# Resistance of infinite nested triangles

#### Mtnbiker

1. Homework Statement
Here is an interesting problem... there is a wire bent in the shape of an equilateral triangle, side length = a and resistivity = rho.

In the center of this triangle is another equilateral triangle (inverted, side = a/2, resistivity = rho) and so on into infinity. What is the overall resistance between points A and B in terms of a and rho? 2. Homework Equations

R = (rho * length)/area

3. The Attempt at a Solution

I started by using the equation for resistivity, R = (rho * length)/area, but I wasn't sure if area would apply here. We aren't given any information about the wire beyond the shape and length. So I'm really asking for help in determining a good starting point... I don't know of any other equations that would incorporate rho and length.

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#### tiny-tim

Science Advisor
Homework Helper
Welcome to PF!

… In the center of this triangle is another equilateral triangle (inverted, side = a/2, resistivity = rho) and so on into infinity. What is the overall resistance between points A and B in terms of a and rho?

I started by using the equation for resistivity, R = (rho * length)/area, but I wasn't sure if area would apply here. We aren't given any information about the wire beyond the shape and length. So I'm really asking for help in determining a good starting point... I don't know of any other equations that would incorporate rho and length.
Hi Mtnbiker! Welcome to PF! I don't think you can solve this on the information given. I suggest you say "let the resistance be R/a times length", and carry on from there. #### Mtnbiker

Re: Welcome to PF!

Hi Mtnbiker! Welcome to PF! I don't think you can solve this on the information given. I suggest you say "let the resistance be R/a times length", and carry on from there. Hi... thanks for the welcome, I'm glad to be here.

I agree with you regarding keeping the area incorporated in the answer. However, I'm still struggling with what exactly would the length be (first triangle is 3a, second triangle is 3a/2, then 3a/4 and so on...). There is a point of diminishing returns, so I need to find that point.

Thanks for the input!

#### tiny-tim

Science Advisor
Homework Helper
Hi Mtnbiker! Yes, you're correct … obviously each triangle has sides half the length of the next one out.

So assume there are n triangles, start from the inside, and work your way outward …

at each stage, get rid of one triangle and calculate the equivalent resistances along the three sides of the next triangle. #### Dick

Science Advisor
Homework Helper
If the network is infinite you can use self similarity. Call the overall resistance between two vertices on the first inner triangle R. Now you have a simple network with three wires of resistance R and and six of resistance a*rho/2. Solve that for the resistance beween A and B in terms of R. Then realize that the outer triangular network is the same as the inner triangular network, but twice as big. So the resistance from A to B is also just 2R. Equate the two values and solve for R.

Thanks guys!

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