I became interested in the patterns in multiplication tables for different base number systems a while ago. Specifically, the pattern made by the last digit of each number in the multiplication table. So, base 10 would look like this:
1|2|3|4|5|6|7|8|9|0 2|4|6|8|0|2|4|6|8|0 3|6|9|2|5|8|1|4|7|0 4|8|2|6|0|4|8|2|6|0 5|0|5|0|5|0|5|0|5|0 6|2|8|4|0|6|2|8|4|0 7|4|1|8|5|2|9|6|3|0 8|6|4|2|0|8|6|4|2|0 0|0|0|0|0|0|0|0|0|0
I thought it was interesting that when you move to number systems with different bases, the patterns don't follow the number. They follow it's relative position in the number system. E.g., in base 12, the pattern is 6,0,6,0,6......
I then realized, I could see the pattern better if I just assigned each number a color. I started with using 10 greyscale colors, with 0 being black and 9 being white. So now base 10 looks like this:
Then, I figured that I really could use as many colors as I wanted to, and see if a larger pattern forms. Using all 256 greyscale colors, I came up with this image representing a base 256 multiplication table:
Or I could go from black to white to black, and smooth out the image:
I decided to animate the pattern to better see what was going on. To do this, I defined my 1-n color scale as
w is white and
b is black. I would create a frame, shift my colors down one
[b,w,w,w,w,b,b,b,b,b,b,b....], and create the next frame. I repeated this until they colors fully cycled and got this animated image.
What is this pattern called?
My question is, what is this pattern called? I'm having a hard time finding anything about it. It seems to be a bunch of hyperbolic curves imposed on each other. There is a bunch of "stars" at the corners of where you would divide the image into 4ths, 9ths, etc.
Any insight into this would be appreciated.