May 12, 2011: It Has No Symmetry | May 11 | May 13 | 2011 | FOTD Home |
Fractal
visionaries
and enthusiasts:
Today's
image is a
view of the curious Julia rectangle as it appears in the Oblate plane,
which is determined by the real(Z) and imag(C) axes. When
viewed
from this angle and given an appropriate amount of stretching and
skewing, the rectangle pretty much retains its rectangular shape.
The name "It Has No Symmetry" accurately describes the image.
At
first glance, the image does indeed appear to have origin symmetry, but
a close inspection will reveal small discrepancies.
The rating of an 8 could have been a half point or so higher if I had
not re-used a previous color palette.
I find the inability of the human mind to visualize the fourth
dimension most curious. Computers have no problem converting
the
math of the fourth dimension to spatial coordinates and manipulating
the resulting spatial figures exactly as they do with everyday 3-D
figures. It would even be possible to construct a monitor
with a
3-D viewing surface, ready to be enjoyed by a 4-D viewer.
Unfortunately, a normal 3-D man with his 2-D retinas would be able to
view such a screen only from the 'edge-on' direction and would be
unable to enjoy the 4-D show any more than he could enjoy viewing a
normal flat screen edge on.
Why are we limited to visualizing three dimensions, when such
fascinating things as double rotation and undistorted Klein bottles
exist only one dimension higher. It is impossible for us to
imagine even a 2-D flatland or a 1-D line without automatically
mentally embedding these lesser spaces in a surrounding 3-D
space. The limitation is not due to the math, which merely
grows more tedious as the number of dimensions increases, with nothing
of a totally new nature appearing.
There is a theory, not entirely scientific, that our inability to
visualize 4-D figures is a limitation of our level of consciousness,
which is stuck at the 3-D level. This would imply that a
being
with a higher level of consciousness might be able to visualize the
fourth dimension, or might even interpret the sensory impressions of
the world he lives in as possessing four spatial dimensions.
If
so, he would not be around too long to enjoy his advantage.
Stable planetary orbits are impossible in a space of four dimensions
with its inverse-cube law, and if our math is correct, a planet in 4-D
space would either spiral into its star or spin off into interstellar
space.
But then again, maybe a 4-D man would have different laws.
The days keep getting better here at Fractal Central.
Wednesday
featured cloudless skies, gentle breezes and a cat-pleasing temperature
of 77F 25C. The fractal cats, totally pleased, spent the day
sleeping and dreaming of what they might do if they ever get to go
outside.
The humans had more mundane things to attend to. The next
FOTD
will be posted in 24 hours. It will likely be another view of
the
rectangle. There is still much yet to be discovered in the
vicinity. Until then, take care, and Nostradamus says the
world
will end in 589 days. It's got ME worried. I
thought we had
5-1/2 billion years.
Jim Muth
jamth@mindspring.com
START PARAMETER FILE=======================================
It_Has_No_Symmetry { ; time=0:03:04.77-SF5 on P4-2000
reset=2004 type=formula formulafile=basicer.frm
formulaname=SliceJulibrot4 passes=1 center-mag=0/0\
/1130/0.02072/0/77 params=0/90/90/90/-1.7435/0/\
0.00019/0.07388/2.003/0 float=y maxiter=15000
inside=0 logmap=62 symmetry=none periodicity=6
colors=000FDHFDJFELFCNFCPFCRGCSIETKGUMIXOK_QLbSNeU\
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EEzEEzDczDczDczDczDczDczDczDczCzzCzzCzzCzzCzzCzzCz\
zCzzBzzBzzBzzBzzBzzBzzBzzBzzBzzmzzmzzmzzmzzmzzmzzm\
zzmzzmzzmzzmzzmzzmzzmzzmzzmzzzzzzzzzzzzzzzzzzzzzzz\
zzzzzzzzzzzzzzzzzzzzzzzzz }
frm:SliceJulibrot4 {; draws all slices of Julibrot
pix=pixel, u=real(pix), v=imag(pix),
a=pi*real(p1*0.0055555555555556),
b=pi*imag(p1*0.0055555555555556),
g=pi*real(p2*0.0055555555555556),
d=pi*imag(p2*0.0055555555555556),
ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g),
sg=sin(g), cd=cos(d), sd=sin(d),
p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd),
q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd),
r=u*sg+v*ca*sb*cg, s=v*sin(a), esc=imag(p5)+9
c=p+flip(q)+p3, z=r+flip(s)+p4:
z=z^(real(p5))+c
|z|< esc }
END PARAMETER FILE=========================================