Pappus's Arbelos

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The arbelos is the area between the 3 red semicircles. Archimedes (287 to 212 B.C.) wrote of its mysterious properties. People are still learning about it.
Pappus in about 320 A.D. wrote about one particular problem. The Arbelos is successively filled with circles from the right. All are tangential to the 2 largest semicicles and successively to each other.

He showed the perpendicular distance of the nth circle to the baseline is n times the diameter of the nth circle.Pappus offered a proof that is too long for this page, the modern proof uses geometric inversion to demonstrate that under each circle n+1/2 circles of the same diameter can be drawn.

A good computer formula to draw the inner circles is radius r=a*b/(n2*a2+b) where n is the number of the circle b=AB/2 ,a=BC/2 and a+b=1.And the center coords y=2*r*n and x=r*(b+1)/a-1.
The system has a number of intriguing properties which are not obvious from the construction but can be easily proven from the computer formula.
The tangents of the lines drawn from point A to the center of each circle are multples of the first and n.

The position phenomina is that the diameter of any circle centered above the line aO is always equal to the length aO.It can be easily shown by moving the cursor until any circle is centered over the line aO.
In the case n=1 the diameter of the circle divided by the radius of the outer arc is (sqrt(5)-1)/2 the famous golden ratio.
general arbelos reference
Thomas Schoch's remarkable Arbelos page
Pappus Chain Reference
A picture of ongoing arbelos research
mail welcome
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