This is another weird example of the bootstrapping I mentioned in my first
post. We were brainstorming a geometry class a few months ago when my friend
Ian asked me, “Can you draw a regular 3
After I mused about this Zen math koan for awhile, he
handed me a ruler and a protractor and said, “Suppose you had only these and
you needed to draw a regular 9-gon? What would you do?”
I told him I’d first figure out what each angle
would have to be. The total angle sum in a 9-gon is (9 - 2) x 180° (because if you draw
diagonals from one vertex of a 9-gon, you end up dividing it into 7 triangles,
and each triangle contributes 180°). That’s 1260°. If the 9-gon is regular,
then its 9 angles are all congruent, so they are each 1260/9 = 140°. To
construct it, I would pick my favorite length, go 140°, and copy the same
length. I would do this 8 times.
Then Ian said, “Okay, just do the same thing for a
I said, “So. . . . How would I figure out each
vertex angle when I don’t even know what it would look like?”
“Just use the same formula you used for the 9-gon: (n -2) x 180°, then divide by n to get each angle measure,” he said.
When I did this, I got 77.14°. Then I drew an
inch-long side, measured 77.14°, drew another inch-long side, measured another
77.14°, and just kept going. Eventually I got this:
Using this same method, for what values of n would you get a closed figure for an n-gon?
Here is the bootstrapping:
started with a definition of a regular
n-gon: a polygon with n congruent
sides and vertex angles.
discovered a rule and justified that it had to work: We proved that each vertex
angle had to be (n - 2) x 180°/n, by putting in all the triangles.
we generalized the rule to a new domain, namely, for fractional n. The only way was to extend the formula,
which works fine if n is a fraction. But
the formula can only be proved for whole-number values of n.
created new mathematical objects. Again, they didn’t have to be defined this
way, but it nicely continued the rule we had gotten fond of.
Another example of the justification structure of
the definition and the proven rule getting switched around and bootstrapped
when we want to generalize to a new domain. This phenomenon occurs when we give
ourselves the chance to express regularity in repeated reasoning (practice 8),
giving us formulas that can then be generalized, weirdly, to new domains.
Want to try a bunch of other fractional gons?
Rob Ely is an associate professor in the Department of Mathematics at the University of Idaho. He likes researching student reasoning and how it relates to historical reasoning, particularly with the infinite, infinitesimal, and algebra. He likes playing hammered dulcimer, bridge, and fetch.