Befuddlement is sometimes my companion. We walk and talk, but rarely understand one another. I tell my students sometimes that confusion does not need to be unpleasant since it is a signal that we might have something to learn.
When I landed in the branch of mathematics called graph theory, I thought it was like home, but without chores. Who really needs to take out the trash in graph theory? It was pure and pristine. Uncluttered by philosophy, religion, politics or money: graph theory was pure math and pure art at the same time. Since it was math it was funded by NSF rather than NEA, and what crazy legislator would ever deny the progress of science in bringing the US to the forefront of world prosperity? No one! Not even backspace backspace backspace. Anti-science and sedition would have been one and the same in 1972, Graph theory was art dressed as science and no legislator would have dared to challenge science in 1972. It would have meant that he (he would have been male) was not only a fascist but a warlock as well! All such fascist warlocks would have had to vie for the same congressional seat and in their zeal, said seat would have exploded under the pressure of their counter incantations!
So graph theory was my refuge. A place where I could play and dabble and imagine and have nary a care about either warp drives or neutron bombs! Imagine my confusion when I discovered that graph theory had applications!! Damn! I can't even tell you about my first patent. Or my first theorem concerning backspace backspace backspace. Or maybe I can. I don't remember.
Since then I have learned to think of applications as like mosquitoes. They sting a bit but they need not prevent the gardener from harvesting a pretty pepper.
So now perhaps you can see how I came to be interested in tiling. Yeah, I can make fissionable alloys out of striate textures that may or may not involve molybdenum (oh do I have a story about the Molybedenum Project of the upper South Platte to tell one day!) -- that's just a mosquito. But I can also tile my bathroom! And if I do my homework I can minimize the number of tiles that need to be cut as I tile around the toilet!
But here's what got me interested in tiling in the first place: there are really simple problems (simple to state, that is) that no one knows the answer to!
No one knows the answer!
Now in today's current political climate if one utters a conjecture followed by the statement "no one knows the answer" , it is likely to cause consternation among two thirds of the American public and 52% of our elected congressfolk. Said consternation is unlikely (in 2015) to result in exile to Belgium, but it may result in reduction of funding to all things mathematical by NSF and kneejerk reduction in funding to 28% of state supported public institutions of learning (and increased assessment to boot - assessment is a secret handshake for backspace backspace backspace)!
So you don't rely on mathematics and science any more: you rely on interior design: the modern engine fueling economic growth for a world increasingly reliant on an increasingly leisurely upper class.
Here is a problem that no one knows the answer to, but which is easy to state:
These are the only known ways to tile the plane infinitely with convex pentagons.
You can count them. There are 14 of them. The number will not change between the first and second time you count them (unless I do some SVG magic while you are looking).
And no one knows if there are more than 14, and the last one discovered was discovered by someone totally untrained in advanced mathematics in (like) 1980 and here it's been 4000 year since humans first started exploring all the different ways that one might tile the plane. And you know what? This is why it is fun!
No one knows the answer!
Oh and I should explain that I got these tiles from Wikipedia (the world's very own encyclopedia) and the code (coming out of Illustrator) was funky so I left behind a better version in repayment for the favor, but the same stuff (in the typically poor web format we have come to expect from the folks behind Mathematica, though they do wonderful math and algorithms!) http://mathworld.wolfram.com/PentagonTiling.html