Higher Geometry

In an effort to improve my mind, I had started on Modern Geometries by James R. Smart.  It is a college level geometry book, so it's a step or two up from what I teach.  And I'm finding it a challenge.

I fear my brain is mush.  I don't know if this is from growing old, being tired, not doing serious math for a year.  Whatever the cause, this book is harder than I thought it would be.

Then again, it's fun.  Okay, I'm only on chapter 1 which deals with finite geometries.  These are geometries that have a limited number of points and lines and a very simple axiomatic system.  The first of these has only 3 points.  It's sounds very limited (and it is) but it is fascinating to see how much can be done and proven with this.  Another interesting part of this is Gino Fano was the first to study these 'miniature' geometries in the 1890's.  I would have thought this had been studied earlier.

A couple of people, teachers actually, have asked me why I bother to work on this higher geometry.  I told them, "Because it was there."  They just roll their eyes and leave it.  They wouldn't understand if I did explain why.  See, what I find interesting is that many high school teachers are not curious about their subjects.  I don't know why, because I do like math.

So, why bother?

• It is interesting in itself.
• Because a lot of geometry is abstract reasoning, this is a challenge for me since I tend towards computational things.
• The more I know about geometry and it's development, the more I can challenge some of my very bright students.
• The higher geometry links various other areas of math together.