how would I go about adding more RAM to the arduino?
you could have an array that gives the pixel positions for each "circle" that needs to be lit up,
fevertheless, he will need to store each ligthed LED, given the examples I estimate around 50 per small circle. Plus some color information. Multiplied by the maximum amount of circles, multiplied by the total number of patterns...
One major consideration, each pixel will need around 60mA at 5 volts when fully turned on, so 422 per panel and 3 panels will need considerable power.
Only if many are turned on.