It occurred to me that a little context might make this make more sense… so in the “Stash credit update and more thimble notes” post I wrote last fall I talked about doing a comparison of thimble threads. Then in my post “Agriculture, math and thimbles” I talked about a decision making tool my DH helped to write the computer programs for.  Now just a few days ago in the “thimble thread analysis part 1″ post I described how I gathered data to assign some rankings to my thimble threads based on some basic criteria.

Now that I have my initial thread data it is time to do some math magic with it. First there is an assumption that some criteria are more important than others.

Order the importance of the criteria

I’ll order the criteria from 1 to 5 with 5 being my first consideration and 1 being my last.  That looks like this:

5.  Availability, 4. Color range, 3. Finished look on the thimble, 2. Cost, 1. Ease of use

That ordering assumes I am more interested in color than in the look on the thimble. I am not sure that is entirely true because color and look are closely related. Another possible ordering is this:

5.  Availability, 4. Finished look on thimble, 3. color range, 2. Cost, 1. Ease of use

Since the decision process is done in a spreadsheet with automated calculations, I can consider both orderings and see if they make a difference.

Do some magic with the rating numbers

DH took my rating numbers and normalized them so that the scale had a max score of 1 instead of 5.  Then he waved his magic wand and wrote a few formulas in a spreadsheet to pull the data together based on the importance of the criteria.  That result give me a list of scores that are like weighted averages.

Make it pretty so I can understand it

The scores themselves are still pretty difficult to makes sense of so he created a graph for me.  It is a line graph but you don’t read it in quite the same way you normally would.  As you move from left to right across the graph you are adding in more and more criteria to the decision.  So the points as the far left of the graph represent a ranking based on only the single most important criteria (availability) and the points on the right represent a ranking based on all of the criteria.  Cost and ease of use are really not nearly as important to me as the other criteria so the center point of the graph is the best place for me to focus my attention.

This is the graph for the second ordering of criteria: 5.  Availability, 4. Finished look on thimble, 3. color range, 2. Cost, 1. Ease of use

Analyze it

From the graph I can tell that the unavailability of the Fujix thread is really skewing the results. If Fujix thread is superior to the others across the board in other areas then I might put in more effort to find a way to get it. So, I redid the numbers assuming that I could get Fujix just to see if that changes the decision.  It isn’t really a surprise to find out that besides the availability, Fujix thread is almost identical to the Orizuru. But, that really doesn’t alter things enough to warrant the extra research I would need to do to find a source. That means I can essentially eliminate Fujix from consideration until the thread becomes easily accessible to me.

Based on these numbers with the finished look being more important than color range, the Orizuru thread does really well and is pretty obviously the best decision.

But, what if I really want a particular color?  What if the choice or range of colors is so important that I am willing to give a bit on the finished look?  How would that alter the decision about what thread to use?  For that I look at the graph using the other ordering of criteria: 5.  Availability, 4. Color range, 3. Finished look on the thimble, 2. Cost, 1. Ease of use.  It looks like this:

Now I can see that Vikki Clayton threads do better initially (availability and color) but when all things are considered Orizuru is still the better decision.

So based on this analysis, the Orizuru would be the best one for me to invest in a stash. For those times when I really do want a particular color, or a shading of colors then I can order specific threads from Vikki Clayton.

A caveat: all of this is based on my subjective rankings of the threads that I explained in this post: “thimble thread analysis part 1″. You might rank things differently, or might have a different ordering of the criteria that would make your decision different from mine.

Next steps?

I need to order some thread! Conveniently I can order a pre-selected palette of Orizuru thread from Chloe Patricia in her etsy shop.

I know it seems like this was probably way too much work for something as simple as getting some thread to use for thimbles, but I like the idea of knowing that I have really considered my options carefully and chosen the best one for me. Now I can comfortably order threads without thinking that I might be missing out on something that would have been a better choice. Besides, the little geek inside of me is really amused that I could apply such a scientific tool to this artistic decision.

Update: I had this all written and ready to publish when Chloe Patricia shared some more information with me about getting either Fujix or Orizuru from Japan. It does not really change the decision to order the initial set of Orizuru threads since having a ready made palatte of threads makes the ordering much easier. But it means that when I do get to the point where I want particular colors, I can order the Fujix  from Chloe Patricia in Japan. That way I can get specific colors for my thimbles and get the fabulous finish that those threads give. Yea!

Captain Babossa from Pirates of the Caribbean

Have you seen the movie Pirates of the Caribbean: The Curse of the Black Pearl (Disney, 2003)?  There’s a delightful scene in it where dear Elizabeth Swan is trying to use the ‘Pirate’s Code’ to negotiate with Captain Barbossa.  He craftily informs her, “…the code is more what you’d call ‘guidelines’ than actual rules.”  Well, that is how I have come to think of any instructions for marking a 42 center temari:  they are more like measurement ‘guidelines’ you can use to help you eyeball your way to a pleasing marking.

Two methods and two different results

I know of two nice ways to mark the 42 centers marking from a C10:  divide the diamond side by 2 or use a Multiple of Three (M3) method.  I’ve suspected that the markings might be different for some time now, but finally got around to doing some testing.  I still have not done the completely rigorous math for the project but I now have enough anecdotal evidence that I may not bother.  It turns out that the two methods give different results that can be quite obvious on larger temari bases (28 cm or so).

42 marking marked with a Multiple of 3 method

42 marking marked with diamond side divided by 2

Note:  For this study I used a single temari marked with a C10, added the lines for a 42 with one method, took pics and then removed the 42 lines so that I could add lines for the other method.  So the pictures really do represent a side by side comparison on the same mari and C10 marking.  Both methods were applied strictly without nudging the lines away from the prescribed distances.  For details on how to use various methods to mark multi-center temari see the multipole study I did on temarimath.info.

The M3 method

To mark a 42 with Multiple of Three you start by adding the lines for a triangle side divided by 2.  After that step you apply the M3 method.  This method guarantees the most even split of the central angles of the hexagon, and subsequently, the most regular shape to the hexagon.  However, to achieve this you sacrifice the size of the pentagons, with them being about 80% of the size of the hexagons.  (That approximation is based upon plane geometry rather than spherical geometry but it is close enough for this study.)

The Diamond side method:

To mark a 42 this way, you are targeting the 1/2 way point on the side of the 4-part diamond of the C10.  That guarantees that the distance from the center of the resulting pentagon to the edge will equal the corresponding distance on the adjacent hexagons.

By its nature the diamond method for marking the 42 creates pentagons that are closely matched in size to the hexagons.  However, the dimensions of the hexagons in the direction adjacent to the other hexagons are not adjusted, with the result of the hexagons being ‘squashed’ with respect to the pentagons.  The hexagons sides that are not shared with the pentagons are shorter and the hexagons do not retain a regular shape.

What’s the big difference?

Because I marked the two markings on a relatively large sized ball, the differences are apparent to the eye and they will make a difference when you stitch a design.  For the M3 marking, the pentagons are clearly smaller than the hexagons.  For the Diamond side marking the hexagons are clearly squashed.  It would seem that you can maximize equal size between pentagons and hexagons, or the regular shape of the hexagons but not both at the same time.

I have a bias towards wanting maximum symmetry in my markings so I lean towards creating the most regular hexagons I can.  However, recently I was (an still am) working on a design where one of the critical features (in my opinion) required an equal distance between pentagons and hexagons.  I marked with a ‘squashed version’ of the 42 and hated the marking even though it seemed necessary for the design.  There has to be something better I think.

To choose or not to choose…

The point here is that for the 42 you can use either method exactly if you are wanting the extreme end of the spectrum (equal size or regular hexagons).  The difference between the two methods will be more exaggerated on larger temari.

How do you know which aspect you want most?  You’ll need to consider the design you are making carefully to determine whether its success relies more upon the equal size of the pentagons to the hexagons or upon the regular and more symmetrical shape of the hexagons.  If you need equal size, do the diamond method.  If you need regular hexagons, do the M3 method.  And if you need something in between, do either method you want, adjusting the lines as you go.

My recommendation

My suggestion for how to start marking a 42. I was not optimizing shapes when I marked this one.

Ultimately I think that the 42 is one of those markings that benefits from a bit of sloppiness in your procedure.  The rules for the diamond method or the M3 method are more like ‘guidelines’ in this case.

Mark your C10 as accurately as you can, but when it comes time to place the points of the 42, eyeball their placement rather than relying on precise measurements.  The optimum placement for the corners of the hexagons will be somewhere slightly closer to the center of the pentagons on the side of the diamond than the 1/2 way point.  Use your eye to determine when you have a pleasing shape to the hexagon while maximizing the size of the pentagons.  I think it is slightly easier to see and judge the hexagon shapes if you start by outlining the smaller pentagons, and then fill in the needed lines to complete the hexagon split.  By using your eye to judge, rather than your ruler, you will get a marking that is best suited to the size ball you are working on.

One last note:  I think there are also implications to starting at the 42 and applying the M3 method to get to larger number of centers.  I have not fully explored that question yet but my suspicion is that you will want to start with the most regular hexagons possible for the best result on your final multi center marking.

I’ve covered some ground in my study of the 42 multi center marking that I am ready to share but it occurred to me that I probably should give a little rationale as to why I even care about this.  The details I am spending time thinking about are things that many temari artists say, “So what?” about.  This is the stuff that makes people roll their eyes at me and think that I am nuts.  I am also pretty sure there are a few out there who think, “If it is not broken, then don’t fix it.”  How can we know if it is broken if we don’t check?

It comes down to a question of procedure.  I tend to think that if you have a procedure for doing something (creating a C10 for instance) then if you follow the procedure accurately to the best of your ability you should get the best possible result.  I think this comes from my Chemistry background… (Gasp!  Not math?  I minored in chemistry in college although I did not go on to teach it or continue to explore it like I did with math.)  In chemistry lab work, when you had a procedure you followed it to the letter or you just didn’t get optimum results.  Now, to be honest, there were times when the procedures were quite loose… dump some of this into some of that … but you didn’t have the loosey goosey procedures mixed in with the precision ones on the same experiment.

However, in temari we mix procedural ideas all the time.  Consider marking a C10:

1. Make your ball as round as possible.
2. Measure the circumference accurately.
3. Calculate the C10 number using a specific formula or chart.
4. Place the lines.
5. Fudge the lines into place and tack the intersections.

Huh?  What’s up with the last step?  We go to all this work to be accurate and then we just fudge things around by eye until we think they look good?  Why bother with all of the accuracy in the first place?

I can see where a small amount of adjustment might be needed because of step 1… since we are dealing with yarn and thread, we are not going to have a mathematically perfect sphere for our canvas.  But the procedural nut in me looks at the other steps as well and wonders are we shooting ourselves in the foot?  Are there inherit errors in those steps as well so that we might as well just eyeball the whole thing and not waste our time with accuracy?

That is why I dig into markings the way I do.  I want to know if the procedure I am following is one that you follow accurately for optimum results or if it is one that is loosey goosey with room for slop.  If it is a sloppy procedure then I am not going to waste my time and effort on accuracy that just isn’t there to begin with.  Rather, I’ll embrace the procedure as guidelines that I can use to help me finesse the lines to what I want them to be.

Now, I mean no disrespect to the methods that come to us from hundreds of years of temari history.  One of the things I am able to do is to validate them for the accuracy that they do give in spite of their limitations.  But, if I can find more accurate procedures using the tools of today, I see no reason not to research and use them.  Increased accuracy in procedures provides for a more guaranteed and consistent result – as long as the accuracy does not come at too great a cost of time and effort.

I have stated in a few places now that the hexagons on a 42 center marking are not regular.  I don’t make statements like that without having proved to myself (and my very picky DH) that they are true.  But, I haven’t provided the proof anywhere outside of my head so it is time for me to write it down.  That way, if someone questions me on the statement I’ll have something to refer them to, and even better, I won’t forget that I have worked this all out in the future.  If you are someone who is willing to take my word for it and are not curious about the math part, you could safely skip the rest of this post.

What makes me think that the hexagons on a 42 are not regular?

I started this by thinking about the hexagons on a 42 center temari marking and how they might differ from those on a 32 center marking.  I know that the shapes on a 32 center marking are regular because that marking is a projection of an Archimedean solid, the truncated icosahedron, onto the surface of the sphere.  It is a special property of the Archimedeans that they have something called a circumradius… that is you can place a sphere around them so that the surface touches each of the vertices of the solid.  That means that when you project the edges of the solid onto the sphere there is no distortion and the shapes of the faces retain their regular shape.

Now on a 42 center marking you have some points where three hexagons surround one point.  A regular hexagon has 120 degree angles between the sides.  Three regular hexagons together at a point will equal 360 degrees creating a flat surface.  You have to have less than 360 degrees around a point to get a solid shape thus there is no solid analogous to a 42 marking.  Does that fact alone prove that the hexagons on the 42 marking are irregular?  Somehow I am not quite comfortable with that; I want something stronger.  So, I am pretty sure that the hexagons on a 42 center marking are not regular….but I need a way to prove it.

A proof by contradiction

There is a method of proof used frequently in mathematics known as proof by contradiction.  Let’s say you want to prove that statement A is true.  Basically you assume the opposite of A is true at the beginning of your proof and show that it leads to a contradiction. Thus, your assumption that the opposite of A is true was incorrect meaning that A itself must be true.  It may seem to be circular but the logic is sound.

Assume (for contradiction) that the hexagons on a 42 center marking are regular.  By definition, all of their sides and angles are equal.  That means the the central angle is 60 degrees because there are 6 angles around a point: 360 divided by 6 equals 60 (marked with a red arc on the diagram).  Also the angle between the sides must be 120 degrees at the point where three of these hexagons meet (marked with an orange arc on the diagram).

42 center marking diagram with some pentagons and hexagons highlighted

closeup of shaded parts of 42 marking diagram

Consider the triangle created by adding a perpendicular line from the hexagon center to the midpoint of an edge and a second line from the hexagon center to the corner of the hexagon (one of the corners with three hexagons around it).  The resulting triangle (shaded grey on the diagram)  has angles of 30 degrees at the central point, 60 degrees at the hexagon corner, and 90 degrees at the hexagon side.  The sum of those angles is 180 degrees.  On a sphere the sum of the angles of a triangle must always be greater than 180 or the triangle has no area.  We have a contradiction.  Thus, my assumption that the hexagons on a 42 center marking are regular must be false, proving that the hexagons on a 42 center marking are not regular.

What are the implications?

Since all of the C10 based multi centers created with the usual methods will have 3 hexagons at a point when you have more than 42 centers, you can use this same method of proof to prove that the hexagons on all of those are not regular either.  Curious.

Who cares?

What does that matter for temari?  For me it means there is a reason why my hexagons always look just a bit squashed when I mark a 42 and try to follow the measurements exactly.  They really are squashed rather than regular.  If I fudge the lines by nudging the pentagon sides towards the pentagon centers a bit I can get them to appear regular enough to fool the eye.  So the suggested measurement of 1/2 the diamond side can get me close, but not exactly to a pleasing marking.  Is there an alternate marking method or measurement that will give a more pleasing marking without all the fudging?  I gotta work on that one but I have some ideas of where to start.

When I really decide to study a temari marking the first thing I ask myself is, “Is it mathematically possible for the shapes created on the ball to be regular?”  It is a jumping off point that generally leads me to conclusions about how to do the marking most effectively.

Let’s start with why you would want to have regular shapes (equal sides and equal angles).  Regular shapes are pretty pleasing to the eye because of their symmetry.  If you are going to have symmetrical patterns on temari, they need to be based on regular shapes.  It isn’t that non-regular shapes can’t be symmetrical… but regular shapes are more symmetrical and so give more possibilities on the temari.

Both of these have a vertical line of symmetry, but which one looks more symmetrical to you?

So what do I mean when I say that shapes on a temari are mathematically regular?  Well, because of the thread, errors on the ball, and just human perception of the round surface it is possible for something to appear regular even if the measurements are not quite right.  That actually works in our favor on temari, making lots of markings possible with ‘regular’ shapes that otherwise would not exist.  There are implications for our marking technique depending on whether a regular marking is possible in the ideal mathematical world vs. we can make it look that way through careful manipulation of the thread.

Knowing that the shapes on a marking cannot possibly be regular based on mathematics means that you are pretty much guaranteed to have to manipulate the lines on the ball somewhat to even out the shapes.  It may be that the method of marking will help with that, but you are still likely to need to do some eyeballing work to be sure that it really looks even when you are done.  However, if you know that the shapes can be regular, and the marking method does not introduce errors, then you try to be as accurate as possible using the method to get the best marking.  I created a flowchart to show how I think about all of this.  It is not a gifted bit of graphic art, but it helps to lay out my thinking.  There’s a pdf of it here: http://temarimath.info/studies/regular_shapes_flowchart.pdf

Before you decide that I am crazy… I do not do this every time I make a temari.  I don’t do it for every marking I do either.  Just once in awhile I get a bug to study a particular marking and see what I can learn about it.  Once I’ve researched the math and come to conclusions about the best practice for marking it I can practice the new techniques if needed and move on.  I don’t need to redo the math each time I choose to use the marking.

So far I have thoroughly researched the C6, C8 and C10 although I have not written my conclusions for them formally.  (I probably should do that someday…)  Currently I am exploring the 32 center and 42 center markings.