A graph-theory game.
“… despite the connotations of the word ‘twin’, a dodecahedron actually has 5 twins. […] But here’s something deeper that Ocneanu claims to have proved, in unpublished work. Suppose you take one of these twins. It, too, will have 5 twins. One of these will be the dodecahedron you started with. But the other 4 will be new dodecahedra: that is, dodecahedra rotated in new ways. […] How many different dodecahedra can you get by continuing to take twins? Infinitely many! This image by Roice Nelson shows the vertices of a dodecahedron, its twins, the twins of its twins, the twins of the twins of its twins, the twins of the twins of the twins of its twins, and the twins of the twins of the twins of the twins of its twins …”
Edward Frenkel talks about math:
From the outside, mathematics might look like one big lump. In fact, it is a huge subject that has many different subfields: algebra, number theory, analysis, geometry, and so on. In the world of mathematics, they look like disconnected continents. But the Langlands program connects different fields and, by doing so, tells us something about the unity of mathematics. It offers a glimpse of something beneath the surface that we don’t understand.
And (among much else of great interest):
Mathematical power is not the power of a bomb. You cannot see its effect as immediately as Hiroshima and Nagasaki. But a formula can be just as powerful in terms of controlling our lives. It can alter the course of history; it can affect millions of people. […] I think we mathematicians are a little bit behind the curve. We are not fully aware of the Frankenstein that we may have already created or could create.
(Wikipedia on the Langlands program.)
John Conway on The Iron Law of Six:
I was astonishingly lucky. I literally remember my former undergraduate teacher telling my then wife that John would not be successful. She asked why. And he said, “Well, he does not do the kind of mathematics that’s necessary for success.” And that’s true. I really didn’t do any kind of mathematics. Whatever I did, I did pretty well, and people got interested in it — and that’s that. I did have a recipe for success, which was always keeping six balls in the air. Now I have had a stroke, so I can’t catch those balls terribly well. But what I mean is: Always be thinking about six things at once. Not at the same time exactly, but you have one problem, you don’t make any progress on it, and you have another problem to change to.