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Spherical Spheres
Article © MAIL User: Inventor

Spherical Spheres

by Inventor

Image: threespheres.jpg

The Epiphany
There are different ways to weave spheres in chainmaille. You can weave triangles and stitch them together or make reductions of increasing frequency to get something fairly spherical, or you can wrap a spherical object with maille. These techniques work well enough given the appropriate weave as the rings will relax into an approximately spherical shape. But what if we want to make really small spheres, or really spherical spheres? What if we want a sphere that is hollow yet solidly rigid instead of flexible? In that case we need to use the right weave, ring, and material combination, and we also need to know the exact number of rings per row that will create a true sphere.

I once created a basket using the reduction method. It turned out nice and round, kind of hemispherical, but not exactly. The night after completing it, I dreamed and awoke with an epiphany! I had realized the math behind making a perfectly spherical chainmaille sphere, or at least as perfect as you can reasonably get. I owe my undergrad calculus teachers a debt of gratitude for teaching me how to set up calculus problems, because the sphere math is based on the way that you set up a spherical surface area calculation. Actually you can understand it with only trigonometry, no calculus required (whew). The math that caused a light bulb to turn on above my head is illustrated in the figure below.

Image: spherederivation.jpg

I'll spare you the derivation details, but basically the four equations on the upper right are obtained from the drawing on the left, and when you solve them you get the equation in the box, which is repeated below:

m = ( 2*pi*r / (kh*d) ) * cos( (kv*d/r) * n )

where the variables are:
r is the radius of the sphere
d is the inside diameter of the rings
kh is the horizontal proportionality constant of the weave
kv is the vertical proportionality constant of the weave
n is the row index
m is the number of rings to put on the nth row.

The proportionality constants are a measure of how dense the weave is in each direction. Well, technically they are inversely proportional to ring density so they are a measure of how loose the weave is. Whatever. You find the proportionality constants for a given weave and ring combination by weaving a rectangular patch and then calculating how much of an inside diameter each ring spans. So the simplest possible weave, ordinary chain link, has a proportionality constant of exactly 1 because each ring spans one inside diameter. For a more practical explanation, let's say you are using 1 inch rings (for simplicity's sake) and you observe that it takes 20 rings to span 10 inches. Then your proportionality constant would be (10"/1")/(20 rings) which equals 0.5. Clear as mud? Good.

An Example
For an example we will use Kingsmaille (European 8 in 2) with 14 ga 3/8" rings. Because we are working with Kingsmaille, which is a double weave, we will call each double ring of the Kingsmaille a pair and do all our calculations in terms of pairs, not rings. Otherwise it gets even more confusing than it already is, sheesh! I have experimentally determined the values of kh=0.74 and kv=0.895 for this weave. We will make a three inch diameter sphere, so r=1.5". The inside diameter of the rings, d, is 3/8". Plugging these into the equation we get:

m = 34 * cos( n / 4.5 )

We start with the equator at n=0 and calculate m which is the number of pairs to weave into the equator. Then we increment n to 1 and solve for the number of pairs to weave into the rows that are adjacent to the equator. Then we do n=2, and so on until we get either zero or a negative number at which point we stop and discard that last value of m because it is non-physical. If we were to solve for theta and plug in those values of n, we would find that m=0 corresponds to theta equal to exactly 90 degrees, and negative values of m correspond to theta greater than 90 degrees. We are only interested in the region of theta from 0 to 90 degrees. The table for this example is shown below:

n m
0 | 34
1 | 33
2 | 31
3 | 27
4 | 21
5 | 15
6 | 8
7 | 1

This means that we will make a sphere starting at the equator with 34 pairs, then weave rows above and below the equator that get smaller and smaller until we reach row 7 with 1 pair. That 7th row presents us with a problem: how is a row with only one pair going to look? It would be kinda weird to just have two rings on a row at each pole of the sphere. The solution is to adjust the radius of the sphere to a slightly smaller value than 1.5, say 1.45 for example. Then we get six pairs on the row six and a negative value for row 7. This will leave us with a circle of rings with a small hole at the poles of the sphere, and due to the tightness of the weave the rings will hold each other in place without the need for a specially sized ring to gather them together. I find that to be kinda kewl that it works out that way.

A sphere similar to the example sphere is shown below:

Image: geometrysphere.jpg

Obviously the sphere will have an odd number of rows. There is a different form of the equation for making a slight variation on the theme, spheres that have an even number of rows such that the equator actually lies on the joining boundary between two rows instead of the center of the largest row. The equation for a sphere with an even number of rows is:

m = ( 2*pi*r / (kh*d) ) * cos( (kv*d/r) * (2n+1) / 2 )

where you begin with n=0 as before, but n=0 corresponds to the two rows that surround the equator. Really, though, you wouldn't want to make a sphere with an even number of rows unless you had a special reason to do so. That's because for a European weave sphere, an odd number of rows allows you to weave the first three rows all at once using the ordinary way of starting a European weave. After that it's pretty much going to be ORAAT (One Ring At A Time), unfortunately. I suppose in special cases if you got to where you were mass-producing a specific sphere design, you could slip in some closed rings at the right locations between reductions, but aside from that it's ORAAT for spheres.

Some Suggestions
There are some details about the weaving technique that will help you make the most spherical sphere possible. These suggestions will help smooth out irregularities in the spherical surface. If you follow these instructions you will be learning from the mistakes of others, namely me, which will reduce the possibility of creating a useless jumble of rings instead of a spherical sphere.

First, you should space the reductions out as evenly as you can around the row.

Second, you should arrange the reductions so that they do not line up on adjacent rows. In fact, it's best if they do not line up on the row after that either.

Note that you can make slight adjustments to the sphere radius in the equation to create a rings per row table that works out better. For example, if you have three reductions on a row and five reductions on the next row, that is not very good because they are both prime numbers and therefore have no common factors. In other words, an equilateral triangle and a pentagon do not line up elegantly if you are trying to prevent their vertices from overlapping. Better would be a situation where you have three reductions on a row and six reductions on the next row because three divides into six and that means that things can line up nicely without overlapping reductions.

You can also "cheat" to make your rings per row table work out nicely by changing the number of rings on one or more of the rows. Although it is not mathematically kosher to add or subtract a ring or two on any given row, you may find that the overall roundness of the sphere is enhanced by doing this. It is especially useful for weaving really small spheres of very few rings. In any case, having things line up nicely is a lot easier when it comes to counting out ring positions and such.

It is possible to find spherical spheres with rings per row such that all of the reductions line up nicely like this, but that is rare. Try to at least get the poles of the sphere to line up well since you have less room to space out the reductions there. Also, when working with Kingsmaille, it is ideal for the last row to have half of the ring pairs as the next-to-last row, and avoid spheres with very low ring count in the last row.

Image: sphereavatar.jpg

Third, with a Kingsmaille sphere you make the reductions by putting a single ring in place of a pair in two adjacent pair positions, or in other words you space the reduction over two adjacent pairs. If you make your reductions by removing a whole pair then you may have trouble getting the rings in the next row to span the gap properly, or at least I did.

Fourth, when you get to the final row or the final few rows at the poles of a Kinsmaille sphere, you can spread out the pairs into twice as many single rings. If you look closely at the photos you will see the last row is woven with individual rings instead of more sparsely positioned ring pairs. Unless you have a huge sphere, there is typically a large number of reductions relative to the number of rings on the last row, so you can do that and it works out well.

Oh, one more trick: for a European weave such as Kingsmaille, start by weaving a band with the number of pairs in the equator, then *remove* rings to form the reductions on the rows adjacent to the equator. If you don't do it that way you'll probably get a tangled mess like I did until I figured that trick out. Along the same lines, you may find it easier to remove those row 1 reduction rings on one side only, add row 2 on that side, then remove the reduction rings on the other side and finally add row 2 to the second side. This way the reduction rings of row 1 help to hold the equator together while you weave row 2 and this prevents unwanted twisting of the equator at the reduction sites.

Image: maille_sphere_2.jpg

Also, unless you are making a hemisphere, I recommend weaving equally in both directions as you move away from the equator. That is, weave the equator and its adjacent rows as a band first, then weave row 2 on both sides of the band before moving on to row 3. This way you will have your reduction spacing fresh in memory and you can make both hemispheres precise mirror images of each other. This also helps the rings relax into the correct positions relative to each other so you get a sphere, not an egg. To further help the rings relax into position, smoosh the work around a bit after adding each row. In addition to causing the rings to equalize in position, smooshing or rolling or squeezing the band is a nice relaxing thing to do in between all that tedious weaving - it's very tactile, like squeezing a stress reliever ball.

Wait, there's one more tip. You may find that using tie wraps to mark the reduction locations is very helpful. That way you only have to count once, not repeatedly count as you go along. This also reduces annoying mistakes, which minimizes the chance that you will throw the dang thing across the room in frustration! I save my bread tie-wraps for this purpose, didn't you always wonder what you could do with those things besides throw them away?

Alternate Weaves
At this time the only weave and ring combination that I have discovered to work is Kingsmaille with 14 ga 3/8" rings, and I have had success with both aluminum and bronze. You can also mix aluminum and bronze on the same sphere. Materials with more springback may not work. I tried 12 ga 1/2" rings but that made a loose sphere that collapsed into a pumpkin shape. An even worse mistake in a European 6-in-1 attempt made such a loose, flexible sphere that I just left one end off and declared it to be a small spherical bag.

My suggestion for experimenting with new weave and ring size / material combinations for spherical spheres is to select combinations that produce a semi-rigid sheet. Too loose and you get a bag, too rigid and your sphere will become too tight to weave at some point and you will end up with something akin to a bracelet. Of course, you might actually *want* a spherical bag or a spherically shaped rigid bracelet, in which case I say go for it!

Image: maille_sphere_3.jpg

If you really want to discover new spherical sphere weave / ring combinations, I suppose you could start by weaving a rectangle of the known good combination (Kingsmaille with 14 ga 3/8" rings). Then you can make rectangles of proposed choices and compare them in terms of flexibility. After all, you're going to need a small patch anyway to measure for the values of the horizontal and vertical proportionality constants. (Shouldn't you be able to base good combinations for the same weave off of the AR? That is, use the AR of your 14 ga 3/8" rings to find other Kingsmaille combos -- Ed.)

In my early attempts I made just tiny patches to estimate the constants, then jumped ahead into making the spheres. This didn't work out all that well because the constants were not accurate since they were measured from too small a patch (or just guesstimated, yikes!), and also because I had no feel for the flexibility of the weave in sheet form compared to the known good weave / ring combination. So once again I suggest that you learn from my mistakes and make patches with about 50 or 100 rings first - save yourself the grief of weaving failed spheres!

Other Shapes
The spherical derivation assumes a constant radius and a row slope with a tangent that passes through the origin, but you can break those rules if you want. If you can mathematically describe another shape in spherical or cylindrical coordinates such as an egg or an ellipse, or a Ming dynasty vase, then you can derive a mathematical expression that tells you exactly how many rings per row to weave in order to create that shape. In my case, college was about 20 years ago and I have forgotten most of the math that I used to know, so I will leave that exercise to the more mathematically inclined among us.

Also, I imagine that you may not need to describe the entire shape mathematically, just each row. For example you could draw a cross-section of a vase and measure the radius at each row height with a ruler, then calculate the rings per row from that by multiplying by 2*pi and dividing by kh*d. For this approach to work, you would have to account for sloped rows not having the full row height because they are leaning to one side or the other. That could be done by simply breaking up the curve into equal line segments of the appropriate length for the proportionality constant that you are using. For example draw your cross-section with line segments equal to the vertical proportionality constant times the inside diameter of your rings. That way you get both the horizontal *and* the vertical measurements correct.

Image: maille_sphere_4.jpg

Now I really should stop there, but I'll go just one step further. If we abandon the cylindrical symmetry altogether, we could possibly create a way to describe just about *any* contoured shape we want. It might be practical to do the shape of a person's head with nose, eyes, mouth, chin, hair, etc. all described as a very complicated set of reductions. A wide variety of sculpture could be created in rigid sheet maille using this method. In fact, I believe that someone with enough education and programming skill could turn any arbitrary mesh description of a surface into chainmaille form. I'll leave that epiphany to someone else though, just doing spheres was challenging enough for me!

You can make really round, rigid-shell, spherical spheres with a little math and a lot of patience by following the instructions in this article. The technique works well even for small spheres that cannot be made with other techniques such as weaving triangular sections together. There are many suggested tricks and techniques for ensuring that your spheres turn out well and don't end up as unfinished lumps of rings. So far the trusted weave is Kingsmaille with 14 ga 3/8" aluminum or bronze rings, but you should be able to utilize your personal mailling wisdom to create spheres of other weaves, ring sizes, and materials. Also, it should be possible to create non-spherical objects such as eggs, ellipses, and vases by using math and/or drawing it out on paper.

Be creative, have fun, and don't weave too much or you'll start dreaming about the intricate dance of math and rings like I did! Enjoy your sphere-weaving adventure!
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