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Math 308 Project: The Shoemaker's Knife Problem - An Application of Inversion

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Pappus of Alexandria (who wrote in the time of Diocletian, 284-305 AD) records in his Collection[i], the following proposition: Let two semicircles be drawn, a larger one B and a smaller one A fitted inside the first, with centres b and a, respectively, lying on a common baseline (fig. 1). This figure Pappus calls an άρβηλος, or “shoemaker’s knife”. Now let the area between the two semicircles be filled with circles C0, C1, C2,... tangent to A and B and to each previous circle in the sequence (fig. 2). Then the following results hold for this figure: (1) The centres of circles C0, C1, C2,... all lie on an ellipse (cyan) passing through q. (2) The points of contact of same circles all lie on a circle (blue) passing through q. (3) Letting the radius of Cn be rn, the height of its centre above the baseline is 2nrn. That the centres of C0, C1, C2,... all lie on an ellipse passing through q is apparent from the diagram below (fig. 3). The foci of the ellipse are points a and b, and

Math 308 Project: The Shoemaker's Knife Problem - An Application of Inversion Robert Hunter Robert Hunter 2 1575 2003-12-21T01:04:00Z 2003-12-21T01:04:00Z 8 1346 7677 63 15 9427 9.2720 1 Robert Hunter rhunter@alumni.sfu.ca Math 308, Euclidean Geometry Term Project, Fall 2003 The Shoemaker's Knife Problem - An Application of Inversion Pappus of Alexandria (who wrote in the time of Diocletian, 284-305 AD) records in his Collection [i] , the following proposition: Let two semicircles be drawn, a larger one B and a smaller one A fitted inside the first, with centres b and a, respectively, lying on

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