Fractal visionaries and enthusiasts: 

Those who have no interest in the fourth dimension should skip to the start of the fractal stuff 12 paragraphs below.  The image is pretty good, at least in my humble opinion.

The fourth spatial dimension, where the Julibrots live, holds many things that defy common sense when they are squeezed into our everyday three dimensional space.  One of the most curious is the strange motion known as double rotation, in which a 4-D object such as a hypersphere may rotate around two absolutely perpendicular planes at the same time, with each plane rotating on itself independently of the motion of the other plane. 

In 3-D space, when two rotations are applied to an object, the rotations combine into a single resultant rotation, but a double-rotating 4-D hyper-object is in an entirely different condition.  This strange motion is impossible to visualize in 3-D space, but with some effort I have worked up a crude visualization of a hypersphere subject to double rotation.  Hopefully, my description will communicate at least some of my visualization.

The trick is to rotate the view in 4-D space.  I begin by visualizing a 3-D sphere like the earth.  On the curved 2-D surface of this 3-D sphere, it is possible to move away from the equator at a right angle in only two directions.  On the earth these directions are straight north and south, and following these two geodesics one will reach the polar points of the rotating sphere.

But on the curved 3-D surface of a 4-D hypersphere it is possible to move away from the equatorial circle at a right angle in any direction in a plane perpendicular to the equatorial circle.  Regardless of the direction one chooses, if he sets out at 90 degrees to the equator and follows a straight geodesic line on the surface of the hypersphere, he will intersect the polar circle.  The direction in which he sets out determines the point of the polar circle he will intersect.

I now drop a dimension and rotate the view 90 degrees.  The equatorial circle vanishes except for the point at which the explorer is standing.  He now sees himself as standing at a pole of the sphere.  The great circle he sees circling the spherical slice of the hypersphere at 90 degrees from his location is the actual polar circle.  This is as close as I can come to visualizing a hypersphere.  I do it one 3-D slice at a time.

Rotating back so that the entire equatorial circle is again in view, the explorer sees that the visible part of the hypersphere is rotating exactly as the earth does, with the equatorial circle rotating on itself.  He now is observing a 3-D slice of a hypersphere in a state of a single rotation, and though he cannot see it, he knows that every point of the the polar axis-plane is remaining stationary while it turns in place.  Now the fun begins.

There is nothing to prevent the polar axis from rotating on itself also, with the equatorial circle as its axis, and when it does, the two rotations are totally independent.  The resulting motion of the hypersphere is called double rotation and like everything else in 4-D space, it is impossible to visualize.

But to make an effort at visualization, let's assume that, just as on earth, the equatorial circle is rotating on itself once every 24 hours.  So far, so good.  But now let's assume that the polar circle is rotating on itself once every hour.  While standing very near the equator, the explorer finds his location on the hypersphere rotating around the closest point of the equator while being carried forward with the equator and tracing out a kind of helix.

The explorer sees the point where he stands rotating around the equator 24 times every time the equator makes one revolution on itself, but the point is not tracing out a simple 3-D helix looped into a ring, only a close approximation.  The farther from the equatorial circle he wanders, the stranger the motion becomes.  If the explorer were standing very close to the polar circle, he would find the point rotating around the entire hypersphere 24 times before it rotated around the polar circle once, tracing out something like a twisted garden hose, and if he stationed himself at a point halfway between the two axis circles, the point would trace out something that could be visualized only by a being with 4-D vision.  The incredible part is that the hypersphere remains rigid while subject to double rotation, undergoing no strain or distortion.

A vague idea of the motion of double rotation can be found by observing the 3-D shadow of a double-rotating skeletal 4-D hypercube.  Such projections are probably on the internet, though I have not yet searched.

The fun does not end here however.  In six-dimensional space, triple rotation is possible, but that's a story for another day.

Now let's get to today's fractal.

To create today's image I cut the portions of Z of yesterday's formula in half.  This keeps the critical point unchanged.  The resulting parent Mandeloid is larger than yesterday's, with more prominent 'wings'.

Today's scene lies near a minibrot northeast of the north wing of its parent.  This minibrot is connected to the main body of the parent by a broken filament, but unlike almost all minibrots, which are connected to the main body at their East Valleys, this minibrot is connected at the southern branch of its Seahorse Valley.

I named the image "Vision in the Sky".  Something about it reminds me of a dream I had long ago, where I saw a similar object in the sky.  The math interest rates only a 6, but the artistic value is a superior 9.

The rather lengthy calculation time of 4-1/2 minutes is a drawback.  The web sites will remove this problem however.

After a night that brought 3.25 inches, 8cm of rain and some minor flooding, today dawned cloudy with light rain still falling.  The rain ended before noon and the sky brightened in the afternoon, but the sun failed to appear.  The temperature of 64F 18C gave no cause for complaint.  The fractal cat, who does not like water, spent the day checking the outside conditions and sleeping under the chair cover.  The humans were thankful that the heavy rain resulted in only a few small puddles in the fractal basement.

The next FOTD fractal will be posted in the proper time, October 13 is the best guess.  Until whenever, take care, and don't be concerned if you fail to understand what I have written say about the fourth dimension and double rotation.  I don't understand it myself.

Jim Muth
jimmuth@earthlink.net


START PARAMETER FILE=======================================

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END PARAMETER FILE=========================================