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"! :rofl:

 

[luiseduardo]

Thanks, but do you know a general formula ?