Math 300 Writing for Mathematics
In this essay you will try to describe some
counter-intuitive properties of infinite sets for a lay audience.
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.
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 Alpha Centauri 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 sports car.
Since the Starter Set is a subset of the Deluxe Set, it could be packed
in the same box used for the Deluxe 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 should include in your essay
some additional examples of your own creation or a discussion of one or more of
the following topics:
a)
The cubes in the sets 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.
Compute how many square feet of paint one would need to buy in order to paint
all the faces of all the cubes in the Starter Set.
b)
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 of the Deluxe Set.
Now suppose the cubes are stacked one on top of another so the stripes
along the edges lineup. As you argue in
the essay, this stripe would be arbitrarily long. Estimate how many square feet of special
paint would be needed to paint this stripe.
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 is due in class the second week of the term.
1. This assignment is a minor variant on earlier assignments of Baldwin, 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.