Mathematica Famous one on circuits - but more mathematical

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The discussion centers on the problem of calculating the effective resistance in a cube-shaped circuit made up of 12 resistors, with current passing between two vertices. Participants explore the potential for solving this problem without relying on symmetry assumptions, suggesting the use of isomorphic graphs for simplification. References are made to Alan Tucker's work in Graph Theory and Bamberg and Sternberg's book on electric circuit analysis, which incorporates graph-theoretic and algebraic-topological methods. The conversation also touches on the properties of symmetrical cubes and generalized n-dimensional cubes, noting that the resistance between opposite vertices approaches a finite value as the dimensions increase. A request for access to a specific mathematical article is made by a participant who is a high school student in India, highlighting the collaborative nature of the discussion.
rushil
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Everyone probably know the famous problem consisting of a circuit having 12 resistors in the shape of a cube such that the resistors make up the edges. We pass current from one vertex and are required to find the effective resistance between this vertex and another one (the difficulty of the problem depends upon which vertexes are considered!)

In Physics, we usually solve this problem by assuming some symmetry considerations like equal partition of current or equal voltages etc. While studying Graph Theory in Alan Tucker's book, I cam across a similar example where he asked whether 2 graphs, one like the cubical circuit above and another circular graph were isomorphic. While his question ended in a negative there, I thought, could it be possible to solve the problem without any assumptions by finding a suitable, simple isomorphic graph that can be easily solved! Before getting down to some brainstorming, I just want to know, has anyone of you (or somebody you know) considered this problem before. Are you aware of a solution to the above problem? Please post what you think and your possible solutions!
 
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There's a chapter or two in Bamberg and Sternberg's A Course in Mathematics for Students of Physics

that describes electric circuit analysis using graph-theoretic and algebraic-topological methods.

http://www.citeulike.org/user/mukundn/article/416002
On the mathematical foundations of electrical circuit theory
Smale S - J. Differential Geometry, Vol. 7 (1972), pp. 193-210.
 
Symmetrical cubes are very easy to solve. The generalized n-dimensional cube is fun problem - I solved it back way back in my 1st year. It has some interesting properties... I believe the resistance between opposite vertices converged to a finite value as n-->infinity.
 
Hey guys, I'm sorry I don't have aceess to the article on mathscinet since I am high schooler in India. Please , can you send me the article by PM or mail!
 
rachmaninoff said:
Symmetrical cubes are very easy to solve. The generalized n-dimensional cube is fun problem - I solved it back way back in my 1st year. It has some interesting properties... I believe the resistance between opposite vertices converged to a finite value as n-->infinity.

That sounds like an interesting question. Can you state that question fully?
 
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