171: Something Euclid Missed

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Before we start, a quick reminder: if you like the podcast, don't forget to go to iTunes and post a good review! I know, I promised not to be one of THOSE podcasters, but I just noticed that Math Mutation has fallen off the iTunes "What's Hot" list of top Science and Medicine podcasts, and it would be nice to get back up there. Anyway, on to today's topic.

Here's an experiment for you. Take a piece of paper, and try using your abundant artistic skills to drawa triangle. Any triangle will do: it can be equilateral, isoceles, right, or none of the above, just the lines connecting three random dots on your paper. Now trisect each angle: draw lines 1/3 and 2/3 of the way across the angle at each corner. There should be three points near the middle of the triangle where pairs of adjacent trisectors intersect each other. Join these to form a little triangle in the center. Assuming you are sufficiently talented to be able to draw straight lines, you will notice something amazing: no matter what triangle you started with, the small one in the middle is equilateral, with all sides the same length and all angles at 60 degrees! How did this happen?

This little equilateral triangle is known as Morley's Triangle. You would think that such a simple trick, drawing lines within a triangle to get cool shape in the middle, would have been part of the classical geometry known since Euclid, but surprisingly, that's not the case. Euclid and his contemporaries may have missed this due to his tendency to concentrate on figures that could be constructed with compass and straightedge, since trisecting angles isn't directly possible with this type of technique. This triangle wasn't discovered until 1899, by Anglo-American mathematician Frank Morley at Haverford College. Morley was actually investigating complex properties of more general algebraic curves, and came across this triangle by accident. He didn't bother publishing it right away, though it spread by word-of-mouth until it eventually appeared in print as a problem in The Educational Times in 1908. He also showed it to his young son, who was fascinated by this magic triangle and later reminisced, "Always, to the eye at least, the theorem, if drawn accurately, proved itself. What caused me considerable annoyance was that I could not for a long time comprehend what purblind examiners might accept as a valid proof. " These recollections also hint that part of Morley's reluctance to publish may have come from the fact that the theorem seemed so simple and obvious (once drawn) that he was sure somebody must have already discoved it centuries ago. But the first actual publication of a proof was by two other mathematicians named Taylor and Marr in 1913, who acknowledged Morley in their paper.

Since Morley, numerous proofs have been discoverered of the theorem. Trying to guess at the intuition behind the Morley triangle, it occurred to me that 180 degrees is a special quantity for triangles, the sum of their angles, so it only makes sense that when messing around with trisected, or 1/3, angles, the quantity 60 degrees, which matches the angles at the corner of an equilateral triangle, would play a special role. Unfortunately, I haven't been able to find a proof that really connects to this intuition as to why this magical equilateral triangle appears. Many proofs are basically solving a set of trigonometric equations to figure out the relations of the lines and angles, fully valid and convincing but not providing much insight. Probably the cleanest proof I've seen online is one discovered in the late 20th century by Conway, where he basically assembles a bunch of small triangles with the right sides and angles, and shows how they fit together to form any larger triangle with an equilateral Morley triangle in the center. You can see nice illustrations of this at the links in the show notes.

But an even more surprising aspect of Morley's theorem is that it can be generalized to find other implied equilateral triangles lurking around. We've been talking about trisecting the interior angles of a triangle-- but what about the *exterior* ones? Actually, if you draw the exterior trisectors of each angle of the triangle, you can come up with yet more equilateral triangles, both by intersecting the exterior trisectors with each other, and intersecting interior and exterior trisectors. You can also come up with slightly different trisectors by adding 360 or 720 degrees to the size of an angle and then dividing by three, yielding yet more implied triangles. There are a total of 18 Morley triangles that can be constucted. One amusing article on the net, linked in the show notes, is from a math enthusiast who wrote a computer program trying to illustrate the central Morely triangle we started with, but due to a bug actually trisected the exterior angles in some cases... and was surprised to produce a equilateral Morley triangles anyway!

I think the coolest aspect of this whole Morley triangle concept is that we had a supposedly well-explored, solidly understood area of mathematics, the Euclidean geometry of planar triangles, and thousands of years later a new and unknown property was discovered. Just draw the trisectors of each corner of any triangle, and their points of intersection determine an equilateral triangle. The ancient Greeks could have discovered the Morley triangle and come up with a proof like Conway's, but somehow they didn't, despite some of them having a literally religious devotion to geometry. How many more surprises are lurking in what we today consider well-understood areas of math? Maybe someday a Math Mutation listener will be the one who discovers something new. Maybe even you.

And this has been your math mutation for today.

References:

• http://en.wikipedia.org/wiki/Morley%27s_trisector_theorem : Morley's theorem at Wikipedia
• http://www.mathpages.com/home/kmath376/kmath376.htm : An algebraic proof
• http://www.cut-the-knot.org/triangle/Morley/ : A nice Morley's Triangle site with an applet letting you draw an arbitrary triangle & illustrating the theorem
• http://www.haverford.edu/math/cgreene/399/morley/morley.html : Site with the 'bug' that uses an external trisector in some cases
• http://www.math.ubc.ca/~cass/courses/m308-02b/projects/hui/proof1.html : Conway's proof, with nice illustrations
• http://www.haverford.edu/math/cgreene/399/morley/morley.pdf : Detailed article on the theorem