A Treatise on the relationship of the Circle and Pi to the solution of Mathematical Problems such as Alhazen's Billiard Problem,

The Billiard Problem simply restated is; In a given circle, find an isosceles triangle whose legs pass through two given points inside the circle. This can be restated as: from two points in the plane of a circle, draw lines meeting at the point of the circumference and making equal angles with the normal at that point.

The great minds of geometry understand intuitively that the circle and Pi are infinite and from the interaction the circle and arc, and other geometric shapes, it is possible to generate the solution to an infinite number of complex mathematical problems. The rub is that often the greats of geometry like Ramanujan are unable to prove conclusively their work and must rely on mathematicians who follow them to prove it for them.

The diagrams depicted in this website http://pg.photos.yahoo.com/ph/edwardchesky/slideshow?.dir=/66d7&.src=ph are just a few of a number of geometric shapes and geometric solution to complex equations that can be formed from the overlap of these circles. Other geometric shapes that can be formed from these figures include the rhombus, equalateral triangle, two equal isosceles triangles and a three sided pyramid, to mention a few.

In these days of computers it is possible to run a reverse analysis of geometric solutions to mathematical problems, however; such analysis takes time and much computer power.

Given the nature of Pi and the circle, the relationship to the divine automatically comes to mind. The great minds of mathematics, science and geometry such as Aristotle have long considered the relationship between the perfection of geometry and mathematics to the divine and whether or not such perfection could be considered proof of the existence of a higher power. Many scholars, such as Sir Isaac Newton and Albrecht Durer, Huygens and Descrates have also pondered this very relationship in their efforts to understand the universe and its workings.

In the classical view of the universe, the uncertainty, (the devil or doubt), with regards to the asking or answering such questions as the Billiard Problem comes from two directions; the first is the framing the question and second is in determining the answer to the question. Formation of a great question takes as much effort and genus as the creation of a geometric solution. Each side of the equation the answer and the solution are plagued by doubt and uncertainty until the solution is revealed at which time the generation of the solution and proof of the correctness of the question is as to the witness’s as if god himself spoke on the validity of the question and the correctness of the answer. In many ways the doubt about the validity of the answer or creation of a correct question, answer mirrors the Hiesenburg Uncertainty Principle, at least in a classical approach and view to understanding the universe, with the random unrepeated composition of Pi representing the will of god in the traditional sense or random factor in quantum mechanics.

http://pg.photos.yahoo.com/ph/edwardchesky/slideshow?.dir=/66d7&.src=ph

Note I used a varient of the Quadratrix of Hippias to Trisect the angles depicted on the website. In terms of process I had to construct the geometric shapes first then use a varient of the Quadratrix of Hippias curve to perform the trisection. All work was done with a compass and straightedge. Not exactly the pure Greek solution to the problem but it fits within existing mathmatical theory. A discussion of the Quadratrix of Hippias is listed below.

Hippias's Quadratrix

This is a curve formed as the intersection of a radius and a line segment moving at corresponding rates. A square and a circle are drawn so that one corner of the square is the center of the circle, and the side of the square is the radius of the circle. The idea is this. A radius falls over from the side of the square to the base at a constant rate. At the same time, a line segment falls from the top of the square at constant rate. Both start moving at the same time, and both hit the bottom at the same time.

Thus, the ratio, change in arc length/ displacement of falling cross piece, represents the speed of the sweeping radius relative to the falling cross piece. Since both move at constant speeds, this ratio is always the same value. With some knowledge of "modern" trigonometry, it can then be calculated. The length of the arc swept out from beginning to end is pi/2 radians(90 deg.) time radius length. The distance the cross piece falls is simple the length of the square's side. But, since the length of the square's side is equal to the length of the radius, (pi/2) times radius length/side length= pi/2. Thus, the curve is a pictorial representation of the irrationality of pi.

Note: The quadratrix was discovered by Hippias of Elias in 430 BC, and later studied by Dinostratus in 350 BC (MacTutor Archive). It can be used for angle trisection or, more generally, division of an angle into any integral number of equal parts, and circle squaring.

I also note an Oxford Don also solved Alhazen's Billard Problem in 1997 by translating it into co-ordinates on an X and Y axis, an insight first obtained by Descartes in the 17th Century. My solution to the Billiard Problem is related to that created but never finished by Albrecht Duur, however was the result of independent research