Or, "How Metric Spaces Taught Me to Walk."

Many of you may be either dubious or confused, but suffice to say: a not-insignificant portion of this weekend was spent staring at the checkerboard pattern of a dance floor, gingerly stepping on only the black squares.

The habit comes from when I was about five years old. In somewhat of an attempt to walk more like an adult, I decided to walk on only the squares that were connected diagonally on our kitchen floor. It worked: ladies and gentlemen, I am now walking like an adult.

But there were far-reaching consequences, and I realize that future employers may be raising a curious brow as to my employability. Be that as it may: to this day, I prefer not to walk on tiles that are not connected diagonally. I don't mean to say that I won't: it's just that I'd prefer not to. And the penchant doesn't hold to strict rules: in an attempt to get to class faster (it started in late elementary school), I figured out that if I walked on every other tile in a row of tiles, I could a) not look so weird walking down the hall, b) get to class faster. I started taking two steps at the same time. As my legs grew ungainly long, I started taking four steps at a time, which proceedes until today, when I find myself looking ahead sometimes five to six steps steps to figure out how, where, and how fast I can get somewhere.

Stop looking at me like that.

But even this required formalization. This happened in college one fateful early morning, walking back to the dorms from the Regenstein Library.

Imagine a tiled floor, say in checkerboard format. Imagine now that every other row of the checkerboard floor were shifted over by half a tile. Thus, stepping on the diagonal tile to any given tile becomes somewhat of a confusing venture.

Fortunately, Paul Sally came to the rescue with the L1 metric. Daniel Biss too--Mr. Biss comes later. That is to say, there's a way to measure how much you've travelled in xy-plane. The cannonical method is to take the difference in the x-shift, squared, added to the square of the difference of y-shift, and taking the square root of the entire sum after you're done. Pythagoras in analytic geometry.

But there are other ways, namely, the L1-metric, which is simply the sum of the x-shift and the y-shift. Thus, in the xy-plane of the checkerboard floor I was walking on in my childhood, I was always taking steps of distance two (or, later, steps of even distance) in the L1 metric: one step forward, one step to the right. Or, I could take two steps forward: one forward, and another forward. Thus, on the crazy Regenstein sidewalk, I could consciously walk horizontally or vertically on every other step and be a-okay. Mr. Sally introduced this to me (in a completely different context, mind you, my first year in college).

Mr. Biss wrapped it all up my fourth year when I finally got around to taking point-set topology (for shame, Sudeep): the L1-metric on R2 (the xy-plane) forms a topology, and all "balls" of radius "r" on this space consists of the steps that I can take, the sum of which are less than radius r. Thus, a ball of one would consist of the square immediately in front, behind, to the left, and to the right. All my steps, therefore, consist of circles of radius 2, 4, 6,..., 2n, n in N, in R2, with metric L1.

Not that it helped me on the dance floor or anything (sadly), but it was still cool to think about.