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.

Webpage and Flash Animation Developed by Aous Manshad
Last Modified: September 22, 2005