Visual Calculus
Encyclopedia
Visual calculus by Mamikon Mnatsakanian
(known as Mamikon) is an approach to solving a variety of integral calculus
problems. Many problems that would otherwise seem quite difficult yield to the method with hardly a line of calculation, often reminiscent of what Martin Gardner
calls "aha! solutions" or Roger Nelsen a proof without words
.
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 (constant-length) tangents used in constructing the ring are translated so that their points of tangency coincide, the result is a circular disk of known radius (and easily computed area). Indeed, since the inner circle's radius is irrelevant, one could just as well have started with a circle of radius zero (a point)—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:
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.
can be calculated by considering the area between it an an enclosing rectangle. These tangents can all be clustered to form a circle. If the circle generating the cycloid has radius r then this circle also has radius r and area πr2. The area of the rectangle is 2r.2πr = 4πr2. Therefore the area of the cycloid is 3πr2, it is 3 times the area of the generating circle.
The tangent cluster can be seen to be a circle because the cyclod 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 angle triangle in the generating circle. This means that clustered together the tangents fill describe the shape of the generating circle.
For another geometric approach to finding the area under a cycloid, see Cavalieri's principle#Cycloids.
Mamikon Mnatsakanian
Mamikon Mnatsakanian is Project Associate at Project Mathematics! at California Institute of Technology.He received a Ph.D. in physics in 1969 from Yerevan State University, where he became professor of astrophysics...
(known as Mamikon) is an approach to solving a variety of integral calculus
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
problems. Many problems that would otherwise seem quite difficult yield to the method with hardly a line of calculation, often reminiscent of what Martin Gardner
Martin Gardner
Martin Gardner was an American mathematics and science writer specializing in recreational mathematics, but with interests encompassing micromagic, stage magic, literature , philosophy, scientific skepticism, and religion...
calls "aha! solutions" or Roger Nelsen a proof without words
Proof without words
In mathematics, a proof without words is a proof of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than more formal and mathematically rigorous proofs due to their...
.
Description
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. (Perhaps surprisingly, no additional information is needed; the solution does not depend on the ring's inner and outer dimensions.)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 (constant-length) tangents used in constructing the ring are translated so that their points of tangency coincide, the result is a circular disk of known radius (and easily computed area). Indeed, since the inner circle's radius is irrelevant, one could just as well have started with a circle of radius zero (a point)—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:
- The area of a tangent sweep is equal to the area of its tangent cluster, regardless of the shape of the original curve.
Applications
Tom ApostolTom M. Apostol
Tom Mike Apostol is a Greek-American analytic number theorist and professor at the California Institute of Technology.He was born in Helper, Utah in 1923. His parents, Emmanouil Apostolopoulos and Efrosini Papathanasopoulos, originated from Greece. Mr...
has produced a very readable introduction to the subject. In it he shows that the problems of finding the area of a cycloid
Cycloid
A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line.It is an example of a roulette, a curve generated by a curve rolling on another curve....
and tractrix
Tractrix
Tractrix is the curve along which a small object moves, under the influence of friction, when pulled on a horizontal plane by a piece of thread and a puller that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed. It is therefore a curve of...
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 cycloidCycloid
A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line.It is an example of a roulette, a curve generated by a curve rolling on another curve....
can be calculated by considering the area between it an an enclosing rectangle. These tangents can all be clustered to form a circle. If the circle generating the cycloid has radius r then this circle also has radius r and area πr2. The area of the rectangle is 2r.2πr = 4πr2. Therefore the area of the cycloid is 3πr2, it is 3 times the area of the generating circle.
The tangent cluster can be seen to be a circle because the cyclod 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 angle triangle in the generating circle. This means that clustered together the tangents fill describe the shape of the generating circle.
For another geometric approach to finding the area under a cycloid, see Cavalieri's principle#Cycloids.
See also
- Cavalieri's principleCavalieri's principleIn geometry, Cavalieri's principle, sometimes called the method of indivisibles, named after Bonaventura Cavalieri, is as follows:* 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane...
- HodographHodographA hodograph is a diagram that gives a vectorial visual representation of the movement of a body or a fluid. It is the locus of one end of a variable vector, with the other end fixed. The position of any plotted data on such a diagram is proportional to the velocity of the moving particle. It is...
This is a related construct that maps the velocity of a point using a polar diagram. - PlanimeterPlanimeterA planimeter is a measuring instrument used to determine the area of an arbitrary two-dimensional shape.-Construction:There are several kinds of planimeters, but all operate in a similar way. The precise way in which they are constructed varies, with the main types of mechanical planimeter being...
External
- Mamikon Main web site
- ProjMath Mamikon
- Proof without Words from MathWorldMathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...