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Ford Circles and Spheres Ford Circles and Spheres Sam Northshield Department of Mathematics SUNY-Plattsburgh Joint Mathematics Meetings, Jan. 2012 Outline • Ford circles: geometric, parametric, barycentric. Outline • Ford circles: geometric, parametric, barycentric. • Generalize up one dimension. Ford circles; geometric Given two tangent circles of radius 1/2, draw (unique) circle between and tangent to them. For any two, repeat this process. Continue forever. Saturated: can’t add or enlarge any circles. Ford circles; parametric For rational number a/b (in lowest terms), let Ca,b denote circle above and tangent to the x-axis at a/b with radius 1/2b2. Ford circles; parametric For rational number a/b (in lowest terms), let Ca,b denote circle above and tangent to the x-axis at a/b with radius 1/2b2. Ca,b||Cc,d ⇔ |ad − bc| = 1. Every Ford circle is of form Ca,b. parametric=geometric Every Ca,b is a Ford circle (enough to show it has parents...) (2, 5) → (2, 3) → (2, 1) → (1, 1) parametric=geometric Every Ca,b is a Ford circle (enough to show it has parents...) (2, 5) → (2, 3) → (2, 1) → (1, 1) (1, 2) ← (1, 1) ← (1, 0) ← (1, 0) (1, 3) ← (1, 2) ← (1, 1) ← (0, 1) C2,5 has parents C1,3 and C1,2. Ford circles; barycentric Let hs, ti denote circle above and tangent to the x-axis at t/(s + t) with radius 1/2(s + t). The set of Ford circles is {hs, ti : (s + t + u)2 = s2 + t2 + u2, stu < 0, gcd(s, t, u)= 1}. An interesting set a2 + b2 + c2 = (a + b + c)2 (1, 0, 0), (2, 2, −1), (3, 6, −2), (20, 5, −4) An interesting set a2 + b2 + c2 = (a + b + c)2 (1, 0, 0), (2, 2, −1), (3, 6, −2), (20, 5, −4) Sum of any two is a perfect square. (not conversely: (0, 9, 16), (0, 25, 169)). An interesting set a2 + b2 + c2 = (a + b + c)2 (1, 0, 0), (2, 2, −1), (3, 6, −2), (20, 5, −4) Sum of any two is a perfect square. (not conversely: (0, 9, 16), (0, 25, 169)). However, (b2 − ab, ab, a2 − ab) solves (s + t + u)2 = s2 + t2 + u2 and gives hs, ti = C(a, b). Ford spheres; geometric Let ω = e2πi/3 so that {0, 1, 1 + ω} form equil. triangle of side length 1. Start with three mutually tangent spheres of radius 1/2 each above and tangent to the x, y-plane at the points 0, 1, 1 + ω. At each stage, for any three mutually tangent spheres, construct the (unique) sphere between and tangent to them. Repeat forever. Ford spheres; barycentric For numbers a, b, c, let ha, b, ci denote sphere above and tangent to the x, y-plane at (a0 + b1 + c(1 + ω))/(a + b + c) with radius 1/2(a + b + c). Ford spheres; barycentric For numbers a, b, c, let ha, b, ci denote sphere above and tangent to the x, y-plane at (a0 + b1 + c(1 + ω))/(a + b + c) with radius 1/2(a + b + c). Theorem: The set of Ford spheres is {ha, b, ci : (a+b+c+d)2 = a2+b2+c2+d 2, gcd(a, b, c, d)= 1}. Part of proof (a + b + c + d)2 = a2 + b2 + c2 + d 2 For U = (a, b, c, d), let U◦ = (a, b, c). Bilinear form: Q(U, V ) = (U · 1)(V · 1) − U · V , Part of proof (a + b + c + d)2 = a2 + b2 + c2 + d 2 For U = (a, b, c, d), let U◦ = (a, b, c). Bilinear form: Q(U, V ) = (U · 1)(V · 1) − U · V , Q(U, U)= 0 Part of proof (a + b + c + d)2 = a2 + b2 + c2 + d 2 For U = (a, b, c, d), let U◦ = (a, b, c). Bilinear form: Q(U, V ) = (U · 1)(V · 1) − U · V , Q(U, U)= 0 ◦ ◦ Q(U, V )= 1 iff hU i||hV i. Part of proof (a + b + c + d)2 = a2 + b2 + c2 + d 2 For U = (a, b, c, d), let U◦ = (a, b, c). Bilinear form: Q(U, V ) = (U · 1)(V · 1) − U · V , Q(U, U)= 0 ◦ ◦ Q(U, V )= 1 iff hU i||hV i. Tetrahedral rule: A, B, C, D mutually tangent ⇒ A, B, C, E too where A + B + C = D + E. A+B+C=D+E An interesting set a2 + b2 + c2 + d 2 = (a + b + c + d)2 (1, 0, 0, 0), (2, 2, 0, −1), (4, 2, 1, −2), (6, 2, 5, −4) An interesting set a2 + b2 + c2 + d 2 = (a + b + c + d)2 (1, 0, 0, 0), (2, 2, 0, −1), (4, 2, 1, −2), (6, 2, 5, −4) Sum of any three elements is of the form a2 − ab + b2 (= |α|2 for some α ∈ Z[ω]). Ford spheres; parametric R := Z[ω] is a Euclidean subring of C. Given α, β ∈ R, let S(α, β) be sphere ‘above’ and tangent to complex plane at α/β with radius 1/2|β|2. Ford spheres; parametric R := Z[ω] is a Euclidean subring of C. Given α, β ∈ R, let S(α, β) be sphere ‘above’ and tangent to complex plane at α/β with radius 1/2|β|2. S(α, β)||S(γ, δ) iff |αδ − βγ|2 = 1. Ford spheres; parametric R := Z[ω] is a Euclidean subring of C. Given α, β ∈ R, let S(α, β) be sphere ‘above’ and tangent to complex plane at α/β with radius 1/2|β|2. S(α, β)||S(γ, δ) iff |αδ − βγ|2 = 1. α ⊥ β iff ∃x, y : |αx + βy|2 = 1. Hence, {S(α, β) : α, β ∈ R, α ⊥ β} is a saturated collection of spheres (with disjoint interiors). Ford spheres; parametric R := Z[ω] is a Euclidean subring of C. Given α, β ∈ R, let S(α, β) be sphere ‘above’ and tangent to complex plane at α/β with radius 1/2|β|2. S(α, β)||S(γ, δ) iff |αδ − βγ|2 = 1. α ⊥ β iff ∃x, y : |αx + βy|2 = 1. Hence, {S(α, β) : α, β ∈ R, α ⊥ β} is a saturated collection of spheres (with disjoint interiors). Theorem These are the Ford spheres. Summary so far Let • G: Start with 3 mut. tan.; choose unique 4th tangent. Repeat ad infinitum, • P := {S(α, β) : α, β ∈ Z[ω], α ⊥ β}, • B := {ha, b, ci : (a + b + c + d)2 = a2 + b2 + c2 + d 2, gcd(a, b, c, d)= 1}. Theorem: G = P = B. Actual proof: • B is maximal. • B < P. • B < G [parent algorithm]. Parametric generalization Any Euclidean subring of C can give rise to a saturated sphere array. E.g., Z[i] also studied [Pickover] Can one classify all such R by classifying arrays of Ford spheres? Sierpinski Gasket −i N SG := 2 Pf (i) : f ∈ {0, 1, 2} . ( ) Xi Sierpinski Gasket −i N SG := 2 Pf (i) : f ∈ {0, 1, 2} . ( ) Xi Can one attach a sphere tangent at each “dyadic" point? Geometric generalization For any three M.T. spheres, there are unique other triple fitting underneath: Geometric generalization For any three M.T. spheres, there are unique other triple fitting underneath: Following SG, get an array of spheres with set of tangencies: Barycentric Closure of this set of tangencies is an Apollonian circle packing! Barycentric Closure of this set of tangencies is an Apollonian circle packing! These spheres are all represented barycentrically by {ha +∆, b +∆, c +∆i : (a, b, c, d) ∈ D} where ∆ := (ab + ac + bc)/3 and D is a set of integral relatively prime Descartes quadruples p ((a + b + c + d)2 = 2(a2 + b2 + c2 + d 2)). (Curvatures of 4 M.T. circles satisfy this) Final thoughts What is previous set of spheres for full D? Z[ζ12]? Annmarie McGonagle (student): Ford circles Rieger, Hellegouarch: Z[ω] geometrically, Pickover: Z[i] parametrically, Diophantine approximation (CFs) with spheres. E.g., for circles, and irrational x, there are infinitely many pairs of integers such that a 1 x − < . b 2b2 .
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