Some preliminary research results (WARNING: 1200x800 jpg)
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Joined: November 20, 2003
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Location: Northbrook, IL

Some preliminary research results (WARNING: 1200x800 jpg)
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Posted on Wed Jan 03, 2007 12:56 am
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Since I didn't bring any of my maille supplies home from school, I decided to start research into some theory. One thing I was interested in was the topologically unique arrangements (TUAs) of a given number of rings. Now, originally I asked myself "how many are possible with x number of rings?" I have since realized that this problem is near impossible to solve and also quite useless. Still, knowing the bases might have some purpose. So here are the 12 unique ways in which four rings can be assembled - I'm pretty sure it's all of them, tell me if you know of another:



Some explaining:
Clockwise means the spiral in the arrangement goes clockwise toward the center, like a storm in the southern hemisphere. Counter means the opposite. Neutral means no direction can be established.

The numbers on the top are probably confusing. In the "x + y" notation, it means the previous set of 3 rings contained x connections, and the new added ring contributed y connections. So in each column, the total number of connections in the pattern is x+y. For example, in the top left, we have a clockwise 3+3. This means we started with a set of three rings with three connections (all connected to each other in a flower) in a clockwise manner, and then we added a fourth ring which added three more connections, to result in a four ring clockwise flower.

Note something interesting: in the last column we have two kinds of arrangements. This is because the 3+1 arrangement of the flower with an extra ring hanging off CANNOT be neutral, because 3 rings with 3 arrangements cannot be neutral. Likewise, 2+1, the straight chain, MUST be neutral. My first idea to come from this is that the persians, as you can see, shouldn't be called "left" and "right", since that doesn't really make sense and isn't defined anyway. I think we should call them clockwise or counterclockwise, based on the two-ring unit spiral direction.

Joined: November 06, 2006
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Weave validation theory
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Posted on Wed Jan 03, 2007 4:52 am
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If you can solve the nearest point(s) of two circles problem,
or the tangents you been working on. A PC could crank out
a multitude of possible valid interactions.

How you go from there to test or prove if two valid weaves
are the same topology or not???

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Do I have it reverse? Are you shooting to define all the
topologies first, then test to see if they are weaveable?

Joined: November 20, 2003
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Location: Northbrook, IL

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Posted on Wed Jan 03, 2007 5:28 am
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Well, as I already said, the problem is impossibly complex. First of all, there are a number of non-inclusive ways to approach it. That is, if you just considered every way in which one ring could connect to a random set of other rings, that doesn't account for the orientation. Even if you did account for that, then you have to worry about whether those connections are physically possible. Maybe not connection angle, but what if some rings have to reach OVER others, like orbitals?

I can say with near certainty that no two weaves in our library are topologically identical, unless we intended it that way.

I don't think I'm going for either of those goals. I'm looking to just see, on maybe only in slightly mathematical sense, how rings pattern together. Like this clockwise, neutral, counter thing.

If you want to try solving for the number of possible unique arrangements, consider this: If you've taken any slight organic chemistry, you've seen how carbon chains are named. You label the longest single chain, then number and name the branches by their position and size. Imagine a maille chain analogue. That is, if you take some pile of rings, make a chain of random length, and then randomly branch off of it, and see how many unique webs you can make. And that's just the SIMPLEST case, with mostly 2-1 connections! I don't even want to think about that.

Joined: November 06, 2006
Posts: 115
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captive orbits
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Posted on Wed Jan 03, 2007 9:49 am
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I see whatcha mean.

Even if we narrow the subset to patterns that repeat within
a finite volume (a chain has at least 2 facets fold & repeat,
but maybe need as many as 8 {Hex tesselation+upNdown})

I can validate if a pattern collides, but I can't tell if unlinked
rings are held captive or orbital. Orbital is probably easier.
We can detect if other rings are linked thru the donut hole?
But captive rings are a whole nuther matter. Not so easy.

Might have to test the freedom of every ring to determine
if they are indeed part of the weave? Perhaps have the PC
shake the weave (we know we can test collisions), and see
if any fall out. Wouldn't exactly garauntee excluding every
loose ring. Suggestions?

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Shaking the weave might also be a good way to tweak the
AR's for desired freedom of motion. Genetic engineering.

Joined: March 3, 2002
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Location: tres piedras, new mexico

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Posted on Wed Jan 03, 2007 12:32 pm
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i suggest you both stop this before you outsource our jobs to computers. Coif LoL

kim


PSA: remember to stretch.
3.o is fixing everything.

Joined: November 20, 2003
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Location: Northbrook, IL

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Posted on Wed Jan 03, 2007 7:21 pm
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I'm pulling away from that, onyx keeps asking me to write some superprogram that calculates the behavior of the sheet hanging in space and shakes it to see if rings fall out, or one that tells which rings are how close to what other rings in the weave. I don't have a computer powerful enough or the language skills advanced enough for that. All we could do is manually write entire weaves in pov-ray and tweak things. A very slow and case-by-case method.

Joined: June 21, 2006
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Location: The Philippines

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Posted on Fri Jan 05, 2007 4:35 pm
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Fwoosh, and that all went over my head...

But it does sound rather interesting. Any tribal elder programmers around? It'd be interesting to see this out.


_________

Being from the third world, BMR claims the right to speak in the third person.

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