Icosahedron
Task 1.
Students make straw icosahedra (see http://emmy.nmsu.edu/breakingaway/Lessons/straw/straw.html), and get familiar with an icosahedron’s vertices (12), faces (20), and edges (30). The instructor shows its shadows on a screen.
Task 2.
Simulate the rotations of an icosahedron on the TI-83 Plus.
An icosahedron has 12 vertices, so you do not want to re-enter the matrix [I] every time you want to start anew. So create a matrix [H] as a backup to matrix [I].
In this 3 by 12 matrix we keep the vertices of a regular icosahedron with edges of length 2. We approximate √(5) + 1 ≈ 3.24. (The computation of the values of this matrix is not part of this lesson. The student will see the icosahedron on the calculator display.)
[[0 0 0 0 2 -2 2 -2 3.24 3.24 -3.24 -3.24]
[H] = [3.24 3.24 -3.24 -3.24 0 0 0 0 2 -2 2 -2]
[2 -2 2 -2 3.24 3.24 -3.24 -3.24 0 0 0 0]]
This is a rather small figure, so to create [I], do
2.6[H]→ [I] ENTER
Matrix [J] has 30 columns, and it looks as follows:
[J] = [[1 3 5 7 9 11 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 7 7 6 6 8 8]
[2 4 6 8 10 12 5 6 9 11 7 8 9 11 5 6 10 12 7 8 10 12 9 10 9 10 11 12 11 12]]
Store angle values in A, B, and C (in degrees):
A is the amount of rotation around the x-axis.
B is the amount of rotation around the y-axis.
C is the amount of rotation around z-axis .
Remark.
More daring students may try to draw this icosahedron by hand. It is easier than it looks if you first draw 6 edges. They are parallel to the three axes of coordinates.
Now use the program OCTAHEDR to
rotate and display the drawing.