An Interesting Triangle Property

I'm considering this one for my "Almost Pythagorean" file. I came across this recently and found it interesting. The Pythagorean Theorem is actually a property about triangles with 90 degree angles; but this one is a property of all triangles that contain a 60 degree angle at vertex A.

The "Then" implied by the above is that

The derivation is not too difficult and might well be presented to a good pre-calc student as a challenge. If you don't want me to spoil it, stop reading now until you have tried it.... It makes me wonder what we could come up with if we started with a thirty-degree angle.

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Surprisingly Constant

Some things never change... and some of the things that don’t change are pretty remarkable. I was recently reminded of two that always seemed non-intuitive, and therefore quite pleasing. Both of these can be proved toward the end of a first year calculus course, and at least one can be explained with a little hand-waving, to a good pre-calc student.

The first is the simple fact that if you take a slice a sphere with a pair of parallel planes that are some distance, d, apart; the surface area of the part of the sphere between those planes is the same no matter where the slices are made. Near the “poles” or near the “equator”, the area is the same. It is nice to be able to tell students that this incredibly interesting fact was known to Archimedes in the first Century of the modern era when he proved that the surface area of a sphere is the same as the lateral area of a cylinder that contains the sphere.

This image is part of an explanation provided at a math site at the University of Regina

The essence of the explanation uses nothing beyond similar triangles. If we approximate the sphere’s surface between the slicing planes as a frustum of a cone, then the slant height (AP) can be shown to be in the same proportion to the height (AB)as the radius of the frustum (QP) is to the radius of the sphere(KP)). This image is also from the U Regina site mentioned above.Another nice thing about this relation is that when you set up the rather fearsome looking integral for the surface area, it reduces through simple algebra to a constant.




The second similarly interesting idea is about the volume of a shape I call a bead, for lack of a more precise term. The bead is the solid remaining when you drill through the center of a sphere. The unexpected constant here is that for any radius sphere you start with, the volume of the bead depends only on the height. And perhaps even less expected, is that the volume is the same as the volume of an un-drilled sphere with the same height (i.e., when 2r = h).

I have not found a simple explanation for pre-calc students that will explain this one. If you have one in your pocket, drop me a note.

RIP Benoit Mandelbrot

Benoît B. Mandelbrot, a maverick mathematician who developed an innovative theory of roughness and applied it to physics, biology, finance and many other fields, died on Thursday in Cambridge, Mass. He was 85. Here is a recent Ted Talk he gave about his work in Mathematics. Enjoy...

More about Timid Testers

I recently wrote about my disappointment with a very small part of Kaiser Fung's new book, "How Numbers Rule Your World." I want to point out again that I have really enjoyed the book. One of the things that makes it really interesting is the little asides of historical note that are just the kind of detail I love...
Case in point:
In the same chapter I spoke of in the last blog, Kaiser explains a little about the history of the lie detector.

William Marston, A Harvard-trained psychologist who was the first to relate truth telling and blood pressure variations, failed to popularize the concept in the early twentieth century, but he ultimately achieved immortality by creating the comic book heroine Wonder Woman, who not coincidentally wieled a Magic Lasso that "makes all who are encircled in it tell the truth".