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Triangles in Wonderland Are there more acute or obtuse triangles? Lewis Carroll/Charles Dodgson Some fun facts: .. as a mathematician, Dodgson was, in the words of Peter Heath: "An inveterate publisher of trifles [who] was forever putting out pamphlets, papers, broadsheets, and books on mathematical topics [that] earned him no reputation beyond that of a crotchety, if sometimes amusing, controversialist, a compiler of puzzles and curiosities, and a busy yet ineffective reformer on elementary points of computation and instructional method. In the higher reaches of the subject he made no mark at all, and has left none since." Lewis Carroll/Charles Dodgson • “Three Points are taken at random on an infinite plane. Find the chance of their being the vertices of an obtuse-angled Triangle.” • Pillow Problems Thought Out During Wakeful Hours in 1893. Problem #58 Solution • We assume that the longest edge is from A=(0,0) to B= (b,0) • (Why can we do this? Can we do this?) Third point must occur where? Right triangle w longest side AB? Pythogorean Theorem: __ ___ __ ||AC || ^2 + || BC || ^2 = || AB ||^2 [Sqrt(x^2+y^2) ]^2 + [sqrt( (x-b)^2 + y^2) ]^2 = b^2 2x^2 + 2xb + 2 y^2 =0 Or [x-(b/2) ]^2 + y^2 = [b/2]^2 Nice precalc/ High school result Need relative areas In green circle: obtuse In orange region: acute Calc 1! • Circle = π b 2 4 • Orange region = 4* So • Obtuse/ total = • ≈0.6393825607119623027857777410193414 1234…. 3 dimensions • 3 points determine a triangle and a plane • Same issue 3 dimensions • Same issue w/ pythagorean theorem. Get a sphere centered at (b/2,0,0) Inside: obtuse Outside: acute 3 dimensions • Sphere = 4/3 • 1/2 of football: b 3 2 b 1 6 3 Go spherical! Calc 3 • 1/2 of ‘football’: So • Obtuse/ total = ( ) 2 0 . 4 5 5 24 b 2 3 b 3 2 3 • More acute triangles in 3d than obtuse, unlike plane Motivation for project Larson pg 573 Essential Calc: n dimensions • Longest edge again from (0,0,0,0) to (b,0,0,0) 2 2 2 2 2 • In 4d, sphere: x y z w b • Pythagorean Theorem: all pts that form right triangles with AB – ‘Sphere’ centered at (b/2,0,0,0) with radius b/2 n-dimensional spherical cap formula n dimensions Dimension 2 3 4 5 6 7 8 9 10 Obtuse/Total 3 8 6 3 2 5 3 32 36 3 8 53 15 640 864 3 16 289 105 26784 3 17920 128 6413 105 71680 114048 3 Decimal 0.6393825611 0.4 0.2468696971 0.1509433962 0.09165800095 0.05536332180 0.03329943290 0.01995945735 0.01192904991 Open Questions • What is the exact probability on the unit square in 2d or n dimensions. (unit disk is known)-simulation • Can be Simulated (unlike my problem) Hyperbolic (Poincare) Plane • ‘straight lines’ are arcs of circles that are perpendicular with the boundary Distance and size • As one approaches the boundary, ‘measuring sticks’ get smaller • Distance formula: Distance and size • As one approaches the boundary, ‘measuring sticks’ get smaller Triangles • Between any 2 points there is a unique line • So we can form triangles. Angles computed similar to plane (use tangent lines) Triangles and area WOLOG • Longest side is (0,±s) • Disks ? are disks, center moves. Pythagorean Thm • If AB is opposite a right angle then: cosh(AB) = cosh(AC)*cosh(BC) Pythagorean Thm • No longer a circle acute obtuse Small edge on left, big edge on right. Proved • As side limits to zero Obtuse/Total limits to • As side limits to 1 (infinity) then Obtuse/Total limits to 0 Next stop unit spheres! • Cookies?