jackrell

I don't think this is going to work but the question is a matter of symmetry. Strangely, there is no cause and effect principle in symmetry, so one has to think in different ways - breaking the symmetry, for example. It occurs to me that pi offers a similar case. Imagine that pi were a rational number instead of irrational. I'll give you as many decimal places as you like but not an infinite number. Now draw me a circle! When pi is a rational number, all equations containing it will be out of balance - if only by the tiniest ammount. As a result the perfect symmetry of the circle is broken and, if you tried to draw one, the ends would not meet up. jackrell

Thanks Sisyphus, I guess the question boils down to "why are length and breadth continuous?" Consider an outside observer who can observe the entire two dimensional series. Because the regular polygons are infinite in number the series appears as a one dimensional line. This is why length and breadth appear continuous to us. If the regular polygons were finite in number, they would stand out as markers or reference points in the previously continuous line. jackrell I should have added that the outside observer of two dimensions is someone located in the third dimension. jackrell

The two dimensional regular polygon series, triangle, square, pentagon, hexagon etc. is infinite. An observer of the entire series will not be able to observe individual members. As a result, the series must appear continuous. Suppose that the regular polygon series was finite rather than infinite. In this case an observer would always see the series as discontinuous. How would this affect our universe? Would one dimensional parameters like length and breadth become discontinuous?

Dear Athiest, Thanks for your comment. I guess the question should be, would any parameters of length and breadth be affected were the regular polygon series in two dimensions to be finite rather than infinite? A symmetry change, impossible of course, is involved, but can we speculate on what the effect might be? jackrell

Length and breadth are generally considered to be continuous functions. If we could make the regular polygon series, triangle, square, pentagon (infinite) a finite series, would length and breadth measurements become discontinuous?