May 24, 2011: Bubbles in Numbers | May 23 | May 25 | 2011 | FOTD Home |
Fractal
visionaries
and enthusiasts:
When
certain people have trouble sleeping, they sometimes count
sheep. When I can't sleep however, I begin mentally exploring
an imaginary four-dimensional world. My imaginary world is
truly four-dimensional; the problem is that I can mentally explore it
only one 3-D slice at a time.
As our planet earth has a curved two-dimensional surface, my 4-D world
has a curved three-dimensional hypersurface. On earth, two
coordinates, latitude and longitude, are necessary to position a point;
on my 4-D world, latitude, longitude and lithitude, (my own invention),
are required. Moving in the lithitude direction does not take
us into the sky or underground; it moves us in the third dimension,
flat along the hypersurface.
To reach a given location on this 4-D world, we start at the
zero-zero-zero point on the equator, which is a great circle that
circles the planet just as earth's equator does, and from there we
travel until we reach the correct longitude.
Now we find ourselves with a problem. Before we can set off
toward the pole, we need to determine in which direction to
go. On earth there are two directions, north and south,
leading toward the two polar points; on my 4-D world there is a full
360-degree circle of directions extending from the equator, each
direction leading to a distinct point on the polar circle.
The direction in which we must travel away from the equator is the
lithitude. Once we have chosen the lithitude, we need the
latitude coordinate, which tells us how far to go toward the polar
circle in the direction we are traveling before we reach our
destination.
But even stranger things than this happen on my planet. There
is also the matter of rotation. On earth, rotation is a
simple thing -- the planet rotates on its axis, a line that extends
through the globe from pole to pole. But on my 4-D planet,
the axis of rotation is a plane that cuts the hypersurface along the
polar circle, a curiosity that causes complications.
The problem arises because the polar plane exists in two dimensions
that are stationary even as the planet rotates around it. The
polar plane is therefore free to rotate on itself, with the equator as
the axis of its rotation. And in a universe with 4-D gravity,
it is unlikely in the extreme that the polar plane does not have at
least some degree of rotation. The 4-D planet is subject to
two independent rotations at the same time. Every point on
the hypersurface as well as every point within the planet is tracing
out what is known as a surface of double revolution, a kind of 4-D
donut.
Things on the surface of my planet move in paths resembling spirals,
which curve back on themselves into closed loops. This means
that the stars in the hypersky must move in helixes. What fun
it must be to set up camp in some dark spot on the hypersurface of my
hyperplanet and point a hypertelescope at the hyperstars and
hyperplanets. The hypertelescopic clock-drive would be a
complex device, but I'm sure that it could be made.
If the two rotations are equal, things get even more complicated, but
this is a story for another day. For now, we have a fractal
to investigate.
Today's image lies in the parent fractal that comes about when Z^(1.25)
is subtracted from Z^(-1.75) before (1/C) is added on each
iteration. This parent is more a field of debris with two
large Mandeloids than a single object. Today's scene is
located in some debris lying between the two largest Mandeloids.
There is not much to be said about the fractal, so I'll let it speak
for itself in visual terms.
The name "Bubbles in Numbers" refers to the bubbly shapes in today's
image and also to the fact that fractals are made of numbers.
The rating of a 6 indicates that the image has
room for improvement.
There were too many clouds at Fractal Central on Monday for the day to
be considered a good one. The temperature of 75F 24C was fair
enough however. The fractal cats spent the day plotting
mischief, but fell asleep before they could put their plot into
action. In the afternoon a tornado warning came over the
radio, but the storm passed over with only a strong wind squall and a
torrential rain.
The day of the humans was one more in an unending series of similar
days. But the older one grows, the more such uneventful days
are appreciated. The next in the unending series of FOTD
fractals will be posted in 24 hours. Until then, take care,
and when do we get to see the all of fourth spatial dimension in one
moment of time?
Jim Muth
jimmuth@earthlink.net
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