A Little Bit of Binary

Saw some flat glass beads lying around my house and figured they had some potential for mathematical craftiness - here are the numbers 0 through 8 in binary (taking the clear beads to be 0's and the red ones to be 1's). I would've liked to go higher but I ran out of red beads, so I switched to graph paper. 

Check it out! I marked off blue lines between powers of 2 and one less than power of 2 (0 and 1, 1 and 2, 3 and 4, etc.) and there's this really neat pattern: if you folded along any of those lines, the boxes on the other side are the exact opposite. Say you folded on the line between 15 and 16, you could take all the numbers you've already drawn from 0 to 15, reflect them over the line, and invert the colors to get the next 16 numbers. This suggests an easy way to quickly color in binary. You'll need some graph paper and a pen that bleeds through (Sharpies work well) 

Step 1: Start coloring and stop when you're one less than a power of 2 (Technically you could start with just 0 and 1 and build up from there). I've done from 0 to 7. Take your paper, fold it along the blue line, and flip it over so that the back of the paper is facing you. Make sure you can still see the squares through the paper though; here I've taped my paper to a window. I also counted down 8 more squares and marked this so that I know where to stop. 

Step 2: Color in all the squares that you didn't already color on the other side. When you're done, it'll look like a solid red block of squares. 

Step 3: Unfold and voila! Now you have twice as much binary coloring - from 0 to 15. You could then take this and repeat the steps again with the new line to get all the way up to 31 and go on generating exponentially more binary (that is, until you run out of graph paper).