Various norms

[ughaibu]

Thanks for the link.

[ughaibu]

I presume I'm taking liberties by mixing norms, but anyway: using an infinity norm the diagonal of a square is the same length as the side, so the square on the diagonal is equal to the original square, even though by comparison of triangles the square on the diagonal has twice the area. As squares of any area from zero to infinity can be constructed by using diagonals as sides or vice versa, this could suggest the conclusion that all natural numbers have the same square root.

This may be a counter intuitive effect of infinity, or just nonsense, but it got me wondering if "things" with zero area have shape. For example, using a 1-norm, the diagonal is conceived as a zig-zag along the sides of sub-squares arrayed on the diagonal. With an infinite number of sub-squares, the zig-zag becomes a straight line but retains it's zig-zag property to give the result that the diagonal is twice the side. If we maintain this zig-zaggedness when the diagonal is used as the side of a subsequent square, we find the diagonal of the new square is a "pure" straight line, and in this square the side is equal to the diagonal. With the infinity norm, if the sub-squares maintain their orientation the square on the diagonal now has a diagonal twice that of it's side. In both cases, if we marry the ratios of the diagonals to the sides in the larger and smaller squares, we get the conventional Pythagorean result.

This tempts me to suggest the idea that squares may exist as rational/irrational pairs. Am I on the verge of re-discovering something so trivial that any mathematically aware readers are laughing from their socks up?

[Qfwfq]

As squares of any area from zero to infinity can be constructed by using diagonals as sides or vice versa, this could suggest the conclusion that all natural numbers have the same square root.

B)

 

This may be a counter intuitive effect of infinity, or just nonsense, but it got me wondering if "things" with zero area have shape.

Possible, when they are the limit of a homogeneous coordinate transformation that reduces them to zero...

 

Am I on the verge of re-discovering something so trivial that any mathematically aware readers are laughing from their socks up?

I just blinked through that quickly, I didn't quite catch your meaning, you might be on to defining some simple fractal objects, or something.

[ughaibu]

Qfwfq: Thanks for the reply. I guess the blue face means that you select the "just nonsense" option, I can live with that, though as nonsense goes, it appeals to me, so I've added it to my repertoire.

I did a brief search for "homogeneous coordinate transformation" but got a bunch of pages couched in further mathematician's slang. As an english speaker my initial expectation is that the expression amounts to 'maintaining angles and ratios, etc, of the outline, regardless of other changes'. If I've caught your drift with this interpretation, would I be correct in thinking that the Sierpinski gasket comes into this catagory? As it goes, I'm not convinced by the Sierpinski gasket, does it satisfy Benacerraf's comment (http://plato.stanford.edu/entries/spacetime-supertasks/ section 2.5) on supertasks?