General Formula Resistors Polyhedron

[luiseduardo]

Hello all,

Does anyone know of any general formula for the equivalent resistance between two opposite points of a regular polyhedron (cube, dodecahedron ..) with all the resistors equal to R?

 

[willem2]

draw a graph of the polyhedron. choose an arbitrary point as one of the terminals. Now mark all points that are 1 resistance away from this, then all points that are at distance 2, etc.
you should get 1 point at maximum distance, which is the other terminal.
(this won't work for a tetrahedron, because there is no opposite point)

All points at the same distance, will have the same voltage, because of symmetry, so you can connect them with wires. The resulting network can easily be solved by replacing series and paralles resistances, because all resistances from an n to an n+1 distance point will now be parallel

for a dodecaedron, I get

3 points at distance 1
6 points at distance 2
6 points at distance 3
3 ponts at distance 4
1 point at distance 5

3 resistances between distance 0 and distance 1 points
6 resistances between distance 1 and distance 2 points
6 resistances between distance 2 and distance 3 points
6 resistances between distance 3 and distance 4 points
3 resistances between distance 4 and distance 5 points

6 resistances between points of equal distance, which will carry no current

so the total resistance is (1/3+1/6+1/6+1/6+1/3)R = (7/6)R

 

[Bill_K]

Careful, I once drew an infraction for answering the cube problem, on the grounds it was "homework"!

 

[luiseduardo]

Thanks, but do you know a general formula ?

 

[PJC01]

I can offer some insights into this question. The general formula for calculating the equivalent resistance between two points on a regular polyhedron with equal resistors is not a simple one. It depends on the specific shape and arrangement of the resistors within the polyhedron.

However, there are some general principles that can be applied. First, it is important to understand that the resistors in a polyhedron are connected in a series and parallel combination. This means that the equivalent resistance can be calculated by considering the resistors as a series of smaller circuits and then combining them using the appropriate equations.

Another important factor to consider is the symmetry of the polyhedron. For example, in a cube, the resistors are arranged in a symmetrical pattern which can simplify the calculation of the equivalent resistance. However, in more complex polyhedrons like a dodecahedron, the symmetry may not be as straightforward and the calculation may be more complex.

In general, it is recommended to use a combination of mathematical analysis and computer simulations to accurately calculate the equivalent resistance in a regular polyhedron. This will provide a more precise result than a general formula. I hope this helps in your research.