# Math Before Coffee

Math is a language. You can learn bit by bit, or you can learn by immersion. I suspect the former is a more successful process. However, this morning my brother tweeted about “fun with math” and linked to a video about the “everything” number. It’s about Tupper’s Self-Referential Formula.

Thus began my introduction to a “simple” formula that can plot itself on a graph. It can also be used to draw any other two-dimensional image.

I mentioned this to Hub, and who looked up the term; this led to a comment, “He’s cheating! He’s using mod and floor.” WTH does that mean?

Now, this is usually where I start feeling dumb and intimidated. But I admitted not knowing, and thus learned that “mod” is short for modular. Basically, it is the remainder in a long division problem. The mod (modular operator) of 5 divided by 2 is 1. Modular arithmetic — who knew?

On to floor and ceiling. Floor refers to mapping a real number to its next lower integer: the floor of 7.1 is 7. The floor of 7.8 is also 7. Ceiling refers to mapping a real number up to its next higher integer: the ceiling of 7.1 is 8. The ceiling of 7.8 is 8. This is different function from rounding. If I round 7.1, the answer is 7; if I round 7.8, the answer is 8. I am told this is used in computer programming and math.

As for the formula, it’s related to computer graphics. Hub went on to tell me about SIGGRAPH, which stands for Special Interest Groups on GRAPHICS and Interactive Techniques. It’s a group of computer professionals who spend their time creating the graphics you see on your computer, phone, tablet, movies, robotics, in emerging technologies, as well as what is used in research. He also suggested I look up the Utah Teapot and Lenna.

Martin Newell, a graphics researcher, created a mathematical model of an ordinary teapot in order to create a 3D computer model. It has since become a standard reference object in the graphics community. Go look at it. I’m amazed.

As for Lenna… it is an image of a woman looking coyly over her shoulder, and it is used as a standard test image for high resolution color image processing experiments. Its detail, shading, texture, and flat regions make it a good subject. As for the source of the image? It’s from a Playboy centerfold. Some controversy is associated with the use of the image because of the underlying sexism. Read more here.

I can’t say I learned actual math this morning, but I did come away with new knowledge. And all before I’d had my morning coffee. (Which I still haven’t had, because I just had to sit down and get this out of my head.)