Visible calculus – Wikipedia

Visible mathematical proofs

Mamikon’s theorem – the world of the tangent clusters are equal. Right here the unique curve with the tangents drawn from it’s a semicircle.

Visible calculus, invented by Mamikon Mnatsakanian (referred to as Mamikon), is an method to fixing quite a lot of integral calculus issues.^[1] Many issues that might in any other case appear fairly tough yield to the strategy with hardly a line of calculation. Mamikon collaborated with Tom Apostol on the 2013 guide New Horizons in Geometry describing the topic.

Description[edit]

Illustration of Mamikon’s methodology exhibiting that the areas of two annuli with the identical chord size are the identical no matter inside and outer radii.^[2]

Mamikon devised his methodology in 1959 whereas an undergraduate, first making use of it to a well known geometry drawback: discover the world of a hoop (annulus), given the size of a chord tangent to the inside circumference. Maybe surprisingly, no further info is required; the answer doesn’t depend upon the ring’s inside and outer dimensions.

The standard method includes algebra and software of the Pythagorean theorem. Mamikon’s methodology, nevertheless, envisions an alternate development of the ring: first the inside circle alone is drawn, then a constant-length tangent is made to journey alongside its circumference, “sweeping out” the ring because it goes.

Now if all of the (constant-length) tangents utilized in establishing the ring are translated in order that their factors of tangency coincide, the result’s a round disk of recognized radius (and simply computed space). Certainly, for the reason that inside circle’s radius is irrelevant, one may simply as effectively have began with a circle of radius zero (some extent)—and sweeping out a hoop round a circle of zero radius is indistinguishable from merely rotating a line phase about one in all its endpoints and sweeping out a disk.

Mamikon’s perception was to acknowledge the equivalence of the 2 constructions; and since they’re equal, they yield equal areas. Furthermore, the 2 beginning curves needn’t be round—a discovering not simply confirmed by extra conventional geometric strategies. This yields Mamikon’s theorem:

The realm of a tangent sweep is the same as the world of its tangent cluster, whatever the form of the unique curve.

Purposes[edit]

Space of a cycloid[edit]

Discovering the world of a cycloid utilizing Mamikon’s theorem.

The realm of a cycloid will be calculated by contemplating the world between it and an enclosing rectangle. These tangents can all be clustered to type a circle. If the circle producing the cycloid has radius $r$ then this circle additionally has radius $r$ and space $π r 2$ . The realm of the rectangle is $2 r \times 2π r = 4π r 2$ . Subsequently the world of the cycloid is $3π r 2$ : it’s 3 occasions the world of the producing circle.

The tangent cluster will be seen to be a circle as a result of the cycloid is generated by a circle and the tangent to the cycloid will probably be at proper angle to the road from the producing level to the rolling level. Thus the tangent and the road to the contact level type a right-angled triangle within the producing circle. Which means that clustered collectively the tangents will describe the form of the producing circle.^[3]