In this episode, a shooter attacks the FBI office and Charlie and the FBI team try to find out why.

At the beginning of the episode, Charlie
explains how to understand the connecting between a child molestor and
his victim by looking at the intersection of the areas that each person
uses. A very simple example is assuming that both the victim and
perpetrator move in circles of radii r_{1} and r_{2}
centered at
(x_{1},y_{1}) and (x_{2},y_{2}).

The following steps would allow you to find a formula to compute the
area intersection of the intersection of two circles, say
with radii r_{1},r_{2}
and centered at (x_{1},y_{1}) and (x_{2},y_{2}).

1. Draw a picture of two
circles with radii r_{1} and r_{2},
and centerd at
C_{1}=(x_{1},y_{1}) and C_{2}=(x_{2},y_{2}).
Draw the circles so the area of intersection
is not zero. Also, name the intersection points I_{1} and I_{2}.

2. What is the distance between P_{1} and P_{2}? Call
this distance D.

3. Find the angles I_{1}C_{1}I_{2} and I_{1}C_{2}I_{2}.
[Hint: Use the law of cosines to
get I_{1}C_{1}C_{2} and notice that I_{1}C_{1}I_{2}=2I_{1}C_{1}C_{2}.
Do likewise for I_{1}C_{2}I_{2}]

4. Make sure the angles are expressed in radians.

5. Find the area of the circular sectors I_{1}C_{1}I_{2}
and I_{1}C_{2}I_{2} [Hint:
Recall that the area of a circular sector is given by

6. Find the area of the triangles I

7. Deduce a formula for the area of the intersection of the two circles.

8. Agoat is tied to a point on the perimeter of a circular field of radius 8 ft. The length of the rope is 2 ft. What is the area of the field accessible to the goat?

Analizing the path taken by the shooter in the FBI building, Charlie noticed that the path describes was a Brownian path. Basically, Brownian motion refers to the random movement of a particle in a fluid. A nice computer simulation can be found in this link. This concept was introduced by the Scottish botanist Robert Brown (1773 - 1858) who used his microscope to look at pollen floating in water. He noticed that the pollen grains moved in a very unpredictable way, and the same occurred with other very small particles. He assumed that the movement was due to physical causes (and in fact it is now know that this unpredictable movement is due to the bombardment of the water molecules). Albert Einstein used Brownian motion to (indirectly) confirm the existance of molecules and atoms, while the mathematician Louis Bachelier used Brownian motion to analyze the stock market (stock options) in the first paper to used advanced mathematics to study finance.

Trying to understand the
perpetrator's path in the FBI rampage, Charlie used the concept of
loop-erased random walk. To describe the concept, suppose one has a
graph G and selects a path in
G; that is, a collection of vertices v_{1}, v_{2},
. . ., v_{n} such that
two consecutive vertices form an edge in G. This path may contain
loops (a path where the
initial and end point are the same). A loop-erased random walk in G is
the a path where the loops
have been removed chronologically.

- Consider the picture below the picture above, suppose a path is
given by 1,3,4,5,5. What would be the loop-erased walk
corresponding to that path?

- What if the path is 1,2,3,4,5,6,7,7,3,7?

- A
*spanning tree*of a graph G is a tree (a graph with no cycles) whose vertex set is the same as G’s. How many spanning trees does the graph above have?

As pointed out in the episode, a
tesseract is a 3-dimensional
representation of the 4 dimensional cube. A tesseract is an example of
a Schlegel diagram, invented
by the German mathematician Victor Schlegel** (1843-1905). A Schlegel diagram encodes a d-dimensional polytope (a polyhedron in more
than 3 dimensions) into a an object of dimension d - 1. To draw a Schlegel diagram,
follow the following steps:
1. Pick a face F and a point P "outside" the center of F (see picture
below).
2. Draw lines between the point P and all the vertices of the figure.
3. Mark the points where these lines intersect F.
4. Imagine F is made out of rubber bands, and move the vertices to
their intersection in F. The edges outside F will correspond to lines
connecting the intersections in F.
The resulting picture is the Schlegel diagram. The example below
describes the construction of the Schlegel diagram of the tetrahedron.
Here the edges outside F are A, B, and C, and they correspond to A, B,
and C, respectively, in the red figure. Notice that the actual Schlegel
diagram depends on our choice of F. In the case of the tethrahedron,
since all the faces have the same shape, the choice of F is irrelevant.
**

- Draw the Schlegel diagram of a 3-dimensional cube.
- Draw the Schlegel diagram of the triangular prism depicted
below.
First use one of the triangular faces, and then use one of the
rectangular ones. You should get different diagrams.

- Convince yourself that a tessaract is in fact the Schlegel diagram of the 4-dimensional cube. Looking back at 1. above may help.