clifford_a-_pickover_surfing_through_hyperspacebookfi-org

notes 227
Appendix C
1. For more information on the Banchoff Klein bottle, see Thomas Banchoff's
exquisite book Beyond the Third Dimension. More accurately, the bottle is actually constructed
from a figure-eight cylinder that passes through itself along a segment. To produce
the Banchoff Klein bottle, you twist the figure-eight cylinders as you bring the
two ends outward.
Appendix D
1. Quaternions define a 4-D space that contains the complex plane. As we stated
previously, quaternions can be represented in four dimensions by Q= a a + a { i + aj +
a } k where i,j and k are (like the imaginary number i) unit vectors in three orthogonal
directions; they are also perpendicular to the real axis. Note that to add or multiply two
quaternions, we treat them as polynomials in i, j, and k, but use the following rules to
deal with products: i 2 =j 2 = k 2 = — I, ij = — ji = k,jk = — kj = i, ki = — ik = — ik = j.
To produce the pattern in this section, "mathematical feedback loops" were used.
Here one simply iterates F(Q,q): Q —> Q 2 + q where Qis a 4-D quaternion and q is a
quaternion constant. The following computer algorithm for squaring a quaternion
involves keeping track of the four components in the formula.
ALGORITHM — Compute quaternion, main computation
Variables:
aO,al,a2,a3,rmu are the real, i, j, and k coefficients
Notes:
This is an "inner loop' used in the same spirit as in traditional
Julia set computations. No complex numbers are required for the
computation. Hold three of the coefficients constant and examine
the plane determined by the remaining two. This code runs in a
manner similar to other fractal-generating codes in which color
indicates divergence rate. rmu is a quaternion constant.
DO i = 1 to NumberOflterations
saveaO =
saveal =
aO*aO
aO*al
- al*al
+ al*aO
- a2*a2
+ a2*a3
- a3*a3
- a3*a2
+ rmuO;
+ rmul ;
savea2 =
savea3 =
aO*a2
aO*a3
- al*a3
+ al*a2
+ a2*aO
- a2*al
+ a3*al
+ a3*aO
+ rmu2 ;
+ rmu3 ;
aO=savead,- al=saveal; a2=savea2 ; a3=savea3 ;
if (al**2+a2**2+a3**2+aO**2) > CutoffSquared then leave loop;
end; /*i*/
PlotDot(Color(i)) ;