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October 19

Euclidean scaling

I believe the following is correct for geometric figures. If an n-dimensional element (like an edge) is scaled by s_n, then its m-dimensional measure will scale by s_m according to the following power law:

$s_{m}\propto s_{n}^{\frac {m}{n}}$

First of all, is this entirely correct? Second, does this law have a name? I know that Euclid and Aristotle proved it for special cases and Galileo showed it for n=1 with m=2 and m=3, but I can't find this generalized form anywhere. Does it have a name? Is it attributed to anyone?

Of course these scaling laws are frequently used in fractal geometry, but explanations of this generally jump from Euclid straight to Hausdorff and Mandelbrot. risk (talk) 12:12, 19 October 2009 (UTC)[reply]

Just to clarify, the figure is m-dimensional, and made up of n-dimensional elements (like a cube made up of 2-dimensional faces). risk (talk) 13:57, 19 October 2009 (UTC)[reply]

I don't understand what you are trying to say. Cube is not made up of 2-dimensional faces, the boundary of cube is. If m > n, then the m-dimensional measure of an n-dimensional object is always zero, and therefore it is pointless to ask how will it scale. Furthermore, the correct formula is that if an object is scaled by s, then its m-dimensional measure is scaled by s^m, there is no other parameter coming into this. — Emil J. 15:40, 19 October 2009 (UTC)[reply]

You can consider a 2D subset of a 3D object. You can then derive the rule the OP describes by applying your rule twice (once in reverse). --Tango (talk) 15:46, 19 October 2009 (UTC)[reply]