**The Murray Polygon**
The entrance of the Jack Cole Building depicts a Murray Polygon. The Polygon
is a space filling curve in the style of other space filling curves proposed
by Peano and Hilbert. The Murray Polygon is a generalisation of Peano
Polygons capable of traversing arbitrary rectangular regions. The technique
developed by Professor Cole was called **mu**ltiple **ra**dix arithmetic, later
shortened to Murray.
Peano’s original definition
takes a point on the [0,1] interval and splits it into two real base-
three numbers by taking all the odd indexed digits in their sequential
order as the *x* value
and all the even ordered digits in their sequential order for the* y* value
to obtain the point (*x*,*y*).
*Murray integers* is a number system in which
each integer is represented as a sequence of digits
d_{n} d_{n-1} d_{n-2}….d_{1}
together with a sequence
r_{n} r_{n-1} r_{n-2}….r_{1} of
radices
such that for each* i* we have
0 ≤ d_{i} ≤ r_{i} - 1.
Professor Cole named the Murray
integers with odd radix *Murray-o* integers and those with
even radix
*Murray-e* integers. He then went on to show that any sequence
of Murray integers could be transformed to describe a polygon such
as the one shown opposite. This program can be generated by the S-algol
program, written by Professor Cole, with inputs
Number of x radices > 2
Number of y radices > 2
x radices > **3, 5 **
y radices > **5, 5**
The resultant curve
contains the Murray integers ranging from *1* to *375*.
All the points with integer coordinates within a rectangle 15 by 25
are traversed with all the steps between consecutive points being of
magnitude one. He named these curves Murray Polygons.
Professor Cole realised that these techniques were not just a mathematical
curiosity but could be applied to problems in computer graphics such
as compaction for storage, transmission and object identification.
This is now a recognised field in computer graphics known as fractal
compression. |