# Geometrical Concepts from Constructions, Models, and Investigations Summary

## Friday, July 5, 2002

Our group is rather unique (we think) in that we have so many varied interests that keep coming up. So, we discussed how to remain focused and produce a product or products while exploring all of the incredible problems posed by group members. In addition, the idea of bringing in experts in Geometry to work with our group was mentioned. Jim suggested that Roger could share things about the 9 point circle and pulling it into 3-space.

Celeste shared a fabulous conversation that she had at lunch with Jeff Lagarias who is a math researcher from AT&T Labs. She explained about Apollonian circle packing and will be putting together a write-up that can be linked to this entry. The idea is that if three circles touch, then there are exactly two other circles that also touch these three circles. Also, the curvature of all of these circles will be an integer and the center times the curvature is a Gaussian integer. Jeff Lagarias wrote an article on this for the April 2002 American Math Monthly magazine. [See other articles: American Mathemaical Monthly]

It was also noted that Jim King has written a book for Key Curriculum and Geometer's Sketchpad entitled Geometry Through The Circle.

Steve has been working on the relationship between factoring Gaussian integers, graphing in the complex plane, and circles. He has been using lattice points to find factors of 20. All of the factors of 20 seem to lie on chords of the circle. He will pursue this and give us a document to link to this entry.

We talked about pursuing Philip Mallinson's notes, but agreed that we weren't as interested in this as our project now that Philip will not be able to come to PCMI this summer.

So, we discussed narrowing down our ideas to workable projects by Tuesday.

Jerry brought up the origami program from the 11:00 to 12:00 hour and expressed an interest in pursuing that. Troy is working on a dynamic sketch of the problem using Geometer's Sketchpad which he will save to the shared folder on the server. Jerry also mentioned a problem that she is planning to write up for the California Communicator which uses adding machine tape to fold triangles that converge towards an equilateral triangle. This is done by continually bisecting the obtuse angle of each successive triangle.

Troy brought in an article entitled The Tangencies of Circles and of Spheres by Benjamin Alvord (written in January 1855 and published through the Smithsonian Contributions to Knowledge.

Other resources that were mentioned today were: Mabel Sykes' book Sourcebook of Problems for Geometry; Hidden Connections, Double Meanings, Herb Clemens book Geometry for the Classroom; Rethinking Proof from Key Curriculum Press; and a new computer program called Cinderella that allows you to do Euclidian, Hyperbolic, and Elliptical Geometry.

We ended with some math jokes.

Here lies Euclid, or at least his elements.
Here lies Fermat, but there's not enough room on his headstone for an epitaph.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.