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An Exercise: Philosophical or Mathematical?

Visualize or draw a triangle....

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Got it?


Odds are quite great that your imagined triangle looks like this....in fact, it is even positioned like this, with a horizontal base at the bottom:

And it is probably equilateral...and if not, then certainly isosceles...and if so, then also a right triangle.

I have tried this with a myriad of students, with the above claims usually confirmed.

The experiment seems to come from the philosopher John Locke's Essay Concerning Human Understanding, and his discussion of "an 'ideational' version of the denotation theory of meaning."

When asked to visualize, Locke suggests that the general term "triangle" encompasses what all triangles have in common. Their different and inconsistent "parts" have been abstracted and left out...making it "neither oblique nor rectangle, neither equilateral, equicrural, nor scalenon; but all of these and none at once" [IV, 7, ix]. Not sure what optins are left by Locke's reasoning...especially since my experiments show it to usually be an equilateral triangle!

In his column "Mathematical Entertainments" in The Mathematical Intelligencer (1992), David Gale differs with Locke. He claims that this "random" triangle is more likely to be obtuse than acute. Note: He is referring to the apex angle.

His logic for imagining or drawing a random triangle involves one of these three cases:

  • One needs to randomly pick three positive numbers for the angles (that sum to 180). Gale then claims it can be shown that three/fourths of these random triangles will be obtuse.
  • Or, one needs to pick three random side lengths (that meet triangle inequality conditions). Gale then claims it can be shown that pi/fourths of these random triangles will be obtuse.
  • Or, one needs to pick three random points on a unit circle (any triangle is inscribable,,,etc.). Gale then claims it can be shown once again that three/fourths of these random triangles will be obtuse.
I must confess that when I imagine a triangle, I "see" a triangle...and do not catch myself internally or unconsciously making the decisions Gale outlines. And, despite Gale's mathematical argument, almost all of the triangle's drawn by my "test subjects" (and perhaps yourself!) have an acute apex angle!

Perhaps one more instance of the mathematics being accurate and true on paper....but not in the real world!


Side note: Mike Naylor (Norway) adds another twist....given the problem of trying to draw all varieties of triangles, why is an acute scalene, non-right triangle so difficult? Mike's blog entry is written in Norwegian, but states that "maybe that makes it more fun to see if you can follow the argument." TAKK, MIKAEL!