So these addition formulas that apply to series and parallel resistors seem more generally to be formulas for finding the new ratio when you have several ratios and want to add them together treating the quantity in the numerator or the denominator as equal for all ratios. Series circuits have the same current for each ratio, and differing voltages, while parallel circuits have the same voltage for each ratio and a different current for each. Similarly for these aspect ratios, where you want the new ratio after adding width or height while setting the other (height or width, respectively) equal.
When I was fresh out of uni, I got asked to build a system to show recommendations to people based on a graph of connections between them. I tried to mentally think about these as a network of resistors, which led me to dimensionality reduction and from there to alternating least squares. Isomorphisms are fun like that: they can help you find techniques when you're stuck on something.
Yep! My problem at the time was that I didn't know I needed a matrix, or which operations would be helpful to perform on one, because I didn't know much about the problem I was trying to solve. I ran into a paper about using SVD on networks of resistors to simplify them, which pointed me in the right direction.
If you made the Piet painting from a substrate with constant resistivity, and put wires between each area, you would end up with a working circuit with all the calculated resistances. This then leads to the insight that it doesn't matter how you divide the painting up, its total resistance only depends on its overall size.
it's overall path length would be the total resistance of each part added together, but if you configure the network with different parallel elements it will change the total resistance of the circuit. for parallel R1,R2: Rt = 1/ (1/R1 + 1/R2)
But if the resistors are constrained to fill the area of the entire painting everything cancels out, reducing to: R = resistivity * (length of painting) / (width of painting * thickness of conducting paint).
An example: If you have N parallel resistors constrained to fill the painting, their width (and resistance) must go as 1/N. Plug into the formula you gave and the Ns cancel out, leaving Rt constant.
Fun fact: a rectangular sheet of resistor with a specific thickness and terminals at both sides has a resistance unit of: ohms per square. Not square meters, just square!
Ok, if the spatial concept of aspect ratio is the equivalent of electrical resistance, and given the analogy in the article of the painting and the resistor network, what is the equivalence of color then? Frequency?
It's not fundamentally related to anything because the choice of color is completely arbitrary. There are no rules that relate color to any configuration of lines unless you choose to impose arbitrary rules.
In the last drawing I noticed the person wrote 2.2k . This is weird for me, for resistors I usually write 2k2 and see others do the same - I also imagine the two red stripes already...
I write 2.2kΩ, but I'm used to seeing people just write "2.2k" on schematics (because obviously it isn't 2.2kH or 2.2kV, and also it's next to a resistor symbol).
I think the 2k2 thing is maybe one of those young whippersnapper things like nanofarads, or maybe a European thing. I grew up with 0.001μF capacitors, not 1nF capacitors.
Some physically smaller resistors are marked 2k2 instead of 2.2k, because this requires less space for the marking and it also avoids ambiguities that could happen when the decimal point is too small or erased and 2.2k could be misread as 22k.
