This is the temari that was talking so loudly to me the other day. Isn’t he a handsome fellow now that he is all stitched?

This is basically a transfer of the design from the recent green and yellow one I did on 32 centers to a C8/14 centers marking. Both are stitched with hhg aka a single continuous path. I think I like this new little guy better than the original version. I am finding that I really like stitching designs with continuous paths. There is something quite relaxing about it.

Stitching was fun because of the way the stitching path goes, and not so fun because the mari was as hard as a rock. I wonder what I put in the center of this one or what I was thinking about when I wrapped it. I definitely did not have the even hand for wrapping that I do these days. The design thread was DMC floche and the marking thread was Kreinik #8 braid. It was one of the vintage colors so is not a sparkly as some are but it suited this temari nicely.

Here’s a pic of the two designs side by side with a ruler for scale. DH tells me that I need to include objects for scale once in awhile so that my readers will know how big things really are. He was mostly referring to my thimbles but really it is true for other things too.

big = 30cm circumference, small = 17.5cm circumference

Ooops. The color in the pictures is off. Neither one is quite right but the one at the top of the post is the closest. The mari wrap is a yellowish green (more green than it appears). The design threads are royal blue and purple.

I found a little temari (less than 2 inches across) in my drawer when I was cleaning out recently. It was one I had just quickie marked with a 14 center marking when one of the chat groups was trying to figure that out. It spoke to me…

That’s right. I could hear the little voice in my head, “You know you want to stitch on me. Don’t you think I would be pretty stitched up in a version of the last continuous path design you did? Maybe you could add a little sparkle? I’m so cute and small…”

(Don’t tell anyone I’m hearing voices ok? They’ll think I’m crazy.)

So, I went digging for thread to stitch with. The colors I found just didn’t work on the wrap the little guy had so to his surprise I ripped out the marking (promising to put it back later) and added a new thread wrap. Then to be true to his wishes, I used a Kreinik braid to do the marking.

So, the marking is 14 centers which just happens to be a multipole based on a C8 rather than a C10. I marked the C8, did one pass of the multiple of 3 technique and voila! A lovely 14 center marking. There are 8 hexagons and 6 smaller squares. It is analogous to the 32 center marking on the C10. You could also get to it by using the multipole method of triangle side divided by 3.

Now I get to stitch on it!

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.