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The Wheatstone bridge is one arrangement of circuit elements that is neither a series nor parallel, nor a combination of series and parallel elements. It makes me wonder what other arrangements are out there. It seems like there could be arrangements of circuit elements that can't be reduced to a combination of series, parallel and Wheatstone bridges.

Do these other arrangements exist? Is there a method for constructing these arrangements (if they do exist)?

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Yes, they do exist.

The examples you gave are examples of planar networks, that is, networks that can be drawn on a plane without intersecting branches (branches can actually intersect but only at nodes).

There are, though, also non-planar networks, that is, networks that cannot be drawn on plane without having branches that intersect. This kind of networks cannot be reduced to a recursive combination of series, parallel and Wheatstone bridge connections.

An example is that of a resistor cube.

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  • $\begingroup$ Is there a nice method for finding and constructing these arrangements? $\endgroup$
    – David Elm
    Commented Apr 17, 2017 at 18:30
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    $\begingroup$ @DavidElm I'm not aware of any algorithm that can systematically generate non-planar networks, but you can look at graph theory to see if such algorithm exist: the topology of any electrical network can be represented by a graph. So, a non-planar network is represented by a non-planar graph. There are criteria to characterize planar graphs, so you can probably "easily" generate graphs that do not satisfy those criteria. $\endgroup$
    – Massimo Ortolano
    Commented Apr 17, 2017 at 18:42
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    $\begingroup$ Are all planar graphs reducible to combinations of series, parallel and Wheatstone? $\endgroup$
    – user126422
    Commented Apr 17, 2017 at 19:54
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    $\begingroup$ @ArmandoEstebanQuito That's an interesting question, and I think that the answer is no, though I couldn't find a clear statement (but I'm not so well versed in graph theory). That is, it doesn't seem that the planarity of a graph can be simply characterized by recursively combining series, parallel and bridge connections. I have a few references on this, but I don't have time now to check carefully. I'll update my answer if I can find an example of planar network that cannot be reduced to those combinations. $\endgroup$
    – Massimo Ortolano
    Commented Apr 17, 2017 at 20:26

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