Monthly Archives: June 2011

Geogebra Beta 5.0 and Riemann Spheres

I know that Geogebra 3D is not ready, but I couldn’t resist downloading to try it out in a class I am taking here at the University of Nebraska Lincoln.  We are talking about Riemann spheres and stereographic projections with the complex plain.  Cool stuff, but the book was no help in visualizing this so I knew it would be worth the risk to download.  This program will be amazing when it is released officially.  The only thing I wish it had was a simple way to intersect two three dimensional objects.  I am sure it is possible, but have yet to see how it is done.

In the image above I show the set Z of all points of projection for a line that does not pass through the origin on the complex plane.  In the program as you pull point A out to infinity, it is possible to watch the intersection circle Z get smaller as it approaches the pole of the sphere at point N.  Sorry to not clean up my description mathematically, but just to excited and thought I needed to share the amazing fact that this problem which was hard to visualize was like putty in my hands with this program.  Great job Markus and the Geogebra Team!!! Can’t wait for the official release.

Here is a copy of my write up for that problem.  Have not had much time to clean it up, but Geogebra makes for a nice presentation Schaben Riemann Spheres Stereographic Projection

The Math of a Leaking Swimming Pool

Our pool at Arapahoe leaks 1 inch in 12 hours.  Local people are outraged by the amount of water used.  Give an argument of why they should or should not be outraged.

Water is a hot topic here in southwest Nebraska.  Farmers are limited on the amount that they can use on their fields, and the cost of it can no longer be ignored for us city dwellers.  It has been told that the local swimming pool leaks 1 inch of water in a 12 hour period.  A community member asked me if this was a significant amount of water and wondered if anything should be done.  I split my students into groups, gave them all tape measures, and asked them the same question.  This time of year the pool is empty, so students can wander in and out of the pool and measure any portion of it they wanted.  I gave them no directions, but made it a competition.  The group with the closest answer and the best explanation would win their name on my wall.

I loaded up a bus full of kids (Having a bus permit has its advantages) and we ran down to the pool.  I gave them the entire period to take as many measurements as they wanted.  I also wanted the capacity of the pool while we were there.   The next day I graded tests and acted uninterested as they worked on the mathematics.  I had four groups and just listened from my desk to all discussions taking place.  The discussions were incredible.  They had several arguments break out.  The bottom of the pool was the most perplexing.  Groups were not sure how to handle the sloped bottom.  Two of the groups decided on the average depth.

After their calculations were completed, we found that the pool leaks about 16 acre-inces a year.  This is not very much water in the grand scheme of things.  The cost of pumping this much water would be small.

Next year I will probably develop this problem farther.  It was a good problem.  I am also thinking that all my teacher prep courses that told me not to use competition in the classroom were full of *&^%.  Competition seemed to make this group come alive.