Material Prep Assignment for Friday, January 27

Bring to class Friday:

• a model of a cube with coordinate labels on the interesting points, as spelled out below
• a loose-fitting cubical box for the cube with an opening top with the same labels

Either bring two models yourself or team with one other person to bring two that work together (one each).

Specifically, assume that the sides of the cube are parallel to coordinate planes and that the center of the cube is the origin. For size, you can assume that one vertex is (1, 1, 1) so the edge length is 2.

On your cube and on your box label these 26 points.

• Each vertex with its coordinates
• The center of each face with its coordinates
• The midpoint of each edge with its coordinates

Mental Prep Assignment for Friday, January 27

Review 2x2 and 3x3 matrices. Specifically,

1. Be able to multiply two matrices.
2. Be able to find the determinant of a 2x2 or 3x3 matrix, especially the determinant of a matrix with lots of zeroes by permuting rows or columns.
3. Be able to write down the matrix of a transformation T and the formula for T(x,y,z) if you know the image by T of e1 = (1,0,0), e2 = (0,1,0), e3 = (0,0,1).

CREDIT for the Models

Those who bring the models to class ready to work will be recorded and awarded points. They will also be in a position to do the task and learn the math.

Those who do the mental prep will be rewarded by being able to do the task and to achieve the learning goals for the day.

The Task in Class Friday

Your task Friday will be to work in pairs to deduce the matrices of some isometries of the cube. To do this, you will align the cube with the box so that same labels are the same. Then apply some isometry to the cube inside and read off the images of e1, e2, e3 on the cube relative to the box, and thus deduce the matrix of the isometry.

Also, you will be asked to investigate some examples of composing two symmetries of the cube.