Resistance of Fractals: Sierpinski Triangle

[mersecske]

Let assume an exact mathematical fractal on a surface,
for example Sierpinski-triangle,
made of material with homogeneous conductivity.
What do you think,
it has zero, finite, or infinite resistance between two points
(for example two corner of the triangle)?

 

[Phrak]

The resistance at each point contact of one triangle to another is infinite. Recursing from the zeroth single to the first order the resistance is infinity at every contact. Every subsequent order gets the same, ad infinitium it seems.

You should be aware that, given any infinite plan or volume of material having nonzero resistivity, the resistance between any two ideal point contacts is infinite resistance. In real life, Ohm meter probes do not contact at an idealized point, but over an area. It is the contact parimeter of the probes that dictates the reading on a DVM rather than the resistivity of the material, beyond the kin of the electrical engineers who normally specify such sorts of measurements.

 

[mersecske]

OK, this is true.
But what about the resistance between
oposite sides of a Menger sponge?
Its existed and finite?

 

[Phrak]

Why don't you try it and see what happens? Start with a solid cube of unit resistivity. Call this cube the zeroth order Menger sponge. Calculate it's resistance. Take out the proscribed 6 cubes out of 27 and calculate again for the 1st order Menger sponge. Then do it for the 2nd order sponge. See if this series of resistance values converges to zero or something else.

 

[mersecske]

Very hard to calculate.
And not possible to measure :)
And the contacts are still not clear!
Maybe we have to take infinite wire with fractal cross section
?

 

[Phrak]

Hmm. The two contacts have to be surfaces or the resistance automatically becomes infinity.

I presumed you intended to pick opposite faces of the cube. For your zeroth order unit cube the contact area is one unit square. The sequence for the contact area is (1, 9/10, 81/100...).

 

[mersecske]

Yes but the current flow is very difficult

 

[Phrak]

If you know your contact areas tend to infinity, it really doesn't matter how you model the rest of it.

 

[mersecske]

Do you now what is fractal?
The fractal has finite surface!
Only its circumference is infinity.

 

[Phrak]

Yes, well, in the case of your 3 dimensional fractal, the volume tends to zero as the surface area increases.

But I see I made an error in my last post. I meant to say "If you know your contact areas tends to infinitely small, it really doesn't matter how you model the rest of it."

Anyway, this is the case with your fractal, and so the resistance for a finite cube is automatically infinite. The series 1, 9/10, 81/100 ... tends to zero.

 

[mersecske]

The series (8/9)^n, but yes.

 

[mersecske]

And what about if you imagine a discrete fractal grid, with finite resistance units, for example a Sierpinski-triangular?