Poll

What would you consider the 3d analog to a spiral as it appears in a Mandelbrot

2d spirals in multiple 3-d vector directions into the center (like a star)
1 (100%)
pathes with 2 orthogonal rotation momentums (some complex motion)
0 (0%)
Something else
0 (0%)

Total Members Voted: 1

Voting closed: June 16, 2020, 07:40:39 PM

 Spiral in 3D

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Offline jukzi

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« on: June 06, 2020, 07:40:39 PM »
As you might know i want to construct a 3d mandlebrot. Now i will model the spirals.
In a 2d Mandelbrot there are basically n spiral Pathes on top of each side mandelbrot.
In 3d i expect somehow n² or m*n spirals pathes somehow meeting at a singularity (probably m in some sensefull relation to n. Coprime integers?).
Whats your expectation?
Any mathematical hint?

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/spiral-in-3d/3561/

Offline hobold

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« Reply #1 on: June 06, 2020, 09:01:52 PM »
I don't know what it is. Options 1 and 2 have the usual problem: they lead to poles and other irregularities. I once played with a variant of "something else", but didn't follow through:

https://fractalforums.org/fractal-mathematics-and-new-theories/28/new-coordinate-system-sphere-wrapped-in-a-noodle/190/msg1061#msg1061

Offline C0ryMcG

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« Reply #2 on: June 07, 2020, 12:15:43 AM »
First of all, I have to admit I don't understand what's being described in the options in the pole... Perhaps I'm too visual in my thinking to pick it up without a picture example.

But I do have thoughts on 3d spirals, unfortunately not backed up by any rigorous investigations. But I've spent a long time thinking about this problem, which comes from many days of when I was working in a cafe that was doomed to fail, and therefore having nothing better to do for hours at a time than to wonder about geometry.

I suspect that a spiral that fits the properties of 2d spirals can only exist in even-numbered dimensions. There are shapes in 3d that are similar, but they're almost all based on 2d spirals extruded or manipulated in some way. For example, a spiral that moves along the y axis at the same rate as it's radius increases will create a 2d spiral projected on a cone. (depending on the type of spiral, I guess).
A snail shell would be an example of this, with the 2d spiral being the path of a sphere-sweep that increases in radius as well as Y direction, and intersects.
A rolled up plane is another option, which is simply a spiral that's been extruded infinitely.

A proper spiral equivalent, I think, (at least for logarithmic spirals) would repeat both by scaling and by rotating, and it would behave like a spiral from any angle. It would also resemble a sphere with smooth and constant curvature changes. Any of the sphere-sweep style spirals I can think of have an axis, like the stem down the center of a snail shell.

I think this is possible in 4 dimensions because 3d extrusion of a spiral could be extruded in a spiraly shape, if it didn't self-intersect in the process... Making use of a 4th dimension would allow this, as far as I can tell. Imagine we take a spiral that's int he x/z plane, and extrude it along a curve... and that curve is a spiral in the w/y plane.
I can't picture this clearly enough to predict anything about dimensions 5 and 6... but intuitively even numbered dimensions make sense.

The closest that the third dimension can do, I think, is a helix, but that still feels like my first example manipulation of a spiral with specific settings. (A spiral that has its radius increase by 0 every turn. Aka a circle being drawn forever.) The nice thing about this in the context of a Mandelbrot analog, though, is that it can maintain a volume of 0. I assume that you're talking about the filiments that come off of the circles of the Mandelbrot set, and are infinitely thin, aside from the infinite number of minibrots contained within...

This became long. I apologize.

Offline jukzi

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« Reply #3 on: June 07, 2020, 01:30:53 PM »
basicly drawing a single spiral would result in:
option1: the spiral always is completely inside the x-y plane. Other spirals could be completely spin inside another plane
option2: based on a spiral with a normal spining rate inside the x-y plane. But also spinning at another speed around the that base:


Offline jukzi

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« Reply #4 on: June 07, 2020, 02:46:40 PM »
Any of the sphere-sweep style spirals I can think of have an axis, like the stem down the center of a snail shell.

My physically tought: Lets start with n rubber bands tight to a single point (the singularity) and each tight to a distinct point on the sphere around the singularity. As next i will start rotating the singularity around an axis. The rubber bands would roll up around the rotation axis. However if over the time we change the axis by rotating itself around one (or two?) orthotogonal axis is should end up with a nice ball of wool.
I guess either  a) each band/spiral would have non constant momentum but they form a solid sphere. Or b) they might have constant momentums but leave a lot of open space in the sphere. Like each has a seperate "phase" to ensure the bands will not intersect.


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