One of the problems they give the more advanced students is to find the resistance between any two points in an infinite resistor array. I wondered how this would turn out in an infinite array was a quasicrystal. I expect there will be surprises but I am too lazy to work it out. Build an array based on a Penrose tile shown below and tap the nodes with an LED. It may be with this additional complication the current will never settle down and each LED will vary in brightness over time or will shut on and off chaotically. Roger Penrose - Wikipedia
Interesting idea, not sure if I fully understand you. For the Penrose tiling, it’s unclear how to equip each vertex with an LED, since an LED requires a + and - pole. It’s also unclear which two vertices in the tiling are the source and sink for electrical potential, i.e., what the source of power is.
If you just mean the classical resistor grid but replaced by the Penrose tiling, I would not expect chaotic behavior. Kirchhoff’s laws provide a direct analogue between resistor circuits and random walks on the corresponding graph, where, given a unit-voltage source vertex A and sink vertex B, the voltage at any vertex equals the probability that a random walk starting there hits A before B. This quantity is time-independent.
That is part of the problem. There is a zillion ways to hook it up. I was thinking the LED would run up from each node perhaps into an identical tiling. Each node could be positive negative or zero depending on the behaviour of each LED or paired LEDs. There are LEDs that are red in one direction green in the other and yellow under alternating current.