### Wednesday, October 27, 2004

Then, out of a manhole...

On a recent airline trip, I got to talking to my neighbor, who was traveling westward for consulting interviews. She posed the following sample interview question: why are manhole covers round?

A few answers have been suggested (feel free to take a moment to come up with your own!).
One is that round manhole covers can be tossed (cavalierly, contributes D.H.) back onto the manhole without regard to orientation.
Another is that roundness promotes ease of transport, as the heavy cover can be rolled from place to place. On the other hand, one risks loss of control, especially near hills.
Yet another suggested response (what I call the "legal realist" answer) is that they are round because they are manhole covers. In other words, the first manholes were round, and since then roundness has become a definitional feature.
Or perhaps too many lawsuits were filed involving people being poked by sharp, protruding corners.

One more: D.H. cleverly suggested that round manhole covers cannot fall through the manhole (assuming a small rim to hold the cover up). So on a cool October evening, one almost-Ph.D. and one future tax lawyer trained in pure mathematics spent a few minutes (okay, maybe more than a few) trying to figure out whether this holds true for other shapes. Disregarding for the moment all irregular polygons (thus preserving at least some of the rotational benefits of Answer #1), we looked at a few sample regular polygons.

Consider the square. Because a square's diagonal is longer than its side, one could (accidentally, of course) drop a square manhole cover, vertically, side first, into its square hole if the cover were dropped through the diagonal. Similar results may be achieved with hexagons, octagons, and even pentagons (though this requires dropping through an off-center line connecting two non-adjacent vertices). Of particular puzzlement was the equilateral triangle. But indeed, an equilateral triangle may be dropped, with one side perfectly vertical, straight down into a hole just next to one of its sides. Diagrams below (acknowledgements to D.H. for her assistance, particularly with the octagonal manhole.)

Today's entry brought to you by the letter Z and the color blue.
With sincerest apologies to any consulting firms or other businesses which may be forced to concoct new questions.

Update: The Original J.W. suggests the following proof of the Manhole Conjecture for Regular Polygons: see here. Turning to other sorts of shapes, however, K.F. posits a generalization of our circle argument. He points out that the inability to fall through a manhole (assuming a rim of positive though negligible width) holds true more generally for ovals of constant width, of which circles are but one type. If you're interested, explanations here and here. (Apparently some British coins have this property -- neat.) Whee!

### Monday, October 04, 2004

Went apple-picking on Sunday with a few friends here -- raspberry-picking too. Fun, I thought -- a gorgeous fall day, ripe red apples clustered on drooping branches, little kids everywhere. Macintosh, Golden Delicious, Stayman, Fuji... now the question is what to do with them all! At the moment, there are three distinct varieties of apple pie-type dessert in the apartment: a quarter of the tarte tatin I made last night, apple pie with cheddar cheese crust (three of them! courtesy of roommate and a couple of other friends -- all I did was eat some ;-) ), and a totally improvised batch of deep-dish raspberry-applesauce tartlets I haven't tasted yet but concocted when I realized I still had quite a few apples left. That, plus a batch of applesauce in the freezer, last night's first version of the experimental raspberry-apple tartlet in my tummy, the last of the apple doughnuts and cider we bought at the orchard, and six apples still unused. So all in all, it's been a good couple of days for cooking. I've been having a good time improvising with kitchen tools (a non-stick frying pan can mimic a cast-iron skillet, if you wrap the handle in foil and don't turn the oven up too high; little ramekins make great mini-pie molds). And I feel oddly housewife-like, finding creative ways to preserve fruit for the cold months ahead. Funny how things come up -- we were talking in one of my classes today about how the English peasantry used beer-making as a way to preserve their excess grain. Um... anyone like some pie?

##### © Paula Levy
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