Mamikon devised his method in 1959 while an undergraduate, first applying it to a well-known geometry problem: Find the area of a ring, given the length of a chord tangent to the inner circumference. The traditional approach involves algebra and application of the Pythagorean theorem. Mamikon's method, however, envisions an alternate construction of the ring: First the inner circle alone is drawn, then a constant-length tangent is made to travel along its circumference, "sweeping out" the ring as it goes. Now if all the tangents used in constructing the ring are translated so that their points of tangency coincide, the result is a circular disk of known radius. Indeed, since the inner circle's radius is irrelevant, one could just as well have started with a circle of radius zero —and sweeping out a ring around a circle of zero radius is indistinguishable from simply rotating a line segment about one of its endpoints and sweeping out a disk. Mamikon's insight was to recognize the equivalence of the two constructions; and because they are equivalent, they yield equal areas. Moreover, so long as it is given that the tangent length is constant, the two starting curves need not be circular—a finding not easily proven by more traditional geometric methods. This yields Mamikon's theorem:
Applications
has produced a very readable introduction to the subject. In it he shows that the problems of finding the area of a cycloid and tractrix can be solved by very young students. "Moreover, the new method also solves some problems unsolvable by calculus, and allows many incredible generalizations yet unknown in mathematics." He also mentions that combining Mamikon's method with the geometric solution yields a new proof of the Pythagorean Theorem. Solutions to many other problems appear on Mamikon's Visual Calculus site.
Area of a cycloid
The area of a cycloid can be calculated by considering the area between it and an enclosing rectangle. These tangents can all be clustered to form a circle. If the circle generating the cycloid has radius then this circle also has radius and area. The area of the rectangle is. Therefore the area of the cycloid is : it is 3 times the area of the generating circle. The tangent cluster can be seen to be a circle because the cycloid is generated by a circle and the tangent to the cycloid will be at right angle to the line from the generating point to the rolling point. Thus the tangent and the line to the contact point form a right-angled triangle in the generating circle. This means that clustered together the tangents will describe the shape of the generating circle.
