Circuit Theory/Symmetry
Some very complicated circuits can be solved by seeing symmetry. Three examples are presented below:
Example 1
editBridge Balance

Original Complicated Circuit

Redrawn up and down

Z impedances are all equal, so middle Z has no current going through it, nodes on either side are at the same EMF

middle Z can be replaced by short or open, since no current is going through it
Example 2
editCube

original flattened cube of resistors

identifying identical EMF locations

redrawing idential EMF as the same nodes, can see three groups of parallel resistors
Example 3
edit2DGrid
Suppose there is an infinite 2 dimensional grid of impedances (Z). What is the input impedance if connected across (in parallel) with any given impedance?

Current due to injection at any spot splits in quarters and then begins to fill the resistive grid like water filling an infinite bucket.

Current removed at another spot splits in quarters and then begins to drain the resistive grid like draining an infinite bucket.

Together the two create a steady state situtation where the question "What is the input impedance" can be asked. The current i is i/4 + i/4 = i/2. v = i/2*Z so v/i = Z/2 = input impedance.
If the grid were three dimensional, the current would split into 6 equal sections, thus the input impedance would be Z/3.
So what does this mean? It helps us understand that the infinity of space has an impedance: 376.730... ohms which is plank's impedance * 4π and is related to the speed of light, the permeability of free space and permitivity of free space.
Perhaps this is related entanglement and to the trinity since: