The Geometry Forum: Coalition of Essential Schools - ces95

Day three - Cindy Schmalzried and the Stella Octangula


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The excitement level seems to be increasing exponentially day by day here at Swarthmore College. The Coalition teachers are really starting to come together as a group, and they're quickly catching on to the ins and outs of the Internet.

This morning we did more work with Netscape. Steve taught the group how to save images and a bit about html documents -- writing hypertext markup language, used to make text readable to Web browsers. We also learned to save images off the Net. In Netscape 1.1N, saving images has been made very easy. You can simply click and hold on an image, and you are given a menu of options including "Save this image" and "Copy this image." Each participant has begun construction of a Home Page. All in all, exploring the wonders of the WWW made for an enjoyable morning.

Stella!

The afternoon's talk was equally engaging. We were led in an activity about polyhedra by Cindy Schmalzried. Cindy is a Swarthmore alum, a teacher at Friends' Central, and a key participant in the Geometry Forum's predecessor, The Visual Geometry Project.

Cindy began by dividing us into four groups. She handed each group a box of different polyhedra and asked us to somehow classify our shapes. Some attention was paid to the actual construction materials -- Googolplex, Zaks, Polydrons, origami, plastic, and Polykit (no longer made) -- but the important thing was the SHAPE of the polyhedra.

When the groups discussed their findings, we found some important classifications of polyhedra. The first was regular polyhedra or Platonic solids. There are five such solids (tetrahedron, cube, etc.), and they are defined as such because their sides are identical, regular polygons (equilateral triangle, square, etc.), and each vertex is the same. We found lots of other solids whose faces were all the same, but they weren't Platonic because different things were happening at different vertices.

There were also Archimedean solids, whose vertices are the same but whose faces are composed of more than one regular polygon. Some examples are the truncated cube (faces are octahedrons and triangles) and the cuboctahedron (triangles and squares).

We also talked about relations between polyhedra, particularly duality. If you take a polyhedron and connect the points at the centers of its faces, you get another polyhedron that is the dual of the first. The dual of a cube is an octahedron, of a dodecahedron an icosahedron. The tetrahedron is dual to itself. In dual polyhedra, the number of vertices and faces flip-flops, while the number of edges stays the same.

Each group was given an activity to work on. The questions were fun and thought-provoking. The first group explored the construction of the stella octangula (a stellated octahedron). The second group looked at the volume of some of its parts, backing up known formulas with visual evidence. The third group played with color schemes on models of the stella octangula. The last group explored the relationship between the volumes of a full stella, its parts, and a cube, using three mirrors.

A lot was learned about geometry, and more importantly about activities that are fun and educational for high school geometry students.

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27 July 1995