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Drawing Apollonian Circle Packings. Complex Descartes’ Theorem. a 2 + b 2 + c 2 + d 2 = ½(a + b + c + d) 2 (b 1 z 1 ) 2 + (b 2 z 2 ) 2 + (b 3 z 3 ) 2 + (b 4 z 4 ) 2 = ½(b 1 z 1 + b 2 z 2 + b 3 z 3 + b 4 z 4 ) 2
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Complex Descartes’ Theorem • a2 + b2 + c2 + d2 = ½(a + b + c + d)2 • (b1z1) 2 + (b2z2) 2 + (b3z3) 2 + (b4z4) 2 = ½(b1z1 + b2z2 + b3z3 + b4z4) 2 • bj = curvature zj = x + yi
Apollonian Circle Packings with First Four Curvatures: -2, 3, 6, 7 Other Apollonian Circle Packings
