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\centerline{\Large Mathematics 420, Professor Feldman}
\centerline{\Large Exam 1: Wallpaper symmetry}

We indicate points per question to show the relative weight of
each part to help students pace themselves.  Once we compute
raw scores we will rank the exams (from first to last) and assign
letter grades on the basis of rank (at least 10\% A's, at least
30 \% B's, at least 40 \% C's.)  Very possibly many student will
find that they do not have time to complete every question, but
this in itself should cause you no concern.

Question 1: (250 points)

You will find six wallpaper patterns attached to this exam.
Choose five of the six. (You will find them arranged in
order of increasing difficulty. 20 extra points if you
choose to do the last five.)
Then for each of the five you have choosen:

(a)  Mark a representative sample of the reflection lines,
glide reflection lines and gyration points.  Show reflections
as bold lines, glide reflections as dotted lines
and indicate the order of any gyration points by writing a
number next the point.

(b)  Outline a template for the pattern (and write the word
``template'' next to it);

(c) On the back of each sheet, describe in words the ``surface with cone
points and
corner points''
associated to the pattern;

(d) On the back of each sheet, give the notation for the symmetry type of
the pattern.

Question 2: (100 points)  

On the blackboard, find a sketch of a polyhedron that consists of a
regular hexagonal prism joined to a cube. 

The prism part by itself would have a regular hexagon for its ``floor''
and ``ceiling'' and six square ``walls.'' Here we remove one of those
walls to join the prism to a cube missing one of {\em its} walls.  You
will want to know that the angle measure of a corner of a regular hexagon
equals 120 degrees. 

(a) Count the number of vertices ($V$), edges ($E$) and faces ($F$)
in the resulting figure. (Do not count the missing interior wall!!)
Compute the Euler number of the surface.  You knew the answer you would
get in advance.
Say why.

(b)  Compute the local angle defect at each vertex.  Having done
that, compute to total angle defect.  Again, you knew the answer
you would get in advance, so say why.

(c)  Say we now declare one of the 4 ``free'' corners of the cube (those
that do not meet the prism) to be a cone point of order 3. 
Recompute the Euler number.  Recompute the total angle defect.

Question 3: (50 points)

Draw a wallpaper pattern with symmetry type $\ast 632$.
(For an extra 10 points, draw a pattern with symmetry type $22\times$
instead and say you are doing so.)

Question 4:  (100 points) 

Briefly explain the Gauss-Bonnet theorem as we have formulated it in our
course.  

(Describe the quantities about which the theorem speaks.
State the theorem.  Explain its significance to wallpaper 
patterns.  Sketch the proof of the theorem,
or just say something about why it is true.)

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