Find a Fractal Object with Known Boundary Term

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For my work, I need to check my calculations with an example of a fractal object. I searched on the internet, there are some examples of fractals with their hausdorff dimensions, but no boundary terms related.
Also found some 1-d examples, but I need d>3 dimensional objects since my calculations are for BEC.

I need a fractal object that its boundary term is known in terms of the volume and the hausdorff dimension.
For example, a regular box with length L in d-dim, the volume as Ld and the boundary term kL^d-1 (k is a const.)
I need such an example for a fractal object or (I don't know if there are any)a regular object with fractal boundary.

Maybe I couldn't express very good, but is there any book, paper, method you can recommend about this subject?
thanks in advance.
 
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They can be, yes, but I also need to know their boundary term in terms of volume element. Is there a rule to generalise any kind of fractal object to d-dimension? Because I may try to apply, I remember I found a fractal with boundary term with dimension less than 3.
thank you very much.
 
ssamsymn said:
Is there a rule to generalise any kind of fractal object to d-dimension?
It's not hard to compute the fractal dimension of the boundaries of the d-dimensional generalisations of these. Have a go.
 
okay, I am thinking on it. thank you.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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