Braids in our lives

MichaelT_123 — Wed, 12 Feb 2025 03:36:26 GMT

Hi Mrs MariaHagnell, Mr TrippyLighting, Mr Drewpan and Sympathizers,

Yes, the sparked braid(ing) debate is also spicy … in the positive term … Mrs MariaHagnell.

Before I attempt to address the topological hurdle of making/modelling a braid, let me stick my nose into the discussion between Mr Drewpan and Mr TrippyLighting (please, don't shut a door).

Mr Drewpan: Total Nonsense? (Can't be done in Fusion)

Mr TrippyLighting: Yes, complete nonsense! (Can be done in Fusion)

Gentlemen, … I agree with both of you … but only with one at a time.

Braids or similar topological objects (like waving's, knots, …) with sophisticated, dense, intricate geometries are complex to model in Fusion and other software. The process starts with the object description, which can be pretty challenging in the case of braids. In my opinion, this can only be done reliably by mathematical abstraction, the area where the F360 modelling environment is … let's say … weak. However, not all is lost. Fusion API offers tools which can be somehow utilized here … to forge the respective mathematical formulas and deliver hardened braid instances to the modelling environment. Also, some simple cases can be modelled in the traditional way, but it would be an arduous undertaking.

However, in general cases, this is an unpassable double-whammy barrier for most, while the Meagre Residue must be FGeniusses plus some. A point for Mr Drewpan.

Nevertheless, the braid feature (like, e.g. coil) is technically possible to implement in Fusion … and within its current constitution, benefiting from as close to the silicon as possible realization (to be practically viable). A point for Mr TrippyLighting.

As I found the problem quite interesting … I decided to wrestle with it … a little bit.

I was whacked from one side to another. A mischievous order mismatch of parameters cost me one day … but after persistent ad hoc efforts … I entangled some braid's forged strands.

I don't think I sustained permanent injuries. Quite the contrary, as the old saying says, 'What doesn't kill you makes you stronger" †. († by Friedrich Nietzsche)

… and this is what I wish you and TF360… in your further battles with Braids in Fusion!

So, what is "a braid"… in a synthetic topological/mathematical view? In the simplistic¿ term, it is a convolution of one or more functions. Traditional simple 'braids' are best described/visualized as a composition of two parametric 3D curves (functions). Still, there is no limit to how many functions and in what way could be 'braided'. Generally, brides can also be constructed in higher dimensions, from objects like vector fields … or even in the social sphere,… as dances … particularly classical ones… check Vienna/Paris dance schools, Moscow ballet, .... Americans, both: simplified things (a line dance) or made it so convoluted (rock&roll) that even brightest mathematicians can't crack the ℬ(t,…).

Thus, let's denote our synthetic braid multiparameter function as:

ℬ(t,…) = ∏ℱᵢ(t,…) ⨂ᵢ here ᵢ >1

⨂ᵢ - is the 'braiding' operator, as even the same function can be convoluted in different ways.

There is the additional condition imposed on ℱᵢ(t,…). In topological term, the consecutive pairs must not 'intersect'. Our braid strands can't cut into each other, so if one is 'in', the other must be 'out' ( or up|down, left|right, …).

The so obvious to many practitioners condition can be … abstracted in a cryptic way … by who? … by crazy mathematicians, as per the Wiki page:

ℬₙ = < σᵢ,… σₙ₋₁ | σᵢ σᵢ₊₁ σᵢ = | σᵢ₊₁ σᵢ σᵢ₊₁, σᵢ σⱼ= σⱼ σᵢ >

But, hey, it is enough of this … the purest abstraction for now. Let's concentrate on describing (still mathematically) a simple helicoidal braid (with sympathy to all genetically influenced by such, particularly to orthodoxies like … Mr. ETFrench and others).

The parametric equation for a helix curve is:

xᵢ = rᵢ*sin(t), xⱼ = rⱼ*sin(t), …

yᵢ = rᵢ*cos(t), yⱼ = rⱼ*cos(t), …

zᵢ = h*t, zⱼ = h*t, …

The question! How do we denote the second helix (with index ⱼ, h will be the same) and modify the two (or more) moving the respective curves 'in' and 'out' at intervals? Remember to change the helicoids' directions. (helix ≠ helicoid)

For those who have not spent too much time intensely picking a nose during math classes and/or their mathematical rebars still not rusted completely away … it should be trivial. For others? … there are more challenging and undoubtedly more enjoyable activities. And, of course, AIGs, in short, Artificial Intelligence Geeks, obviously only those who are connected can try to make a call to AI Master.

The exercise could be pretty straightforward by looking at the following examples.

Note: The last 'squarish' example has some flaws. There are strands' intercuts … but I was too lazy … to massage the respective function well enough.

The additional inspiration for the solution can also be drawn (with some imagination) from WheelOnWheelOnWheel trickery:

https://forums.autodesk.com/t5/fusion-design-validate-document/how-to-create-a-perfect-or-atleast-very-close-sine-wave/m-p/10082974

The unexpected assignment… was easy, wasn't it?

The helix makes our life convoluted by birth. Some even complain about how complicated it is. Oh hey, … do not grumble. It could be even worse.

The examples below show how lucky (or not) we are that Nature has not selected, e.g. the trefoil knot braided on itself as our generic genetic convolution ℱᵢ(t,…) ⨂ᵢ.

… or perhaps Some … are already equipped with such … ???!!!

Attached files:

Braid_B_arcd_150.jpg 5K_stereo ( 0.7MB) https://a360.co/4gncXOJ

Braid_B_mono_150.jpg 4K_mono ( 0.3MB) https://a360.co/42CTx56

Braid_C_arcd_150.jpg 5K_stereo ( 0.7MB) https://a360.co/3CyBYZp

Braid_C_mono_150.jpg 4K_mono ( 0.3MB) https://a360.co/3Q3FBcL

Braid_G_arcd_150.jpg 5K_stereo ( 0.7MB) https://a360.co/4hJxcXE

Braid_G_mono_150.jpg 4K_mono ( 0.3MB) https://a360.co/3Q5KYrU

Braid_H_arcd_150.jpg 5K_stereo ( 0.7MB) https://a360.co/3CrZgQF

Braid_H_mono_150.jpg 4K_mono ( 0.3MB) https://a360.co/3CFwhJ9

Braid_ ABDCE _arcd.mp4 4K_stereo (236MB) https://a360.co/4aOnlgZ

Braid_ ABDCE_mono.mp4 4K_mono (145MB) https://a360.co/4jZ1yHZ

Braid_FGH_arcd.mp4 4K_stereo (38MB) https://a360.co/4jOCNy3

Braid_FGH_mono.mp4 4K_mono (30MB) https://a360.co/41cfze1

To be viewed on 4K media devices (monitors, UHD TVs, projectors...) of reasonable performance. For the best experience, use stand-alone media applications (WMP, VLC) and the native resolution 3840x2160 - full screen. The '_arcd' files require an anaglyph red/cyan glasses, while '_al' is for active shutter glasses 3D hardware (~30°viewing angle is recommended). Download the files over a network, where the cost of doing so is not a concern. The files are to be used for private, non-commercial purposes only.

Regards

MichaelT

Re: Braids in our lives

TrippyLighting — Thu, 06 Feb 2025 11:41:16 GMT

For whatever reason there is no "like" button so this is a "like" in text form.

Re: Braids in our lives

MichaelT_123 — Tue, 21 Oct 2025 10:11:41 GMT

Hi Braids_Lovers,

As asked by Mr/Mrs M_LegrasJJGZK in

https://forums.autodesk.com/t5/fusion-design-validate-document/how-to-model-braided-rope/td-p/13255253

I have attached Braids_Sample.f3d file (only simple ones!).

Regards

MichaelT

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Braids in our lives

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