SOAR Homework Eleven
These homework problems are meant to expand your understanding of what
goes on during class. Any you turn in will be graded and returned to
you. Answers may or may not be posted on the web, depending on demand.

Finish the proof in class that there are only five Platonic solids
(with Euler number χ = v  e + f = 2). That is, given that
4r = v (2d + 2r  dr)
where d is the common degree of all the vertices, and r is the
common number of edges per face, prove that (d2)(r2) < 4.
Hint:
All we really need from the equation above is that 2d+2rdr >
0..


Calculate χ(P^{2} # P^{2} # P^{2}
# P^{2}) by drawing the same sort of picture we had in
class.

Describe a cell division of P^{2} #
· · · # P^{2} (n copies) in terms of v (the
number of vertices), e (the number of edges), and f (the number of
faces).

Use part (b) to calculate χ(P^{2} # · ·
· # P^{2}) (n copies).

For this problem, let (T^{2})^{n} = T^{2}
# · · · # T^{2} (n copies).

Calculate χ((T^{2})^{3}) by drawing a cell
division of (T^{2})^{3} as shown as dotted lines
in the crude diagram below (see pdf file for pictures).
(There should be eleven dotted ellipses indicating cuts.)

Calculate χ((T^{2})^{4}) by drawing a similar
cell division of (T^{2})^{4}.

How does adding an extra torus affect v, e, and f?

Use your answer to part (c) to write down v, e, and f for a
similar cell division for (T^{2})^{n}.

Use the previous question to calculate
χ((T^{2})^{n}).
These problems are also available as a pdf file.
Solutions
Please email Peter
if you are interested in answers or solutions for the web. Thanks.
SOAR Winter 2003 Course Homepage