Tuesday, 1 March 2011

Not My Kind of Geometry

I'm reading Alex Bellos' "Here's Looking at Euclid"... a catchy title, and yet... I'm a little put off by a few things. Maybe I've gotten picky in my old age, and maybe I'm not catching on to some literary style that I don't understand. Perhaps a few examples will explain. I apologize in advance if I come off too critical. Read the book and make your own decision, but read before you recommend it to a student.

All of these examples are from the third chapter, which he calls chapter two since he starts at chapter zero.  Okay, that is example one.... or maybe he would prefer I say that is example zero..but counting doesn't start at zero.   Too cutesy.  Or am I just being picky?

Then around page 60 he mentions centers of triangles; "In fact, there are four ways we can define the center of a triangle... " ???   I think Clark Kimberling has 1000+ on his triangle centers web site.  Part of what bothers me about this kind of language is that the book is not written for experts or even experienced mathematicians. The nature of much of the book seems far too simple for that. But if we are writing for young mathematicians, it would seem a more appropriate use of language might be "Four triangle centers that I will illustrate.." or "four of the many possible triangle centers"; or somethings similar that allows that there could be other choices. In fact, the four he uses omits the common incenter formed at the intersection of angle bisectors and includes the nine-point circle, which he calls the "midcircle", a name I had never heard applied to the circle (and have more frequently heard applied to a different circle). Of course, it would be hard to use the term nine-point circle if you only mentioned six of the points for which it was named. His selection leads to the "oh-wow" that they all lie on a single line, the Euler line.  It seems there is no problem switching from three centers that are the points of intersection of cevian, and then to one that is determined by points on the triangle. As the Sesame Street characters sing, "One of these things is not like the others..."

After a few pages on triangle centers, he drifts off into Origami. There are some interesting discussions of the different views of origami from the Easter and Western cultures, and then he introduces a rather unusual character, Kazuo Haga, whom he describes as an origami rebel. Why? Because traditional origami has only two first folds. Folding adjacent points of the square together to bisect the sheet parallel to a side, or to fold opposite corners together to form a crease on a diagonal. What has this "rebel" done to upset the origami world? He has folded a corner to a midpoint of one side.  Wait... think.... so how do you find the midpoint of a side? Well, you do one of the traditional folds which bisects the side...

Okay, so then he extends the "rebels" work to an amazing discovery. If you pick a random point along one of the edges and fold one corner on the opposite side to this point and crease, then fold the other corner on the opposite side to the same point and crease, behold the creases intersect... and they "always" intersect on the middle line of the paper... and the distance from the random point to this intersection is the same as the distance from the two corners of the paper that were folded up to meet it....

But wait, every kid who has played with folding paper in geometry class knows that when you fold one point onto another and crease, the crease is the perpendicular bisector of the segment joining the two points. If you do that two times you have found two of the perpendicular bisectors of a triangle formed by the random point and the points at the bottom corner of the page...and the intersection of the perpendicular bisectors meet at the circumcenter, a point equidistant from all three vertices and thus the center of the circle that just contains the triangle. And the third perpendicular bisector??? That's the one that is made by that conventional first fold that produces the line up the center of the paper where the creases "mysteriously" intersect.

It seems at times the book just gives away simple geometric ideas with no evidence (hence, no geometry).   Regiomontus' statue problem is presented in one short paragraph, then answered in the next by simply stating, "The angle is largest when a circle that goes through the top and bottom of the statue touches the dotted line (through the observer's eyes)..." By touches he must mean tangent.  There would be a circle through the three points anywhere the viewer stood..but again the language is a little too unspecific for my instructional taste. Why is that a solution? It's a beautiful problem to hook kids into more geometry. Why is the viewing angle (inscribed angle) the same from any point on the circle? Probably a good three page explanation on its own.

Perhaps my problem is that I think like a teacher and not like a popular author. I guess I would spend a chapter on just the beautiful concurrency theorems.  Euler's line gets another whole chapter of teasing out..and the variations of the statue problem, one more chapter. But like too many of today's textbooks, this books seems to send up a flare with a little light and a lot of noise and says, (the title of chapter three..oops sorry, two ) "Behold."  Brahmagupta presented a beautiful theorem with the single word, and it is a challenge to think. I fear that this chapter might better be titled BEWARE.
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