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Ring Interaction Model
Article © MAIL User: mithrilweaver

Imagine - every time you start a chainmaille project, you start with one ring. This one ring is the corner stone of all chainmaille weaves. Then, you connect another ring to that first ring. These two rings have to connect to each other, there are no other combinations of connections. Then you add a third ring. Now there are more possibilities for interactions. You can make Straight 2 in 1, you can make a Mobius Knot, or you can make an Orbital interaction. Depending on the aspect ratios of the rings, you can make an infinite number of outcomes, but they will always be a version of Straight 2 in 1, Mobius, or Orbital. There are no other mathematical combinations for 3 rings. Then you add a fourth ring variable. Now there are many more combination possibilities. The table I created, called the "Ring Interaction Model (R.I.M)" shows all of the connection possibilities at levels 1-4 (level = # of rings in the interaction). The R.I.M shows levels 1-4, but will hopefully someday include level 5 as well. The mathematical possibilities for ring interactions seems to grow exponentially as the level increases. So, level 5 will likely include 100-500 possible combinations and the table that includes all of these would be huge. On my website, you can download a free pdf of the R.I.M here to look at as you read this article:

The R.I.M. was inspired by the periodic table of elements. Think of each interaction as an element. It is meant to be a guide to the building blocks of all ring interactions where the resulting interactions are stable - in that no ring can fall off.  As chainmaillers, we know that a weave is a repeating pattern of possible connections as we add rings or levels. At what point we choose to start repeating, with what connection, and with what size ring, is what makes weaves different from others. The R.I.M takes into account every possible connection (whether repeating or not) with every possible aspect ratio. In other words, every weave must start with a connection listed in the R.I.M. No weave falls outside the R.I.M. Interactions that are not possible are not listed in the R.I.M.

It is debatable whether Orbital is a “connection” since the orbiting ring is not connected to any ring.  This is why the table in called "Ring Interaction Model" and not "Ring Connection Model."  The model simply shows and labels all interactions that are mathematically possible/stable and lets the user decide for themselves how they want to interpret it for their own weave theory.  For example: someone could hold the belief that a right handed Mobius and a left handed Mobius are different interactions. The R.I.M. doesn't say whether they are or not, it just shows a picture of a 3 and 4 ring Mobius Knots and labels them with an (h) that indicates that it has right and left handed versions.

The R.I.M. doesn't name weaves. Weaves are called by many different names. The R.I.M. strictly adheres to mathematical possibilities of connections. Emample: you can see Half Persian 3 in 1 start to form in “4-4” of the R.I.M., but there is no name listed in the picture. The naming of Half Persian 3 in 1 is up to the user.

The R.I.M. is intended for use by the international chainmaille community. Instead of naming a weave, it lists a combination possibility in a table that every can use and agree on. The R.I.M. is not intended to replace weave naming or current categorization attempts. In my opinion, it does have potential for weave categorization, but current methods that have stood the test of time should be honored. I don't think anyone would like to see European 4 in 1 renamed as something like “5-1.” I think the R.I.M. can best be used as a tool for creating new weaves. As the user attempts ring combinations, he/she will start to see new and interesting ways to connect rings in repeating patterns. The R.I.M. can also be a tool for understanding ring connections: through eye (te), around eye (ae), no eye (ne), and orbital (o). There are other ring connections that do not yet present themselves in levels 1-4, so are not listed in the R.I.M. This is another reason why completing level 5 in the R.I.M. would be valuable.

Explaining level 4: The interactions in level 4 are all the mathematical combinations of 4 rings.  The columns are lined up in a way that shows how they can be built from the interactions above them in level 3.  If the column is situated between two interactions in level 3, then those interactions can be built from either interaction in level 3.  The columns are also organized by "h" handed and "s" symmetrical.  If an interaction is handed, then it has no symmetry and the ring orientations have a structure of left or right handed.  Example: left and right handed Half Persian 3 in 1 or right and left handed Mobius Knots.  If an interaction is symmetrical, then it has at least 1 plane symmetry – the image in size and arangement of rings is the same on either side of the plane that splits the interaction in a given direction in 3 dimensions.

The model shows the interactions where all rings have the same aspect ratio.  It lists the mathematical minimum for the ring interaction when all rings in the connection have the same aspect ratio (example: >2).  I am still working out the mathematical accuracy of these figures. Currently, the values are rounded to the nearest whole number. The R.I.M. lists the Orbital connection as needing rings that are >5, but in actuality the more accurate value is >5.4. The model is not limited by - all the rings being the same size though.  Yes, the minimum aspect ratio value is no longer valid when the rings' aspect ratios change, but the number of possible interactions remains valid.  There are 2 exemptions to this that I have found in the model. They are in level 4 and indicated with a (*) and a (**).  The first (*) exception is if you have 3 rings in a row with “ne” connections and 1 ring orbiting between one of the connections (as in “4-15” and “4-19” of the R.I.M.).  If the middle ring shrinks and goes down in aspect ratio, eventually the orbiting ring will be able to slip over it and be able to move to the other side (as in “4-19”), yet is still bound in the overall interaction of rings.  This exception is visible in Orbit, Dragon Tail, and Shenanigans.  The other (**) exception is in the bottom right corner “4-18”.  It shows 2 rings with (ne) connections, 1 orbiting between them, and one ring with a (ne) connection to the orbiting ring.  If the interaction has all rings that are between aspect ratio 5 and 7, the ring that connects to the orbiting ring is locked into a quadrant and the connection becomes (h) handed.  If the ar of the interaction is above 7, then the ring that connects to the orbiting ring can move freely in all 4 quadrants and is then classified as (s) symmetrical. That is why the symmetrical/handed box contains (**) - it is conditional. As far as i can tell, these are the only exceptions in level 4. I would like to get feedback from the community as to whether these are the only exceptions in level 4. Level 5 would have many more exeptions. This posses a problem for completing an accurate table. A few exeptions are acceptable and even interesting, but 20 exceptions would be rediculous and make the R.I.M cumbersome to interpret. By completing level 5, I may find that there is a new and better way of categorizing exceptions, but there is no way to know until it is tried.

My predictions: level 5 would be a huge undertaking and could possibly account for 90% -100% of all interactions and weaves in the database of M.A.I.L.  I'm guessing that the number of interactions in level 5 would be in the hundreds - somewhere between 100 and 500.  I've tried to figure, mathematically, how many interaction possibilities would be in level 5 and I can't seem to get an accurate answer.  There are just too many variables. This is largely due to an effect that presents itself in level 4 of the model.  As the ring number/level rises, the possible number of multiple ring interactions with infinite aspect ratios exponentially grows.  Since there are an infinite number of aspect ratio combinations in an interaction, it is difficult or impossible to know what interactions and exceptions it will yield without actually trying them.  It is much easier to figure the possibilities if all the rings have the same aspect ratio and then jump into the exceptions of different aspect ratios within an interaction later.  

What could this model do for weave cataloging and organizing?  If you look at this model purely as an interaction model, then the answer is "nothing."  If you look at this model as a guide for how all weaves begin, then it can do a lot.  For example, if you look at each level individually, you can say "all weaves come from these interactions."  Level three is the first split where the interaction options are more than 1.  So, you could categorize all weaves from the 3 interactions on level 3 and call them “3 families.”  Then, you could say that the options in level 4 are "sub-families" of level 3.  All exceptions would be accounted for, no weave would or could fall outside of the R.I.M. Since there is no naming of weaves in the R.I.M., it wouldn't matter when stable European 4 in 1 presented itself because it would be present in many different interactions. Example: you can see European 4 in 1 start to form in “4-3,” but only because of the way the rings are layed out in the picture. In actuality, “4-3” is just two rings connected to 2 rings with (ne) connections. Whether or not European 4 in 1 is represented in the R.I.M. is irrelevant. It only shows every mathematical combination of rings. Seeming complications like - weaves that display multiple interactions - are easily explained as having multiple origins that are clearly displayed in the R.I.M.  I don't think many people would be excited by this though. Weaves would no longer have names, instead they would have number signatures in multiple interactions within the R.I.M.  Since the model is purely mathematical, it shows no preferential treatment to historical or cultural weaves. I happen to think history and culture are very important, and I think others do as well.

Please let me know your thoughts about the R.I.M. and any constructive criticism on table organization would be great.  Thanks so much for listening and for your feedback.

-The Mithril Weaver
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