# Chapter 4: Reaction and Exercises

## Alexis Nogelo

Reaction:

This chapter dealt primarily with drawing cubes, hypercubes and their shadows, as well as counting various elements of higher-dimensional cubes and simplexes such as edges, vertices, faces and n-cubes. There was also quite a thorough treatment of one application of our class's newly acquired methods of slicing and projection that is used by the Geological Sciences Department at Brown to study the climate changes over thousands of years in an area of the Midwestern United States. I enjoyed reading a more detailed analysis of how one can count all of the elements of higher-dimensional objects, since the mathematical beauty in the patterns is significant. I also appreciated the small reference to group theory, because I need those refreshers to remind me to consider the geometric connections to algebra that I have studied. (Or vice versa--algebraic elaborations on geometry I am studying.)

I do have one request to ask of Professor Banchoff--could we view some more movies in class? I would like to see The Hypercube: Projections and Slicing as well as some more of your world-famous geometric films.

Exercises:

Familiarize yourself with counting using combinations. (a)In how many ways can a teacher arrange 28 students into groups with 4 students per group? Suppose there are 12 girls and 16 boys in the class. (b)In how many ways can the teacher form 4 groups with 3 girls and 4 boys in each group? (c)In how many ways can just one group of 4 students be formed in a class of 28? (d)How about the teacher's options for the first of the coed groups of 7 students?

I know that we are to do our own exercises, but I was losing confidence in my solutions to these problems and would like to check them outside of the CIT where the air is more clear and I will be less likely to make a foolish mistake. I will add my answers A.S.A.P.

Alexis Nogelo