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Date Uploaded: December 26, 2009, 10:31 pm Last Edited: December 16, 2012, 4:04 pm |
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European X in 1
(Expanding/Contracting)
Article © MAIL User: ArmoredDrake
Expansion and contraction are two vitally important aspects of mailling, and they ought to be as familiar as the grip of a favorite pair of pliers. This is the basic European 4 in 1 expansion that we are all familiar with:
Remember, if I were to flip this image vertically, this would be a picture of a contraction, as these two techniques are just different versions of the same thing.
However, I have recently noted a definite, reliable fact of nature, and it is this: at least once every couple of months, someone will ask whether it is possible to expand and contract in, say, European 6 in 1. So in the hopes of discouraging this repetition from continuing, I am here to tell you the answer: MOST DEFINITELY, YES!! And better yet…HERE’S HOW!
Put simply, this is what a Euro 6:1 expansion looks like:
Note that the ring passes through only two above. In a larger sheet, shown below, it passes through two above and the normal three below, like so:
If your goal is to contract within the weave, it will look like this:
…in which the ring passes through four above, or in a wider sheet as below, four above/three below:
So, can this work for a Euro X:1 weave where X is not 6, but rather 8, 10, etc…? Theoretically, yes (practically, I don’t know why you’d want to…sculpture maybe). If those pictures were all you needed to figure out the pattern for higher-count weaves, great! But if you feel up for a slightly more mathematical and less pretty-picture type explanation, read on.
EXPANDING IN HIGH-COUNT EURO X:1 WEAVES
There is a very simple formula for expanding in any even-numbered Euro weave. This is it:
Y = X – 1
What does this mean? Y is the number of rings that the idler passes through. X is the same as in your Euro X:1 ratio. So, an idler will always pass through one ring fewer than those around it (the “normal” rings). To determine how many rings the idler passes through above, use A for rings Above and keep the same X:
A = (X/2) – 1
For rings Below (B), it is:
B = (X/2)
So, again, what’s that saying? Basically, the idler ring of any expansion goes through the same number of rings below as all the others around it, but one fewer ring above it. (If you add together your A and B, you should see that it is equal to Y…reinforcing the first relationship)
Let’s prove this now, using Euro 4:1. Our X = 4. So,
Y = 4 – 1 = 3
…and indeed, an idler ring in Euro 4:1 only goes through 3 rings. Plug it into the other equations, and you’ll get A = 1 and B = 2, exactly what we’d expect (because a 4:1 idler only goes through 1 ring above, but the normal 2 below). For Euro 6:1, our numbers are Y = 5, A = 2, and B = 3…again, exactly what we noted in the pictures above.
So, yes, these three equations can be applied to higher-count Euro X:1, and you can theoretically do any European variant you like (as long as X is even). Euro 8:1 would go through 7 rings, 3 above and 4 below. Euro 10:1 would go through 9 total, 4 above, 5 below…etc.
If you want to do a contraction, this is basically the same animal, but the equations do need to be modified. Simply:
Y = X + 1
A = (X/2) + 1
B = (X/2)
So, in Euro 4:1, the idler goes through 5 total, 3 above and 2 below…for Euro 6:1, it’s 7 total, 4 above and 3 below…however high you want to go.
Whew. So that question is answered. But…some people don’t like that idler rings cause the weave to bulge. The discontinuity is annoying. What can we do? Well, option A is “deal with it,” but that’s a little cold-hearted and doesn’t really solve any problems. Option B…well, there will always be a little discontinuity visible, but the following is a thought I had about somewhat lessening the visible bulge (pardon me if it’s been thought of before – this is pure speculation). I haven’t actually tried it in a real application yet, but here’s the thought. I call it king-ring expansion, and in Euro 4:1, it looks like this:
What you do is pass the idler ring through two rings above, laying it right on top of one of the adjacent rings, as if you were kinging that ring. However, as you add the row below, you treat every ring as a separate entity. So, it’s as if you added in the idler in the traditional way, hooking through only three rings, but after working down a few rows and finishing the sheet, you went back and hooked the idler through one extra ring above. This minimizes the discontinuity in the weave because that idler isn’t being forced to sit lower than the others in its row. And it even seems to work in higher-count weaves, too:
Problems solved? Questions answered? I hope so, but if not, PM me or contact me somehow, and I’ll try to help you out. And now, hopefully, the next couple of months will be blissfully absent of the question, once again, of how high-ring-count expansion can be done. Thanks for reading.
Remember, if I were to flip this image vertically, this would be a picture of a contraction, as these two techniques are just different versions of the same thing.
However, I have recently noted a definite, reliable fact of nature, and it is this: at least once every couple of months, someone will ask whether it is possible to expand and contract in, say, European 6 in 1. So in the hopes of discouraging this repetition from continuing, I am here to tell you the answer: MOST DEFINITELY, YES!! And better yet…HERE’S HOW!
Put simply, this is what a Euro 6:1 expansion looks like:
Note that the ring passes through only two above. In a larger sheet, shown below, it passes through two above and the normal three below, like so:
If your goal is to contract within the weave, it will look like this:
…in which the ring passes through four above, or in a wider sheet as below, four above/three below:
So, can this work for a Euro X:1 weave where X is not 6, but rather 8, 10, etc…? Theoretically, yes (practically, I don’t know why you’d want to…sculpture maybe). If those pictures were all you needed to figure out the pattern for higher-count weaves, great! But if you feel up for a slightly more mathematical and less pretty-picture type explanation, read on.
EXPANDING IN HIGH-COUNT EURO X:1 WEAVES
There is a very simple formula for expanding in any even-numbered Euro weave. This is it:
Y = X – 1
What does this mean? Y is the number of rings that the idler passes through. X is the same as in your Euro X:1 ratio. So, an idler will always pass through one ring fewer than those around it (the “normal” rings). To determine how many rings the idler passes through above, use A for rings Above and keep the same X:
A = (X/2) – 1
For rings Below (B), it is:
B = (X/2)
So, again, what’s that saying? Basically, the idler ring of any expansion goes through the same number of rings below as all the others around it, but one fewer ring above it. (If you add together your A and B, you should see that it is equal to Y…reinforcing the first relationship)
Let’s prove this now, using Euro 4:1. Our X = 4. So,
Y = 4 – 1 = 3
…and indeed, an idler ring in Euro 4:1 only goes through 3 rings. Plug it into the other equations, and you’ll get A = 1 and B = 2, exactly what we’d expect (because a 4:1 idler only goes through 1 ring above, but the normal 2 below). For Euro 6:1, our numbers are Y = 5, A = 2, and B = 3…again, exactly what we noted in the pictures above.
So, yes, these three equations can be applied to higher-count Euro X:1, and you can theoretically do any European variant you like (as long as X is even). Euro 8:1 would go through 7 rings, 3 above and 4 below. Euro 10:1 would go through 9 total, 4 above, 5 below…etc.
If you want to do a contraction, this is basically the same animal, but the equations do need to be modified. Simply:
Y = X + 1
A = (X/2) + 1
B = (X/2)
So, in Euro 4:1, the idler goes through 5 total, 3 above and 2 below…for Euro 6:1, it’s 7 total, 4 above and 3 below…however high you want to go.
Whew. So that question is answered. But…some people don’t like that idler rings cause the weave to bulge. The discontinuity is annoying. What can we do? Well, option A is “deal with it,” but that’s a little cold-hearted and doesn’t really solve any problems. Option B…well, there will always be a little discontinuity visible, but the following is a thought I had about somewhat lessening the visible bulge (pardon me if it’s been thought of before – this is pure speculation). I haven’t actually tried it in a real application yet, but here’s the thought. I call it king-ring expansion, and in Euro 4:1, it looks like this:
What you do is pass the idler ring through two rings above, laying it right on top of one of the adjacent rings, as if you were kinging that ring. However, as you add the row below, you treat every ring as a separate entity. So, it’s as if you added in the idler in the traditional way, hooking through only three rings, but after working down a few rows and finishing the sheet, you went back and hooked the idler through one extra ring above. This minimizes the discontinuity in the weave because that idler isn’t being forced to sit lower than the others in its row. And it even seems to work in higher-count weaves, too:
Problems solved? Questions answered? I hope so, but if not, PM me or contact me somehow, and I’ll try to help you out. And now, hopefully, the next couple of months will be blissfully absent of the question, once again, of how high-ring-count expansion can be done. Thanks for reading.
Original URL: http://www.mailleartisans.org/articles/articledisplay.php?key=423
