Lab 2

This lab has two parts.

• Part One is an introduction to a paper and student worksheets on more polyhedra related to the cube.
• Part Two is a Sketchpad lab about centers of mass.

Part 1. Build a Rhombic Dodecahedron with Zomes

Go to this link at the Math Forum and study these two papers:.

• An Amazing, Space Filling, Non-regular Tetrahedron
• Rhombic Dodecahedron - Hidden Within or Surrounding the Cube?

During the lab you can look over the papers, but the lab activity is to build the Zome model illustrated of the rhombic dodecahedron.

Part 2. Study Centers of Mass with GSP

Make ONE sketchpad file with multiple pages illustrating these points. You can add your work to this Lab 2 file

• Given a segment AB, and two lengths x and y, show how to divide the segment AB internally by a point U on segment AB so that AU/UB = y/x. Hint: Think parallel lines and trapezoids.
• On a new page, repeat the construction using Dilate in the Transform Menu.
• Is U the center of mass of a system with mass x at A and y at B or with mass y at A and x at B? Explain.
• Use coordinates to check that if we set M = x+y, then the center of mass U = (x/M)A + (y/M)B.

Now with 3 points, carry out these constructions. (Suggestion: Make a tool that will do the job. It can take two points and two measurements or quantities. If you don't know how to make a tool, ask.)

Given masses x, y, z at points A, B, C find the centers of mass of each of these pairs:

• x at A, y at B; call the center of mass U
• y at B, z at C; call the center of mass V
• z at C, x at A; call the center of mass W
• x at A, (y+z) at V; call the center of mass E
• y at B, (z+x) at W; call the center of mass F
• z at C, (x+y) at U; call the center of mass G

Later: What are the coordinates of these points?

Work through the copied handout on balance points.