# This Is Your Brain. This Is Your Brain On Infinity. Any Questions?

Happy Geek/Nerd Pride Day. Happy World Planking Day. Oh ya, and Happy Towel Day. So much to celebrate, so little time.

Anyway, as I mentioned in yesterday’s post (here), today is a triple threat of nerdiday goodness. To celebrate, I present to you the following brain-bending, get your nerd on, mathematical thinkable.

By this point, dear readers, I’m assuming you have read enough of my rants and raves about $\pi$ that you know the following: $\pi$ is

1. irrational,

2. transcendental, and

3. awesome.

Being irrational, we know that $\pi$ cannot be represented as a ratio of integers. It can, however, be represented (at least partially) as a non-terminating decimal.  I write partially, because although we could begin to write out the infinite digits of $\pi$,

3.1415926535897932384626433832795028841971693993…

we’d never be able to complete the task, unless we had an infinite amount of time and never, ever, ever stopped writing. Clearly this is something that us mere mortals do not have. Or do we?

Consider the following:

Imagine that you have impeccable penmanship. In fact, imagine your penmanship is so good that people think your writing is a perfectly formed font. Imagine further that this font is legible at any size. That is, no matter the page size you are working with, you could increase or decrease your font size such that one single letter or digit would occupy the width and height of the page.

Next, imagine that you want to write each digit of $\pi$. Being the nerd that you are, you would know immediately that this would be a fruitless effort because, as a mere mortal, you wouldn’t have enough time to write out each and every single one of the infinite digits after the decimal.

But what if, by writing each digit of $\pi$ on a smaller page using our perfect almost font-like penmanship, we cut our writing time as well? I mean, it would seem reasonable to think that the time required to write a digit occupying a smaller page would be less than the time needed to write a digit occupying a larger area, especially if the speed of writing remained constant.

Specifically, let’s assume that your penmanship is so good that it would allow you to write each digit using half of the area of the previous digit, and requiring half the time to write. If the first digit takes 1 minute to write, and requires 1$m^{2}$ of space, then the second digit would require 30 seconds to write and only 0.5$m^{2}$ of space. Clearly the third would require half that time and space again (ad infinitum).

My question for you is this: how long and how much space would be needed to write out all of the digits of $\pi$?

I’ll leave you to ponder that for now, but fear not; I’ll provide the answer tomorrow.

Happy Towel-Geek-Nerd-Planking Day all y’all.

Oh ya, I almost forgot! I found this gem on the intertubes the other day and figured, given that today is Geek/Nerd Pride Day, it was my nerdly duty to share it with you. I take my nerdly duty very seriously.

Basically, what I’m trying to say is that $9x-7i>3(3x-7u)$.

Moral of the story: When you can’t say it with words, say it with math 🙂