they banished from consideration all polygons with coinciding vertices. This was done despite the fact that the various results concerning angles, areas and other properties of regular polygons remain valid regardless of the relative primeness of n and d. Moreover, allowing polygons with coinciding vertices is essential if one wishes to have continuity in the combinatorial types of polyhedra. One other consideration requires admitting polygons – and in particular, regular polygons – with coinciding vertices. In the present paper we are concerned with unoriented polygons; however, in some situations it is convenient or necessary to assign to each polygon an orientation. This yields two oriented polygons for each unoriented {n/d} (except if d=0 or d=n/2). Among regular polygons it is convenient to understand that the rotations through 2πd/n yielding {n/d} are taken in the positive orientation; then the polygon oppositely oriented to {n/d} is {n/e}, where e=n−d. Thus, oriented regular polygons {n/d} exist for all n>d≥0, and these n polygons are all distinct. The appropriateness of such a convention is made evident by its applicability in many results concerning arbitrary polygons. It would lead us too far to describe these results, which can be interpreted as consequences of the possibility of expressing every n-gon as a weighted sum (in an appropriate sense) of regular polygons. The results range from Napoleon-type theorems to the elucidations of limits of iterations of various averaging operations on polygons. Detailed information about such applications, which would not be possible under the Poinsot restriction, may be found in [2], [13], [14], [19], [39], [43], [46], [47], [48], and in their references.

References[edit]

[2] Elwyn Berlekamp, E. N. Gilbert and F. W. Sinden, A polygon problem Amer. Math. Monthly 72 (1965), pp. 233 – 241; reprinted in Selected Papers on Algebra, Mathematical Association of America, Washington, D.C., 1977
[7] Augustin-Louis Cauchy, Recherches sur les polyedres; Premier memoire. J. ́Ecole Poly-tech. 9 (1813), 68 – 98. German translation by R. Haußner, with comments, as “Abhandlunguber die Vielecke und Vielflache”. Pages 49 – 72 and 121 – 123 in Abhandlungenuber die regelmaßigen Sternkorper, Ostwald’s Klassiker derexakten Wissenschaften, Nr. 151. Engelmann, Leipzig 1906.
[13] P. J. Davis, Circulant Matrices Wiley-Interscience, New York 1979.
[14] J. Douglas, Geometry of polygons in the complex plane J. Math. Phys. 19 (1940), 93 – 130.
[19] J. C. Fisher, D. Ruoff and J. Shilleto, Polygons and polynomials, In:The Geometric Vein: the Coxeter Festschrift, C.Davis, B. Grunbaum, F. A. Scherk, eds., pp. 165 – 176. Springer, New York, 1981
[20] A. Girard, Table des sines, tangentes & secantes, selon le raid de 100000 parties Avec un traict ́e succinct de la trigonometrie tant des triangles plans, que sphericques. Ou son plusieurs operations nouvelles, non auparavant misesen lumiere, tres-utiles & necessaires, non seulment aux apprentifs; mais aussiaux plus doctes practiciens des mathematiques. Elzevier, a la Haye 1626.
[23] Branko Grunbaum, Polyhedra with hollow faces In POLYTOPES: Abstract, Convex and Computational, Proc. NATO – ASI Conference, Toronto 1993. T. Bisztriczky, P. McMullen, R. Schneider and A. Ivic Weiss, eds. Kluwer Acad.Publ., Dordrecht 1994, pp. 43 – 70.
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[25] Branko Grunbaum, Isogonal decagons In: The Pattern Book. Fractals, Art, andNature, C. A. Pickover, ed. World Scientific, Singapore 1995, pp. 251 – 253
[30] S. Gunther, Vermischte Untersuchungen zur Geschichte der mathematischenWissenschaften Teubner, Leipzig 1876.
[32] Edmund Hess, Uber gleicheckige und gleichkantige Polygone Schriften derGesellschaft zur Beforderung der gesammten Naturwissenschaften zu Mar-burg, Band 10, Abhandlung 12, pp. 611 – 743, 29 figures. Th. Kay, Cassel 1874.
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[41] A. L. F. Meister, Generalia de genesi figurarum planarum et inde pendentibusearum affectionibus. Novi Comm. Soc. Reg. Scient. Gotting. 1 (1769/70), pp.144 – 180 + plates.
[43] Bernhard Neumann, Plane polygons revisited. InEssays i n Statistical Science:Papers in Honour of P. A. P. Moran, J. Gani and E. J. Hannan, eds. Also in J. of Appl. Prob., Special volume 19A(1982), pp. 113 – 122.
[45] Louis Poinsot, M ́emoire sur les polygones et les polyedres. J. ́Ecole Polytech. 10 (1810), 16 – 48. German translation by R. Haußner, with comments, as“Abhandlunguber die Vielecke und Vielflache”. Pages 3 – 48 and 105 – 120 in Abhandlungenuber die regelmaßigen Sternkorper, Ostwald’s Klassiker derexakten Wissenschaften, Nr. 151. Engelmann, Leipzig 1906.
[46] Isaac Jacob Schoenberg, The finite Fourier series and elementary geometry. Amer.Math. Monthly 57(1950), pp. 390 – 404.
[47] W. Schuster, Polygonfolgen und Napoleonsatze. Math. Semesterberichte 41 (1994), 23 – 42.
[48] W. Schuster, Regular isierung von Polygonen. Math. Semesterberichte 45 (1998), 77 – 94.
[55] C. Wiener, Uber Vielecke und Vielflache. Teubner, Leipzig 1864.