Math
300 Writing for Mathematics Spring 2003 Baldwin Essay 1.
In this essay you will try to describe some
counter-intuitive properties of infinite sets for a lay audience.[1] Think of this essay as an article that might
appear in popular magazine devoted to science or as an article in a Sunday
paper. You may assume your audience has a background in precalculus mathematics
but no calculus.
As all essays in this course, your essay should have
a title, introduction, and summary.
The
Blocks Unlimited store sells various sets of toy blocks. One set, called the Deluxe Set, consists of
infinitely many cubes, the first of which is 1 ft. by 1ft. by 1ft., the second
cube has edge length 1/2 ft, the third has edge length 1/3 ft, and the nth
cube has edge length 1/n ft. A second
set of blocks, called the Starter Set, is a subset of the Deluxe Set. It consists of infinitely many cubes, the
first of which has edge length 1 ft, the second has edge length 1/2 ft, the
third has edge length 1/4 ft, and the nth cube in the Starter Set
has edge length 1/2n-1 ft.
Give
a convincing argument that if all the blocks in the Deluxe Set were stacked one
on top of the other, then the stack would extend beyond the orbit of the moon
but that it is possible to pack the Deluxe Set into a box that would be small
enough to easily fit inside the trunk of a sportscar. Since the Starter Set is a subset of the De Luxe Set, it could
be packed in the same box used for the De Luxe Set. Argue that if the cubes in the Starter Set were stacked one on
top of another, then the stack would be not very high at all and the exact
height of this stack can be determined.
The
examples given in the previous paragraph are to be discussed in detail in Essay
1. You might want to include in your
essay some additional examples of your own creation or a discussion of one or
more of the following optional topics:
a)
The cubes in the Deluxe Set are sold unpainted. The Blocks Unlimited store also sells a special paint that can be
used to paint these cubes. The paint is
special because it has zero thickness.
This paint is sold by the square foot.
Estimate how many square feet of paint one would need to buy in order to
paint all the faces of all the cubes in the Deluxe Set.
c)
If instead of painting the entire cube, suppose only a thin stripe is painted
on one edge of each cube. Now suppose
the cubes are stacked one on top of another so the stripes along the edges line
up. As argued in the essay, this stripe
would be arbitrarily long. Estimate how
many square feet of special paint would be needed to paint this stripe. Would an infinite number of square feet of
paint be needed?
d) Can
you explain the apparent paradoxes?
You
may assume your reader knows basic algebra and will remember, when reminded,
formulas for geometric series.
However, the reader does not know about definite integrals or about
divergent series. Thus, the challenge
in writing this essay is to justify the claims about these sets of blocks using
only the mathematics the reader already knows.
An
outline for this essay should be place in my mailbox or under my office door by
January 23, 2003 (note no class on the 20th). The outline will be returned with comments
on January 27. The first draft is due
on February 10rd and the final draft on February 24.
[1] This assignment is a minor variant on earlier assignments of Berman and Radford and is based eventually
on an exercise in “Writing in the teaching and learning of mathematics”, J. Meier and T. Rishel, MAA 1998.