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I have a question related to higher dimensional topology and electrical engineering, It is incredibly common for engineers to come across schematics and diagrams for circuits which cannot exist in 2 dimensions but which can exist in 3 dimensions, often we use jumper wires in our diagrams to illustrate the point that two wires are not connected (see example shown below)

schematic for a high gain amplifier.

So from this fact it stands to reason that you can design a circuit which can be described in 3 dimensions but which can only really exist in 4.

So my question essentially is broken down into three parts:

a.) Is this assumption correct?

b.) If this assumption is correct is it possible to build a circuit in three dimensions whose behavior would differ in a 4 dimensional world from that of a three dimensional world?

c.) If b is possible is it possible to do such and test models of theoretical physics like string theory which describe higher dimensions being closed in on themselves?

any insights would be well appreciated as I am by all means well outside my own depth of knowledge in this train of thought.

Ethan
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    @Ethan Your assumption is is actually incorrect: no matter how complex your network, 3 dimensions is always enough. See this post for one argument to that effect. – Ben Grossmann Aug 08 '23 at 18:39

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There may be some subtlety depending on exactly what you mean by "a circuit", but my suspicion is the assumption is not correct.

  1. Physically, a circuit diagram is intended to be an abstraction of... well, circuits. It stands to reason that circuit diagrams which cannot be realized by any circuit in our 3D reality are less likely to be interesting physical ideas, and more likely to be a bad definition of a circuit diagram. Of course, this line of reasoning is not rigorous at all, and indeed historical counterexamples exist (black holes, famously). So I would not find this argument convincing if it were not bolstered by:

  2. Mathematically, every network of nodes and edges (often called a "graph" by mathematicians) can be embedded in $\Bbb{R}^3$ (3D space). My understanding of circuits is that they can all be modeled as decorated graphs where the edges are wires, and the nodes are either physical elements (like a battery, resistor, etc.), or junctions.

Note that if the edges can, themselves, be connected e.g. into triangles, squares, ..., then we are now trying to embed a surface. It is well-known that some surfaces cannot be embedded in $\Bbb{R}^3$ but can be embedded in $\Bbb{R}^4$, for instance the Klein bottle. However, if an essential "2-D wire" corresponds to anything remotely electical, it's certainly beyond my pay grade :P

Eric Nathan Stucky
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  • +1 for the last paragraph imagining "2-D wires". You are close to inventing cell complexes and studying their embeddings. – Ethan Bolker Aug 08 '23 at 19:31
  • I don't know how to say this without it sounding snarky, but I mean it very genuinely: the instructor in me is very pleased that apparently I can still write in a cadence that suggests I am a student :) – Eric Nathan Stucky Aug 08 '23 at 20:51
  • Not snarky at all. From your profile I knew you were more advanced than "inventing" would suggest, but went with it anyway. – Ethan Bolker Aug 08 '23 at 22:05