First, I drew out a few levels of f(x,y)=x+y on this scale, filling in the z-axis with the binary place value (a cube set on z=1 is in the 2^1=2 place value, so it has a value of 2).
Here is a diagram, rotated so that the pattern is easier to see:
This is not very informative, as all the information below the surface layer is hidden. To remedy this, I switched to considering each place value layer separately, as shown below for all the cubes at z=0 (or the place value 2^0=1)
In order to produce these digitally with my limited graphical abilities, I used a bird's eye view of each place value layer. I wrote a quick program to generate the patterns for my using the formula I worked out a while ago. Here is f(x,y)=x+y at z=1, 2, and 3:
Stripe-y, but not very interesting.
f(x,y)=x*y, on the other hand, is just plane awesome. Shown here are the first 5 layers:
I don't know if this really qualifies as a fractal, but each decreasing level is a one quarter size approximation of the one that came before. If you try mentally stacking these views, it looks a bit like Cantor's comb, if it was a stool instead of a comb, and a bit more complicated.
The fact that binary produces fractally patterns (fractally is now a word) is not surprising. The most obvious binary pattern, just the numbers 1-31:
(isometric diagrams made with this)
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The x*y patterns are awesome, but monochromatic. A suggestion: take the z layers 2 at a time and map to 4 colors instead of black and white. Equivalently, do the x*y in base 4 instead of binary.
ReplyDeleteIt seems to me that if the layers are similar for binary, they ought to be similar for base 4 (or base 8, etc.). Base 3 or base 5, who knows?
Anyway, keep up the good work. I've been following you since the fractal teddy bear.