Nowhere. Everything in nature that is close to circular isn't quite a circle.

That's irrelevant, however. While scientists and engineers use mathematics to describe reality, mathematics itself is not bound by reality. That a "pure circle" does not exist in physical reality doesn't matter to mathematicians.

Even if a "pure circle" did exist in physical reality, it's circumference would not be exactly 2πr because physical reality is not Euclidean. Space-time is instead, as far as we know, pseudo-Riemannian. Rhetorical question: Did special and general relativity disprove Euclidean geometry? The answer is no. While those theories did show that Euclidean geometry isn't a perfect descriptor of our universe, they did not disprove Euclidean geometry itself. Mathematics is not bound by reality.

That's nonsense. Jumping ahead to your next post where you expanded upon this,

You are (a) spouting nonsense, (b) talking about the positive integers, and © assuming the positive integers are "perfect". I suggest that you read about Gödel's incompleteness theorems.

Continuing with this next post,

Yes, one or both of the radius and circumference must be irrational.

The degree is defined as 1/360^{th} of a full circle. You can't prove or disprove a definition.

Space-time or GR contradicts the 5th euclidean postulate, which makes it wrong.

Engineers use mathematics for real purposes, though maths itself isnt bound to reality the only maths we actually need to know, understand and discover are bound to reality, all the rest are pointless probabilities that didnt and wont exist, else again they are bound to reality (so all maths that isnt relative to reality is void of purpose (also if we create an equation or formula mathematically that isnt directly related to reality it could be very detrimental if used in certain physical situations like the hadron collider or a nuclear submarine)).

You claim im "spouting nonsense" then continue to say that im defining positive integers, explain whats nonsensical about my briefly informal definition?

I'm currently working on a thesis that deals with this perfect mathematical system i purpose, it has a direct relation to shapes and topological math aswell as the nature of circles and time. Perhaps once ive finished i'll drop it on here and you can pick it apart, but for now positive integers will do to show how circles dont work within the nature of reality.

Just a little side question im confused with...from origin on the double positive part of a polar circle graph, how do we calculate the co-ordinates for the first half of a sine wave in terms of degree's? as in were working with a 90 degree right angle origin and were trying to account for or calculate the points of a semi circle, which is 180 degrees?

if the answer is splitting the angle into .5's how small can we split angles?

if the answer is relative to one axis representing 3-d (some measurement of energy (mass, speed, force etc)) and the other representing time (so this right angle is a representation of 4-d) then why are the vectors connecting the vertices curved and not 2-d lines? (as the crow flies so to speak)

I still cant quite comprehend how a circle fits into reality, even based on the fact that only having 3 digits of pi is a good enough approximation of reality to use for engineering purposes, how can it be infinite?? there must surely be some cut of point where a fractal pattern emerges? such as an infinite regression based on recursion (a single base unit).

**Edited by DevilSolution, 11 March 2013 - 05:50 AM.**