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.67(b) where a sequence of successively larger equilateral hyperbolictriangles are represented in the Euclidean plane.Note that, as the sides of thetriangle become unbounded, the angles approach zero while the area remainsbounded.I.e., limA’!À ¸ = 0. Mathematical Properties 75(a)(c)(b)Figure 2.68: The Hyperbolic Plane: (a) Embedded Patch [302].(b) PoincaréDisk.(c) Thurston Model [20].[138]Property 84 (The Hyperbolic Plane).The crochet model of Figure 2.68(a)displays a patch of H2 embedded in R3 [302].As shown in Figure 2.68(b), it may also be modeled by the Poincaré diskwhose geodesics are either diameters or circular arcs orthogonal to the bound-ary [33].In this figure, the disk has been tiled by equilateral hyperbolic trian-gles meeting 7 at a vertex.This tiling ultimately led to the Thurston model ofthe hyperbolic plane shown in Figure 2.68(c) [20].In this model, 7 Euclideanequilateral triangles are taped together at each vertex so as to provide noviceswith an intuitive feeling for hyperbolic space [318].However, it is importantto note that the Thurston model can be misleading if it is not kept in mindthat it is but a qualitative approximation to H2 [20].Property 85 (The Minkowski Plane).In 1975, L.M.Kelly proved theconjecture of M.M.Day to the effect that a Minkowski plane with a regulardodecagon as unit circle satisfies the norm identity [192]:"||x|| = ||y|| = ||x - y|| = 1 Ò! ||x + y|| = 3.Stated more geometrically, the medians of an equilateral triangle of side"3length s are of length · s just as they are in the Euclidean plane.Midpoint2in this context is interpreted vectorially rather than metrically.Property 86 (Mappings Preserving Equilateral Triangles).Sikorskaand Szostok [281] have shown that if E is a finite-dimensional Euclidean spacewith dim E e" 2 then f : E ’! E is measurable and preserves equilateraltriangles implies that it is a similarity transformation (an isometry multipliedby a positive constant).Since such a similarity transformation preserves every shape, this may beparaphrased to say that if a measurable function preserves a single shape, i.e.that of the equilateral triangle, then it preserves all shapes.In [282], theyextend this result to normed linear spaces. 76 Mathematical PropertiesFigure 2.69: Delahaye ProductProperty 87 (Delahaye Product).In his Arithmetica Infinitorum (1655),John Wallis presented the infinite product representationÀ 2 2 4 4 6 6 8 8= · · · · · · · · · ·.2 1 3 3 5 5 7 7 9In 1997, Jean-Paul Delahaye [74, p.205] presented the related infinite product2À 3 3 6 6 9 9 12 12" = · · · · · · · · · ·.2 4 5 7 8 10 11 133 3"The presence of À together with 3 suggests that a relationship between thecircle and the equilateral triangle may be hidden within this formula.We may disentangle these threads as follows.The left-hand-side expression,pR2À"= , is the ratio of the perimeter of the circumcircle to the perime-p3 3ter of the equilateral triangle.Introducing the scaling parameters Ãk :="(3k-1)(3k+1), Delahaye s product may be rewritten as3k2 2 2pR · Ã1 · Ã2 · · · Ãk · · ·lim = 1.k’! "pThus, if we successively shrink the circumcircle by multiplying its radius by2the factors, Ãk (k = 1,., "), then the resulting circles approach a limitingposition where the perimeter of the circle coincides with that of the equilateraltriangle (Figure 2.69). Mathematical Properties 77Property 88 (Grunsky-Motzkin-Schoenberg Formula).Suppose thatf(z) is analytic on the equilateral triangle, T , with vertices at 1, w, w2 wherew := exp (2À1/3).Then [69, p.129],"3f2 2 (z) dxdy = · [f(1) + wf(w) + w2f(w2)].2TWhile this chapter has certainly made a strong case for the mathematicalrichness associated with the equilateral triangle, it runs the risk of leaving thereader with the impression that it has only theoretical and aesthetic value or,at best, is useful only within Mathematics itself [ Pobierz caÅ‚ość w formacie PDF ]