Yesterday in church I asked myself this question: if I had a square piece of paper that I folded in half on an angle, and then folded the tip back to the crease, and then repeated that over and over, what equation would describe the placement of the tip after n folds?
If the square has sides of length equal to one, then the tip will always be on a diagonal line that has length equal to radical two. So I wanted to write an equation where I plugged in the number of folds and what I got back was how far along that diagonal line the tip would be.
Well, I'm not good enough of a mathematician to do that. Instead, all I could do was write an equation that requires you to know position from the previous fold. What I ended up with was this: the position (as measured along the diagonal line) can be represented by the fraction
where l is 2 to the (n-1) and k is the previous value of k, multiplied by 2, then with one either added (for even-number folds) or subtracted (for odd-number folds). So this would look like this:
This works for all n greater than zero. I can't quite figure out how to get it to work for n=0. I know the answer needs to be radical two. The denominator (l of zero) would be two raised to the negative first power, which is one half. So I need the numerator to equal one half. But now I need to know the position of the tip for fold (n-1), which seems like it should be negative radical two, but negative radical two multiplied by two and then added to one doesn't equal one half. So I gave up on being able to have my equation give the position for the zeroth fold.
Also, it appears this sequence is limiting to one third, though I don't remember how to solve that. Like I said, I'm not good enough of a mathematician to do that.
Meanwhile, my Sunday School class was defining "miracles" to be whatever they are currently experiencing in their lives so they didn't have to deal with the implications of Moroni 10:24. And thus we see why I set myself math problems during church: it's much harder to be charitable when you're paying attention (Fundamental Truth of Life #7).