Finding equal potential points in electric circuits

[anachin6000]

Hi! I have a real problem in understanding the use of symmetry for finding equal potential points in nasty electric circuits. There are lots of problems were the solution simply says: "due to symmetry reasons, nodes X and Y have equal potential", but I rarely really understand the so called symmetry. So far I haven't found someone to properly explain to me how this really works.

To better explain myself, I annexed a cubic circuit. The problem asks to find the equivalent resistance when the circuit is connected to a voltage source between points S and A. The values for R1 and R2 are considered known. The solution says that the points I, M and the points H, B are at equal potential, so you can connect them together. Why those points?
Bare in mind, this is no homework, just an example (though I would be grateful if anyone would help me understand it).

So, can anyone explain me how to find symmetry in similar cases or suggest a paper on this topic?

 

[mfb]

You can quickly find candidates just by looking at the vertices, checking them is more work:
Are there vertices with 3 R1? T and A, but as the voltage source is connected to A those are not at the same potential.
Are there vertices with 2 R1 and 1 R2? I, M, B, H. Both I and M are connected to S via R2, both B and H are connected to A via R1. Possible candidates.
Are there vertices with 1 R2 and 2 R2? No.
Are there vertices with 3 R2? C and S. Same argument as with T and A: not at the same potential.

Can I/M and B/H be pairs at the same potential?
I is connected to S via R2, to T via R1, to H via R1.
M is connected to S via R2, to T via R1, to B via R1. They are at the same potential if B and H are at the same potential.
The same analysis for B/H shows that they are at the same potential if I and M are at the same potential. In other words, the points are symmetric, and at the same potential within the pairs.

 

[anachin6000]

You can quickly find candidates just by looking at the vertices, checking them is more work:
Are there vertices with 3 R1? T and A, but as the voltage source is connected to A those are not at the same potential.
Are there vertices with 2 R1 and 1 R2? I, M, B, H. Both I and M are connected to S via R2, both B and H are connected to A via R1. Possible candidates.
Are there vertices with 1 R2 and 2 R2? No.
Are there vertices with 3 R2? C and S. Same argument as with T and A: not at the same potential.

Can I/M and B/H be pairs at the same potential?
I is connected to S via R2, to T via R1, to H via R1.
M is connected to S via R2, to T via R1, to B via R1. They are at the same potential if B and H are at the same potential.
The same analysis for B/H shows that they are at the same potential if I and M are at the same potential. In other words, the points are symmetric, and at the same potential within the pairs.

Thank you!This was really helpful and revealing.