The Graph of Robo

Since I've finally gotten my feet sunken into the realms of math education, I've been enthralled by the power and elegance of polynomials. They're predictable, yet so powerful when it comes to working in several bases at once.

One beautiful use of them in within the "Everything Formula", which is a beautiful inequality that is able to graph absolutely any picture you want it to, given you can fit it in certain parameters. I will explain how it works in this post, but Matt Parker will always give a better mathematical explanation than I can.

The formula looks something like this:

1/2 < floor(mod(floor(y/17)* 2^(-17*floor(x) - mod(floor(y),17)), 2))

Ok, the formula is not so beautiful, but the outcome is so cool! For instance, I can graph my little nickname "Robo".

So let's find out how polynomials happen to be involved here and how I feel this could be a fun project in the classroom.
The main key of finding anything in the "everything formula"s graph is finding the elusive "k" value you see on y-axis. Every k-value will be different for each picture. For instance, the k-value in this graph is:

31,906,094,523,385,994,368,481,299,331,958,673,414,312,090,258,671,068,452,891,354,163,026,957,591,747,931,239,823,646,614,166,751,734,581,823,167,481,905,809,115,761,714,115,538,319,734,188,902,939,157,355,078,026,353,195,148,215,842,562,150,854,534,727,330,647,214,594,307,840,044,204,981,488,844,449,687,130,096,476,891,888,444,808,101,888

Which is incredibly large! How in the world can we expect to find this number?

Well, we use binary! Or a polynomial of base 2. The binary number is easy enough to find, though very laborious to come up with. All you do is start at the bottom left of where your "picture" begins and if you want it to be a black square, you put a 1 and a white square is a zero and you go up the picture. Then start at the bottom again and go up again; putting ones and zeros wherever applicable. 

For example, the binary number that works for my ROBO graph is the following:

Convert that to base 10 then multiply by 17 and VOILA! You have your very own ridiculous k-value for your graph.

It's an interesting use of polynomials and definitely can be a fun thing to do in the classroom. I wouldn't go completely in depth as I've done here, but you mathy people would probably would like to know how it works. 

For instance, rather than writing out the entire binary number, maybe it's better to do only a portion of the binary. But whatever binary numbers they do happen to come up with, have them write out the polynomial in base 2! It's definitely worth seeing what crazy math is going on in the background. Besides, most of the squares will probably be white anyway, so the polynomial should still be relatively short. Keep it to may the first 20 squares? But then at the end of the day, they will have a number that is literally theirs!

To create a graph and find corresponding k-value, I used this website. There is no way I expect me, you, or any student to figure it out long-hand.

Enjoy and happy math!