Kanji and Linear Approximations

Firstly, I've noticed that nobody has submitted any ideas on that topology problem from the last post. I'll assume that you're all just stumped, as I am.

Anyway, I was making kanji cards today and I started thinking about some things (warning: math ahead). Let me just explain the process first. I have been making kanji cards from the paper in a sketch book left behind by my predecessor. If you are curious or would just like to hear me brag, I'd say I know around a thousand at this point. First, I draw a grid of 3 cm x 3 cm squares on the paper with the aid of a ruler, which works out to be 13 squares by 10 squares with some space left over, if I recall correctly. Then I write/draw a kanji in each square. Then I cut the squares out and write the readings and meaning on the back side.

So, I noticed while cutting squares out, which I imagine is obvious to anyone who has cut things out of a book before, that cutting along lines for a short distance is much easier than cutting along them for a long distance, especially when one is cutting parallel to the binding. It occurred to me that this is due to the fact that when we cut with scissors in this way, we have to lift the page, which actually means rotating it along the binding, so that the scissors can fit. When we do this, what we are actually doing is warping the plane (an approximation of a plane, if you want to be technical) of the paper. So, our straight line is no longer straight, but some weird curve, both because we try to hold the paper flat so as to cut, though it is angled up due to the binding, and because paper has nonzero mass, so the far end bends down due to gravity.

Scissors (鋏 [hasami] in Japanese, if you would like something more social-science based out of this post), however, are really only good at cutting in straight lines. So, what we are really doing here is linear approximation. If you don't get what I mean by that, you should brush up on your calculus. It makes sense, then, that our best bet in cutting is to make small cuts, which would approximate the curve more easily. Another little trick I noticed that is probably obvious to anyone else, though I had never thought of it, was to cut past the first line parallel to the binding when making the perpendicular-to-the-binding cuts. Then the curve is shorter and the stiffness of the paper allows for less bending, meaning our approximation is going to be better.

So, yeah.