Thursday, February 25, 2010

Find me a circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the center. The common distance of the points of a circle from its center is called its radius.

But Circles can not, and do not exist in reality, anything that seems to be a Circle is actually a polygon with a very high number of sides... the closest to a Circle that we can get is to have each side of the polygon the size of quantum distance.

Just try and draw a Circle in The Gimp (or in Paint.NET).

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Now zoom 600% in to it. You will see it is actually a polygon, composed of thousands of little pixels, the same is true for "circles" in reality, they are composed of thousands of little quantum pixels.

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So, Circles do not exist, they are just a math abstraction. And that is why we end up needing unattainable numbers like Pi to deal with the area inside of a circle... because there is no such thing as the area inside of a circle (because there are no circles).

The area inside of anything in reality, that looks like a circle, can be calculated by dividing it in "quantum pixel areas", and the adding those areas to get the total area inside of the supposed circle. Now, that would require a lot of effort, that is the reason we believe in circles, because they are a very useful abstraction (even if it is an abstraction that fails and produces Pi as a result of its failure)

Among Archimedes mathematical accomplishments is the computation of pi, he used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a polygon of a greater and greater number of sides. The quotient formed by the area of this polygon divided by the square of the circle radius can be made arbitrarily close to Pi as the number of polygon sides becomes large, proving that the area inside the circle of radius r is Pi * (r*r),Pi being defined as the ratio of the circumference to the diameter.

When a circle's diameter is 1 unit, its circumference is Pi units. Why Pi is such simple concept but such a strange "number"? because Circles do not exist, Pi is an example of a Mathematics (Geometry?) abstraction leak.

As we increase the number of sides, we approximate more, and more to the circle, but we can never really have a circle.

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This is why we can never know the complete value of Pi, because there is now way to have a circle in a discrete quantum based reality, "circles" are a good model for this things with too many sides to count easily, but circles are just a model, not a reality

That is something, that as software developers we should always keep in mind: The Map (the Model) is not The Territory (the Reality)

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