Utah Population by County

I found the 2000 U.S. Census population data for each county in Utah last night (I guess it was early this morning). I used GEGraph to create a 3D bar chart of population by county. I did it several times to explore the options, and came up with two that I really liked (one I adjusted after importing it into Google Earth).

The one that I will say was naturally occurring was one where both the height and diameter of the cylinder are scaled relative to the distribution of the population across counties. This one really gives you a good sense of how small the populations of most of the counties are when compared with those counties on the Wasatch Front (Utah, Salt Lake, Davis, and Weber).

The second one had only the height of the cylinder changed with respect to population distribution. The diameters of all of them were the same. Once I got this imported into GE I selected each of the 29 individually and edited their points so that the shape was no longer cylindrical, but instead matched the general shape of the county itself. This view is nice when trying to read the data because the polyhedrons provide a solid background for viewing the text.

The .kml files for both of these are available, but you must email me for them, as I have not yet figured out where I can post them online. I have a couple of options, but they are down at the moment. When I find something I will update this post.

My next project is to figure out how to get all my iTunes library into Fathom. I can get the music in easy enough, but am looking at a way to include the podcasts and the movies as well.

Google Earth and Math

This morning I shared with my co-attendees here at PCMI some information about GoogleEarth (hereafter GE) and how I have used it in the classroom. I told them how my classes had looked for circular objects in and around Salt Lake City. We used these objects as our examples when we looked at calculating the area of a sector and the length of an arc. A couple of the circles we found allowed us to see some practical applications of these calculations.

As the day wore on I was playing with the polygon tool on GE and I noticed that a path drawn between two points did not line up with the side of a triangle that shares endpoints with the previously drawn path. As our discussion continued I learned a little more about what constitutes a polygon in spherical geometry (all sides are portions of great circles). We came to the conclusion that the polygons were not true polygons in the spherical mathematics sense. It does look like a path between two points will be the shortest distance spherically, but for some reason the paths and the polygons don't seem to use the same construction when dealing with segments between the two points.

I started browsing the web to find out what I could about GE and how it constructs its paths and polygons. I still have not found the answer to this, but I did run across some interesting software. GEPath (see GE Blog article here or go to download page here) allows you to export a set of placemarks from GE as a KML file and then open this file in GEPath and create a polygon. GEPath will also calculate the area and perimeter of the polygon. This turns out to work nicely.

Using this method, I created a spherical triangle. I then zoomed in to each of the corners and copy/pasted the pictures into Geometer's Sketchpad. I calculated the angles of the triangles and they sum to greater than 180 degrees (181.69 degrees). This type of activity can be used to show students the importance of understanding the importance of the assumptions that are made in problem solving, particularly in geometry.

The other program is GEGraph, which allows you to plot data in a 3D view in GE. I have not tried it yet, but hope to try it soon. I am particularly interested in how GEGraph data can be transferred to and from Fathom.