Fractal Related Discussion > Fractal Mathematics And New Theories

(New Coordinate System) Sphere wrapped in a Noodle

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hobold:
This is just a heads up that there is yet another kind of spherical coordinates.

It is possible to wrap the surface of a sphere in a noodle. Some might remember that once I asked what the 3D equivalent of a spiral was. My vague intuition back in the day was that the shapes which we had seen thus far still lacked variety. There were various spirals, helical surfaces, scrolls of parchment, but no (well-defined) tangled ball of wool.

As I was playing around with the sphericon and its relatives
http://www.interocitors.com/polyhedra/n_icons/index.html
I began to realize that this particular construction might lead to yet another coordinate system on the sphere. It has many flaws (discontinuities and poles), but it has one big thing going for it. The figurative noodle is a single closed loop, topologically just like a circle. If "length along the noodle" is one of the coordinates, it can easily be doubled to yield another point along the noodle. That would be equivalent to doubling an angle in the complex plane. In other words, the squaring step in the good ole' Mandelbrot formula.

Not sure if this will ever yield an interesting new variant of the Mandelbulb. But it would certainly be a novel approach. I'll probably try that some time, but don't hold your breath.

Anyway, I wanted to share the idea here, so it doesn't get lost. With any luck the fractalforums community finds it useful.

msltoe:
From Riemann spheres to noodle spheres? https://fractalforums.org/Smileys/default/wink.gif
I've been looking at a similar thing lately: Fibonacci spiral spheres. They have nice even coverage of surface of the sphere and they can plotted with a single top to bottom 1-D spiral. However, it does not have the nice property like your noodle sphere, where the ends are connected and it can be wrapped around.

v:
That's a very tricky surface, I would love to know if there is a parametrization for it.

One can trace the sphere with an arithmetic spiral in two parts:
f(t) = (cos(t/4)*cos(3*t),cos(t/4)*sin(3*t),sin(t/4))
and
f(t) = (cos(t/4)*cos(3*(t+pi)),cos(t/4)*sin(3*(t+pi)),sin(t/4))
for the second part.

But thickening the line of this curve to a cylinder does not work out without overlap.

How did you create that image, by the way?

hobold:
The construction is obscure, but not complicated.

1. Pick an even number N >= 4. In this illustration I'll use N = 4; this will yield the "classical" sphericon.

2. Regard the regular N-gon (that would be the square in the example case N = 4), rotated such that it balances on one vertex. In that case it has a vertical axis of symmetry through its center. (For the example square, this means the axis is one of the diagonals.)

3. Jump into 3-space and rotate the flat N-gon around the vertical axis of symmetry. The resulting object is made of slices of cones, where the slicing planes are all parallel, and the plane's normal vectors are aligned with the cones' central axes. (In the square example here, we now have a double cone, with the angle at the apex being 90 degrees - because that's the inner angle at a vertex of a square.)

4. Viewing the axis of rotational symmetry as "up" direction, cut the stack of cone slices along this axis such that you obtain a front half and a back half (planar cut). Both halves are identical, and the surface resulting from this cut is the regular N-gon from step 2 (for example a square balancing on a corner).

5. Keeping the cut planes of both halves parallel, rotate front and back halves in opposite directions (around the "forward" axis) until both N-gons match again. (In the example case you rotate both squares by 45 and -45 degrees, respectively. For larger N, one can twist more than a single vertex, but that has weird and often undesirable effects for the application of "noodle wrapping".)

6. Glue front and back halves together again. The conical slices meet continuously at the N-gon edges, with continuous tangents. The resulting object is no longer rotationally symmetrical. (In the example case N = 4, you arrive at the original "sphericon".)

These N-sphericon objects have interesting properties. In particular, they are developable, which means the can roll on a flat plane such that every point of the N-sphericon surface touches the ground exactly once per cycle. If the sphericon was covered in wet paint, it would draw its own construction plan on the ground while rolling. (This is what got me thinking about trying to find an application for spherical coordinates.)

Going from here to the noodles is just an additional replacement step. Between steps 2 and 3 above, replace each edge of the N-gon with a circle centred on the edge's midpoint, with a radius such that adjacent circles just touch. This replaces each cone slice with a torus, so the end result of step 3 is now a stack of tori. The subsequent twist turns it into a single long noodle.

My picture above uses N = 22 if I remember correctly, and was rendered in PoV-Ray ( www.povray.org ). PoV-Ray is script based, so the modelling was done essentially by writing the above instructions as a program in some geometrical language.

hobold:
I produced a bit more imagery of this. Not fractal - I am hoping for your forgiveness :) - but still eye candy. These don't play smoothly from my server due to bandwidth constraints. So save to disk and play the animation loops from there.

A noodle sphere:
http://www.vectorizer.org/3doodle/noodle_16_1_ultra_001.mp4

The world's longest marble run:
http://www.vectorizer.org/3doodle/MarbleRunGroundHD001.mp4

On a more serious note, a variant of this construction could have real applications for "evenly" distributing points on a sphere. Consider for example, Dave Rusin's "disco ball" (https://bendwavy.org/pack/pack.htm); connecting adjacent latitudinal bands (by twisting front and back hemispheres) allows combining leftover space, occasionally making room for an additional point.