Vol. 10 No 6 - Pi Mu Epsilon

468 PI MU EPSILON JOURNAL
VISUAL REPRESENTATION OF THE SEQUENCE SPACE 469
Define Yn: (En, d.,) -
(R", dG) by
y n (a 0
a 1
••• a.,_J = (s 0 , s 1
, ••• , sn_ 1 )
ai
w h ere s 1 = 1
.
2
We will prove that y n is an isometry and thus that its image is a model
of :E.. in (It", dG) •
For example, consider El . The image of L 1
points, as pictured in Figure l(a.).
l D D
Figure 1
contains only two
Now let us consider ~ . Its image consists of four points in the plane
arranged as shown in Figure l(b.). y 3 (L ) 3
is pictured in Figure l(c.).
Continuing this way we could easily model Ln in n-space. However this
n-dimensional model is more difficult to visualize for large n, and the
analogous model for :E would be a subspace of Reo, hardly an easy thing
to visualize. So we would like to find a way of "compressing" the entire
space into three or fewer dimensions. In fact we will be able to fit it into
two dimensions.
To do this we should first consider y 3 (L 3
). Think of it as a box sitting
on the floor in front of you. The box has length one unit, width one half
unit, and height one quarter unit. You can find the distance between any
two points in ~ by measuring the length of any shortest path which travels
along the edges of the box between the corresponding points in y 3 (L 3
)·
Now take your Swiss Army knife out of your pocket and cut off the 1 /2 by
1.4 sides. Step on the box gently. Don't bend the sides; let it bend at the
edges. You have just transformed a three dimensional model into a two
dimensional model. In an analogous manner we can "crush" our higher
dimensional models of :E.. into R 2 •
In this paper we develop a model which "crushes" :E into the real
e in an analogous manner. However, before we consider this model we
look at a model for a much simpler space to develop our intuition for
we mean by a "geometric model."
Modeling the Discrete Metric
To illustrate the kind of geometric model we have in mind, let us first
ider a similar model for a simpler metric space: the discrete metric on
a set X. Under the discrete metric the distance between any two distinct
- ts of X is one. Our model will have four key ingredients: 1) a set of
• ts, p (corresponding to the points of X),2) a set of line segments
viding paths between any two points),3) a definition of a unique
per" path between any two points, and 4) a metric induced on P (by
ring the Euclidean lengths of the line segments in the proper path
·een two points). In the case of the discrete metric we can define these
items as follows.
Points
Let the set of points, P C R 2 , consist of the n equally spaced points on
circle of radius 1 /z (that is, in polar coordinates
p = { ( t , 6) I t = ~ and 6 = 2 : , x E 0, 1, ... , n - 1})
shown in Figure 2(a.) for x = 8.
•
• •
• •
•
•
•
(a.) p
(b.) P and S
Figure 2 ·.