continuous nature of length and breadth

[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?

[timo]

Continuous and discontinuous with what parameter?

[jackrell]

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

[Sisyphus]

Why don't you explain your own reasoning, because I'm having a little trouble grasping what you're talking about. How would you account for such a change, and how do you think that will make length discontinuous?

[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

[Sisyphus]

Sorry, what does a series of polygons have to do with continuity?

[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