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Homework 8
Due April 5, 2006
1) Find a formula for the angle of an n-sided regular polygon. You will do this in two
steps:
a) The sum of the angles of a triangle is 180 degrees. With a diagonal slice, a square can
be cut into two triangles, each of which has an angle sum of 180 degrees, and so the angle
sum of a square is 180 + 180 = 360 degrees. If you have a pentagon, then with two
slices, you can cut it into three triangles, each of which has an angle sum of 180 degrees,
and so the angle sum of a pentagon is 180 + 180 + 180 = 540 degrees. Continue this
argument to get a formula for the angle sum of an n-sided polygon.
b) Now, knowing the angle sum of an n-sided polygon, and knowing that all the angles of
a regular polygon are equal, find a formula for an angle of an n-sided regular polygon.
You can check your formula against the cases you currently know (when n = 3, 4, 5, 6,
the angle is 60, 90, 108, and 120 degrees, respectively).
Now recall our five rules that a regular/semiregular tiling must follow (cheerfully retyped
for you here):
Rule 1) The angles meeting at a corner must add up to 360 degrees.
Rule 2) There must be at least three polygons meeting at a vertex and at most six.
Rule 3) No semiregular tiling can have four or more different types of polygons meeting
at a corner.
Rule 4) No semiregular tiling can have a notation of k.n.m where k is odd and n is
different from m.
Rule 5) No semiregular tiling can have a notation of 3.k.n.m unless k = m.
2) Use problem 1 to justify Rule 3.
3) Justify Rule 4 when k = 5. If you can do it for 5, then you should be able to convince
yourself that it is true for any odd number.
This is a partially filled out chart of notations that satisfy Rule 1:
Notation
3.3.3.3.3.3
3.3.3.4.4
3.3.4.3.4
4.4.4.4
# of polygons
6
5
5
5
4
4
4
4
4
4
4
Notation
6.6.6
4.8.8
3.7.42
# of polygons
3
3
3
3
3
3
3
3
3
3
All of these, I gave you in class, except for 3.7.42, which just happens to work (you can
check it with your formula from problem 1). Note that just because a notation satisfies
Rule 1, that doesn’t mean that the notation represents an actual tiling. It has to satisfy all
five rules.
4) Finish filling out the chart.
Hint 1: Remember that order matters: each notation with four numbers has a twin on the
list with the same four numbers, just in a different order.
Hint 2: You will only need to use regular polygons of size 3, 4, 5, 6, 8, 9, 10, 12, 15, 18,
20, and 24.
Hint 3: You have seven notations to find with three numbers in them. The first two will
have a repeated number. The next two will have a 4 in them. The final three will have a
3 in them.
5) Your table now has 21 entries for possible regular/semiregular tilings. Only eleven of
these are really tilings. Use Rules 4 and 5 to cross out the 10 bad notations (you will be
left with one of length 6, three of length 5, four of length 4, and three of length 3).
6) Of the eleven good notations, we have already drawn six of them in class. Draw a
sampling of the remaining five. To help you draw, I have included a page of regular
polygons, all with the same side length, on the web page.