Famous one on circuits - but more mathematical

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Discussion Overview

The discussion revolves around a mathematical exploration of a circuit problem involving 12 resistors arranged in the shape of a cube. Participants consider the effective resistance between vertices and the potential for solving the problem without relying on symmetry assumptions. The conversation also touches on graph theory and its relation to circuit analysis.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant describes a circuit problem with 12 resistors in a cube and questions whether it can be solved without symmetry assumptions by finding an isomorphic graph.
  • Another participant references a book that discusses electric circuit analysis using graph-theoretic and algebraic-topological methods.
  • A participant mentions that symmetrical cubes are easy to solve and shares their experience with generalized n-dimensional cubes, suggesting interesting properties related to resistance as dimensions increase.
  • A participant expresses a request for access to an article on mathematical circuit theory due to restrictions on their access as a high school student.
  • One participant asks for clarification on the question regarding the generalized n-dimensional cube that was mentioned.

Areas of Agreement / Disagreement

Participants express varying levels of familiarity with the problem, and while some agree on the ease of solving symmetrical cubes, others are uncertain about the implications of their findings or the methods proposed. The discussion remains unresolved regarding the best approach to the original problem.

Contextual Notes

Some assumptions about symmetry and graph isomorphism are not fully explored, and the discussion does not resolve the mathematical steps or implications of the proposed methods.

Who May Find This Useful

Readers interested in mathematical approaches to circuit analysis, graph theory, and those exploring the intersection of physics and mathematics may find this discussion relevant.

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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