### Infinite Helix Distance Estimator? ...Helix from helix from helix from helix...

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#### John

• Fractal Freshman
• Posts: 5

#### Infinite Helix Distance Estimator? ...Helix from helix from helix from helix...

« on: November 21, 2020, 07:27:50 AM »
Imagine a helix that coils itself into another helix, which coils itself into another helix, etc. onward to infinity. Surely there's a mathematical way to represent this, but I have no idea how. Here's a primitive 3d render of one helix wrapped around another for context.

Now imagine if that large helix shape was much longer, and it wrapped itself into another helix, and that continued onward infinitely. And imagine if that solid gray helix strand (that's being wrapped around the bigger black helix) was actually itself a much smaller helix instead. And onward in both directions to infinity.

How would you mathematically represent such an object? Is there a distance estimator for it?

Basically, it would be infinite in both directions, so for any given helix radius you're observing, it would wrap itself into another helix with radius N times the current radius, and the current helix you're observing would be formed from a smaller helix with a radius of 1/N the current radius, where N is a positive real number (probably greater than 2, but I don't know how big it would need to be to avoid collision).

Does a distance estimator even make sense for this? Is there some way to represent it mathematically?

• 3f
• Posts: 2269

#### Re: Infinite Helix Distance Estimator? ...Helix from helix from helix from helix...

« Reply #1 on: November 21, 2020, 08:30:26 AM »
Imagine a helix that coils itself into another helix, which coils itself into another helix, etc. onward to infinity. Surely there's a mathematical way to represent this, but I have no idea how. Here's a primitive 3d render of one helix wrapped around another for context.

Now imagine if that large helix shape was much longer, and it wrapped itself into another helix, and that continued onward infinitely. And imagine if that solid gray helix strand (that's being wrapped around the bigger black helix) was actually itself a much smaller helix instead. And onward in both directions to infinity.

How would you mathematically represent such an object? Is there a distance estimator for it?

Basically, it would be infinite in both directions, so for any given helix radius you're observing, it would wrap itself into another helix with radius N times the current radius, and the current helix you're observing would be formed from a smaller helix with a radius of 1/N the current radius, where N is a positive real number (probably greater than 2, but I don't know how big it would need to be to avoid collision).

Does a distance estimator even make sense for this? Is there some way to represent it mathematically?
Interesting idea.
Not sure I understand how you can tell if a point is part of this thing or not.
Also, if you start say with a helix with 100 windings, how many windings can you make out of that at the next level? Not very many so I think you'll get flatter and flatter helixes if you'd allow them to scale size nonuniformly.
I don't know how to visualize what you describe.

• 3f
• Posts: 2269

#### Re: Infinite Helix Distance Estimator? ...Helix from helix from helix from helix...

« Reply #2 on: November 21, 2020, 09:50:29 AM »
I can't think of how to do it with a helix but maybe with flexible cylinders like gardenhoses.
At a certain scale you have cylinder, but when you zoom in by a lot (say 1e10) you see the surface of the cylinder is made up of a coiled tiny cylinder and when you examine that one closer, ..., etc.

If you then replace all the cylinders by helixes in some specific way we have maybe what you meant?

I don't know how to apply math to this, but it looks graphics programmable.

• 3c
• Posts: 860

#### Re: Infinite Helix Distance Estimator? ...Helix from helix from helix from helix...

« Reply #3 on: November 21, 2020, 11:11:51 AM »
This reminds me of the organization of DNA in helices and chromatin. Maybe you can find some mathematical insight in computational biology (computing hydrophobic surfaces of DNA molecules)..

"D Marenduzzo. The Physics of DNA and Chromosomes". (e.g. fig 1 and 2)
https://iopscience.iop.org/book/978-0-7503-1602-6/chapter/bk978-0-7503-1602-6ch1

#### xenodreambuie

• Strange Attractor
• Posts: 87

#### Re: Infinite Helix Distance Estimator? ...Helix from helix from helix from helix...

« Reply #4 on: November 21, 2020, 09:19:53 PM »
I don't know how to do a distance estimator for it. Finite sections can be done with IFS. Here is a quick test using XenoDream. It's a single iterator with a transform that replicates in a helix, but I have to do the size and orientation visually and it's not quite aligned.

• 3f
• Posts: 2269

#### Re: Infinite Helix Distance Estimator? ...Helix from helix from helix from helix...

« Reply #5 on: November 22, 2020, 08:41:46 AM »
Another way I can think of it is to start a video showing a garden hose, one side in the window, the rest going off screen (imagine infinitely long). Then animate it to coil up to form a cylinder and at some point when coiled up enough and the right viewpoint it will look again like the garden hose in the first frame and then you loop the video.

An actual self-similar fractal could perhaps be defined as something to which when you do this the thing at frame 1 is actually the same as in the end frame in some mathematical sense.

• 3c
• Posts: 860

#### Re: Infinite Helix Distance Estimator? ...Helix from helix from helix from helix...

« Reply #6 on: November 22, 2020, 11:39:05 AM »
Rough thought: The points on a thin axis-parallel helix are described by (sin(A), cos(B), t) with A,B some function of the height t, the windings per unit length and the radius of the helix. If you describe your visible helix as a collection of slightly shifted point helices (and rotated/translated appropriately) that form a solid object, maybe you could compute the distance of an arbitrary point to those by a minimum analysis (derivative of Euclidean distance function (point, specific helix) / dt = 0) and then take the minimum of those values. A sphere of that radius should then fit around the point and could be used as the overall distance.

• 3f
• Posts: 1686

#### Re: Infinite Helix Distance Estimator? ...Helix from helix from helix from helix...

« Reply #7 on: November 22, 2020, 11:51:23 AM »
As there is only 1 iterator, running the IFS in reverse with derivatives matrix should be able to turn it into escape time with distance estimate.

(I have a helix DE on another computer, but I did not succeed yet in making it fractal. It's also a double helix for reasons I haven't figured out...)

• 3f
• Posts: 1686

#### Re: Infinite Helix Distance Estimator? ...Helix from helix from helix from helix...

« Reply #8 on: November 22, 2020, 10:12:22 PM »
Cleaned up my non-fractal (double) helix frag a bit, attached.

• 3f
• Posts: 1686

#### Re: Infinite Helix Distance Estimator? ...Helix from helix from helix from helix...

« Reply #9 on: November 23, 2020, 01:18:25 AM »
figured out the single helix, still not fractalized though.  the line from the point to the surface is in the same plane as the axis of the helix.
EDIT only an approximation, valid only in the small rise per revolution case... tricky problem...
« Last Edit: November 23, 2020, 01:29:19 AM by claude, Reason: imperfection »

• 3f
• Posts: 1686

#### Re: Infinite Helix Distance Estimator? ...Helix from helix from helix from helix...

« Reply #10 on: November 23, 2020, 04:58:30 AM »
got something promising.  the matrix m is an approximation valid for tight spirals, for looser spirals it needs to depend on the angle...

• 3f
• Posts: 2269

#### Re: Infinite Helix Distance Estimator? ...Helix from helix from helix from helix...

« Reply #11 on: November 24, 2020, 04:45:03 AM »

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