Tuesday, October 27, 2015

Unizor - Geometry2D - Circumference of a Circle

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Notes to a video lecture on http://www.unizor.com

Circumference of a Circle

1. Definition

The first question to discuss related to a circumference of a circle is "What is a circumference of a circle?"
Unit of measurement used for segments is not good for measuring length of curves. So, we have to come up with a different definition.

Typical approach to this lies in approximation. We will rely a lot on intuitively obvious statements with full understanding that proving them rigorously might present some problem.
It's tempting to define the circumference of a circle as follows:
a) start a process by inscribing a regular hexagon into a circle and setting the first approximation to a circumference of a circle as a perimeter of this hexagon;
b) on each step we draw perpendiculars to all sides of a polygon obtained on a previous step and take intersections of these perpendiculars with a circle as new vertices of a polygon with twice as many edges as on a previous step;
c) continue this process of doubling the number of edges to infinity, and the limit of the perimeters of our polygons is, by definition, a circumference of a circle.

There are a few problems with this definition. For example, it is intuitively obvious that, if we start with an inscribed square instead of a hexagon, we should also gradually approach some limit by doubling the number of edges of a polygon. But is this limit the same as if we start with a hexagon?
Issues of existence of a limit and its uniqueness can be rigorously addressed, but are rather complex, and we just point the result that, no matter how our process is arranged, as long as the maximum distance between neighboring points goes to zero, the limit of the perimeters of polygons exists and is the same. That justifies the definition of a circumference as such a limit.

2. Similarity of All Circles

Therefore, if a circumference of one circle is greater than its radius by some factor, the same factor is applied to any other circle.

3. About π

Traditionally, the factor between a circumference and a diameter of a circle is designated a Greek letter π. It is a constant for all circles. It's an irrational real number and can be approximated to any precision. To four decimal places it's equal to 3.1416.

4. Iterations

If d[N] is the length of a polygon, inscribed into a circle, on Nth iteration of replacing each of its sides with a pair of equal but smaller ones, this recursive equality is held:
d[N+1] = √{2 − √[4 − (d[N])²]}

5. Conclusion

As you see, our first process started with hexagon and we got one set of formulas for a perimeter of polygons obtained by doubling the number of edges.
The second process started with a square and formulas were different.
But in both cases a few first steps of iterative process led to very close results. This is exactly how it should be, because, no matter how we organize the process of approximation of the circumference of a circle with perimeters of polygons, as long as the maximum edge length tends to zero, perimeter tends to the circumference of a circle.

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