COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

376 Harvey Cohn
In the case of Fig. 1, the only transformation was z’ = - 1 /z. Here,
however, there can be very many, but we set them up into a minimal number.
The computer program discovers the transformation
Z’(z) = (az+fwW+ 4
which maps z into its transform z. (or ZJ by listing the eight rational parts
of
a = a+a’*2’, /I = b+b’.2’, y = c+c’*2’, 6 = d+d’*2’ (4.5)
as well as listing zo, the transformed point (with z:) for each given point
on the floor. (To keep the machine program free from symmetrization,
sometimes z. was listed and sometimes zI, depending on whether (4.3) or
(4.4) happened to be theoretically correct.) The machine stored the transformations
as
a+64 a’+ . . . +646d/647d
so that repeating transformations can be assigned the same identification
numbers on each occurrence.
For each point of the floor subject to transformation L’(z), we have
11 yz+6 11 = 1 yz+s 12 1 y’z’+6’ I2 = 1 (4.6)
the analogue of 1~12 = 1 in Fig. 1. There can be several such surfaces meeting
at lower dimensional submanifolds of the floor but the pairing of
points is more important than the transformation which does the pairing.
(Thus, such banalities as round-off errors can change a transformation by
slightly shifting a point, but this is not important by itself.)
In the calculation pursued here, 34 different transformations Z occurred
and the machine assigned numbers from 1 to 34 in the order of occurrence.
We group them for later purposes in accordance with congruence classes
(mod 2) as in (2.7). They are as follows:
Congruent to Si:
z32 =
Congruent to S3:
Algebraic topology on bicomplex manifolds
z+;-::). z3=(y 4;):
~ 34
Congruent to S4:
Congruent to S,:
&s = ( -lr2’ m;+2;), 227 = (: -:I;):
377