# Geometrical Concepts from Constructions, Models, and Investigations Summary

## Thursday, July 11, 2002

We have an amusing group. It was suggested that PCMI should not only have tee-shirts with logos and pictures, but also PCMI underwear. We thought this would help with the problem of laundry facilities and time to do laundry. We will share this with the program committee!

We spent quite some time discussing what kind of products we would be producing as a group. We have divided ourselves into five groups.

Jerry, Susan, and Marilyn will be working on the write up of the coffee cup problem that we investigated on Monday, July 8, 2002. Jerry has already written a preliminary paper which they intend to prepare as an article to submit to a math journal. Jerry has posted this on the discussion group and is interested in suggested questions and critiques from members within the Geometry group and from outside the Geometry group.

Steve has already presented some of his work on multiplying Gaussian integers and factoring Gaussian integers using circles. He plans to continue working and perfecting this topic and hopes to submit it to the Mathematics Teacher (the NCTM journal).

Troy will continue to work on his project from last year involving balance points in triangles and other polygons using the vertex model, perimeter model, and area model. Jim will be working with Troy on this.

Celeste is working on Appolonian circle packing and is now using the April 21, 2001 Science News article as a reference. She will be writing up her discoveries for us.

Joyce and Peg will be working on writing up an article to be submitted to a math teacher's journal. It will be on generating the 2-D net for a 24 piece 3-D tetrahedral dissection of the Rhombic Dodecahedron (which is also a 48 piece dissection of the Stellated Rhombic Dodecahedron). This is the lesson that Joyce shared with the high school teachers during the 7:30 - 9:30 evening block.

We then used Polydrons to make regular tetrahedrons and octahedrons. We built stage 1, 2, 3, and 4 models like the kite from last week's parade entry. We then filled out a chart and worked through the numbers to calculate the number of up facing tetrahedra, octahedra, and down facing tetrahedra. This is the same information that Art shared with us in the chart write-up about the kite problem.

We also discussed using the program "Poly" with our classes since it gives the 2-D net, and a revolving 3-D picture of each of the Platonic Solids, the prisms and antiprisms, the Archimedian Solids, and the Johnson Solids. The program is available through their website at www.peda.com. We also discussed that the orientation of the octahedron when used in the kite model makes it appear as a triangular antiprism (which it also is!).

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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.