(Poorly) Applied Geometry Posted on February 15th, 2009 by Lisa Heldke

Lisa writes…

Last summer, our pal V.B. presented us with an applied geometry example that he uses to stymie his high school math students into paying attention to him. He presented it to us, a group of adults who’d bisected our last angle anywhere from thirty to sixty years earlier, and set us to work solving it. Most of his pupils, lounging on wicker chairs on a beautiful Maine afternoon, stopped chewing on their pencils before they’d even given the problem half a go. Not me; geometry-lover from way back (seriously), I struggled mightily for a very long time. I don’t recall, now, what the problem asked, but I do know that I managed to find the surface area of the ring that is formed when you draw one circle inside of another.

Turns out that particular piece of information was utterly irrelevant to the solution of the problem. The real answer could be found via some ridiculously easy but not necessarily intuitively evident process. I got partial credit for effort and creativity, but lost big on needless complexity and failing to find the answer.

Life lately has been presenting me with a host of these applied geometry problems. My scores aren’t getting any higher. (Interestingly, V.B. almost always seems to be around to grade my papers.)

Just this morning, I decided that the difference in volume between an eight inch cake pan and a nine inch cake pan wasn’t sufficient to warrant spooning out some of the dough in order to compensate for the fact that I had the smaller pan when I should have had the larger. No worries; the excess batter easily ran over the sides and onto the floor of the oven, where it burned and stank.

I might just as easily have spent five minutes with calculator and kitchen scale, carefully computing the percentage difference in volume and then removing that percentage of the weight of the dough-but my previous experiment in applied geometry seemed to have taught me that these things have a way of solving themselves, don’t bother with the calculations. On that occasion, I baked a perfectly successful circular pan of brownies for a games night-at V.B.’s house, as it turns out. The brownies were supposed to be baked in a nine-by-thirteen cake pan, which I don’t have, for some inexplicable reason. So I chose the biggest round pan I had, and figured “what the heck; must be close.”

“Not even close,” was V.B.’s pronouncement, upon finishing the calculation he did when I explained why we were having “round brownies.” (He whipped out his carpenter’s rule to measure the diameter of the pan-how cute is that? Ask me, next time you see me, what’s the chief thing you want to do to avoid rendering your carpenter’s rule inoperative. I know, because V.B. taught me.) The nine-by-thirteen was “way” bigger. The brownies, inexplicably, were not way thicker-and no, I hadn’t been dipping into the batter before putting it in the pan, thank you very much. I reached the only logical conclusion one could reach, from the fact that these brownies were just the right thickness; pan size doesn’t matter.

Then there’s the Lilac Project, an applied geometry problem if ever there was one. Well, okay, I’m the only one who sees it that way, but I absolutely am convinced I’m right, and all I have to do is find the right Renaissance painter, and they’re absolutely going to agree with me that this is all about vanishing points and all that cool perspective stuff that those guys were figuring out, which required them to use lots of tile floors in their paintings, just so you could see how well they’d figured it out.

We have a patch of lilacs in our front yard about twenty, maybe thirty feet long. (And no, thank you, I will not be taking my carpenter’s rule out there to find out just how long. I’m saving it so it doesn’t get broken.) The lilac “hedge” apparently used to grow all the way out to the road, but before we bought the house, someone chopped a four- or five-foot (leave me alone) swath along the road edge, to improve visibility for the car trying to turn left out of the adjacent driveway. Good idea, but it didn’t improve the visibility enough. First of all, the lilacs kept sending up volunteers, and second, the ground underneath (an old stone wall, it turns out) was just too high in the first place. Dramatic measures were called for.

I consulted V.B. “If I want to increase the visibility to the left, for the driver of a car sitting in the driveway adjacent to this patch of former lilacs, do I want to dig up a level patch-or does the part further away have to be lower or higher, to account for perspective?” I think I described it to him as a problem in “applied perspective”-surely a sub-branch of applied geometry-and I think I threw in “vanishing point.” I also appealed to the way the road rises as it moves off to the left. I don’t remember just what all I appealed to, but I made a very convincing case that this was a problem in applied geometry-a problem whose solution required the resources of just the sort of guy who knows how to stump a classful of ninth graders and how to keep the tip of your carpenter’s rule from breaking off.

V.B. was unimpressed. “This is not a problem in applied geometry,” I think were his exact words.

He’s wrong of course. Give me some time and I’ll have worked out why. I think it involves the surface area of a ring.

P.S. I’ve finally finished the Lilac Project, just in time to go back to Minnesota and not be able to enjoy the fruits (well, really the grass) of my labors. Turns out that it’s a good thing V.B. refused to help me calculate the precise incline to create in order to achieve maximum visibility. Applied geometry, with its lovely straight lines, its precise curves, its lovely Q.E.D. conclusions, was no match for eighty to one hundred and fifty square feet[1] of twisty, gnarled lilac roots, wrapped around big rocks,[2] medium rocks and small rocks. As Barb the Brief likes to say, there were three rocks for every dirt.

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[1] Boy, that’s a big difference in area, isn’t it? Good that I didn’t have to bake this thing!

[2] It turns out that God can make a rock so big that I can’t lift it. Apparently he was in the habit of it, even.