Vol. 10 No 6 - Pi Mu Epsilon

482
PI MU EPSILON JOURNAL
CHING FOR INFINITE FAMILIES OF 2-TRANSITIVE SPACES 483
Example 1. K 4
C=3
B=l
LENGTHS
AB =DC
AC • DB
AD= BC
Note that the actual distance does not matter, only which edges are
equilvalent, such as AB and DC. These edges can also be shown
numerically with AB = [0,1] and DC = [4,3]. Then the three different
types of edges represented here can each be described by the following sets:
= {[2i,2i + 1] mod 4 I i = 0,1,2,3}
= {[2i + 1,2i + 2] mod 41 i = 0,1,2,3}
= {[i 2 ,i 2 + 2] mod 4 I i = 0,1,2,3}
Acting on these sets of points are permutations known as similarities.
Definition. A similarity is a permutation on a set of points which
preserves the distance relations.
A similarity maps one point to another in a shape. For example, the points
of a rectangle can be rotated using the similarity j(X) = X + 1. Remember
that since we have only 4 points, we operate in mod 4.
Example 2. 1 rotation on K 4
0
1
f(x) = X + 1 (mod 4)
ft.O) = 1
.ftl) = 2
.ft2) = 3
.ft3) = 0
Note that while rotating the points, the similarity preserves the distance
relations; that is, the horizontal tops and the bottoms remain equivalent in
length, the vertical sides remain equivalent in length, and the diagonals
ranain equivalent in length. This quality of preserving distance relations is
essential for a space to be singly or doubly transitive.
Before we can begin to search for 2-transitive spaces we tnust
rstand the meaning of !-transitive.
Definition. A space is said to be ]-transitive if, given any point A in
space, there exists a similarity defined by the function/that sends A to
y point, say A', in the space.
This means that if we choose any point in the space, we can map it to
every other point in the space while preserving the distance relations.
If the similarity j(X) = X + 1 (mod 4) is applied to the rectangle three
times , each point is mapped to every other point while preserving the
distance relations each time. Thus, the rectangle is !-transitive.
We can expand this idea of !-transitive and move two points instead of
only one.
Definition. A finite set of points is said to be 2-transitive if, given any
two points (A and B) in the space and two images for these points (A' and
B'), there is a similarity/such thatf(A) =A' andf(B) = B'.
That is, if it is possible to choose any pair of points and map them to any
other two points in the space while preserving the distance relations, the
space is 2-transitive.
Before attempting to find examples of 2-transitive shapes, there are
certain conditions necessary for 2-transitivity which should be considered:
(1) the number of different edge lengths in a shape must divide the total
number of edges; (2) the number of different edge lengths at a point must
divide the total number of edges at that point. With the use of these two
conditions, it is much easier to narrow down which shapes are candidates
for 2-transitivity .
Now that we know of the necessary conditions for 2-transitivity, all we
need is a way to prove that something is indeed 2-transitive; we need a
theorem that states when a space is 2-transitive. Hence, the following very
crucial theorem developed from the definition of 2-transitivity. •
Theorem. (S, =) is 2-transitive if and only if