Activities for Class 6

Today you are going to expand your knowledge of Maple's graphics capabilities. Also, you will learn how to take graphics that you create and put them on your home page on the World-Wide Web. First, here are some comments on the homework from last week.

• A nice image of the Borromean rings in three dimensions can be found at http://www.geom.umn.edu:80/graphics/pix/Video_Productions/Not_Knot/Rings.html. This image also appeared on the cover of the Notices of the American Mathematical Society for October 1996.

• There were several popular solutions to the problem about the polynomial that takes integer values at integer points. Recall the problem as formulated by Maple:

     > p:=unapply(interp([1,2,3,4],[2,3,5,9],n),n);
  
                                3        2
                 p := n -> 1/6 n  - 1/2 n  + 4/3 n + 1
  
      Show that p(n) is an integer when n is.
  

  1. Some people gave a proof by mathematical induction.
  2. Others observed that p(n) - n - 1 = n(n-1)(n-2)/6.
     • The right-hand side is an integer because the product of three consecutive integers is always divisible by 6 (since the product contains one term that is a multiple of 3 and at least one term that is a multiple of 2).
     • Alternatively, the right-hand side is an integer because it is a binomial coefficient: the number of ways to choose 3 objects from n objects (without regard to order).

  A couple of people generalized the problem. For example, a polynomial of degree n that takes integral values at n+1 consecutive integers necessarily takes integral values at all integers.

• There are oodles of links to information about Lewis Carroll at Joel M. Birenbaum's Lewis Carroll page.

• The home page for the Ninth ICTCM Conference is located at http://www.aw.com/he/ictcm/.

After we hear presentations from the people who did not get a chance to speak last time, we will go on to today's computer exercises on graphics. (That page will turn on after the talks.)

Comments to Harold P. Boas.
Created Sep 24, 1996. Last modified Oct 9, 1996.