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.


Steven Colyer said...

"Maybe I've gotten picky in my old age"

First, you've survived as long as you have so I do believe you've earned the right. Second, what's with this "old" stuff? Are you 70? because you know how the world's greatest Demographers, the English, define "old", don't you?

Young = Age 0-35
Middle-Aged = 36-69
Old = 70 +

Which I've never understood, since I reckon a 36 yr old has more in common with a 33 year old than he/she does with someone 68.

As far as popular books go, you have a point. I think no book is perfect and no two people think the same way so you know what the solution to that is, don't you? Write your own. ;-)

I was at my local Borders Bookstore CLOSE-out sale yesterday and came this close to buying Hawking's "God Created the Integers." What do you think of that book? Lots of Euclid in there as well.

Pat's Blog said...

I like "....Integers" I think my Jeannie picked it up for me in a charity shop for a buck or something.. It is a nice resource for historical readings.
I think I'll hold out for the Okinawan view of aging. Maturity begins at 80.

Sue VanHattum said...

The Tsilagi (Cherokee) say you become an adult at 51. I've been grown for a few years now.

I like thinking about your comments, Pat. It would be great if you ever decided to write a book.

If you'd like to critique mine before it's in print, that would be super.

Anonymous said...

Not being a geometer, I can't comment on your distaste for the geometry coverage. But starting chapter numbering at 0 seems perfectly natural to me as a computer scientist. We usually start counting at zero.

As a bioinformatician, it amazes me to see how fearful biologists are of zero. DNA base positions around a start site are numbered +1, +2, ... and -1, -2, ..., with no base 0 (+1 and -1 are adjacent). I had a couple of sabbaticals in a med center building that had floors numbered -2, -1, 1, 2 (again, no 0).

I'm surprised to see a mathematician uncomfortable starting at zero though.

Pat's Blog said...

I think zero is a perfectly good number, but it is not an ordinal number.. perhaps this is the difference in view that leads to so many conflicts in the definition of "natural" numbers.

dan.mackinnon said...

Hi Pat,

Alex Bellos responds to those (few?!) who pointed out the missing triangle centers on his blog:


Bellos points out that he's not alone in underestimating the number of triangle centers.

Personally, I didn't learn about the multiplicity of triangle centers until second year university - I took a course that used Coxeter and Greitzer's "Geometry Revisited" as its text.

I thought you might like the quote from E.T. Bell that the text begins with:
"With a literature much vaster than those of algebra and arithmetic combined, and at least as extensive as that of analysis, geometry is a richer treasurehouse of more interesting and half-forgotten ghings, which a hurried generation has no leisure to enjoy, than any other division of mathematics."

Pat's Blog said...

Thanks for the comment, and the quote... I didn't want to suggest that everyone should know there are an infinite number of centers (after all, any point in the plane can be the intersection of three cevians).. but I just guess I had my expectations a little higher for an Oxford Math/Philosophy Grad who publishes... I hope it wasn't MY comment he was responding too...