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What is so special about multi-faceted temari?

Before I start, I would like to give a brief explanation of terminology. For a long time this type of temari and marking have been commonly referred to as a multipole or multiple center. Recently I had the opportunity to get a translation from the Japanese book Edo Temari (ISBN4-8377-0394-1) pages 70 and 71 about multipole markings. The temari were referred to as multi-faceted temari. Although all of the terms are referring to the same sort of temari they have a slightly different focus. The multi-faceted term emphasizes the idea of the hexagonal and pentagonal shapes which closely matches the mathematical idea of faces of a polyhedron. The multicenter or multipole terms emphasize the center of the faces. Mathematically that is the vertex in the center of each hexagon or pentagon that is a center of rotation for the face. Essentially all three terms: multipole, multi-facet and multi center, can be used interchangeably. Sometimes I will even shorten it up to plain old multi when there is no chance for confusion.

Some methods to create the markings

There are two methods that are commonly used to create multi-faceted markings. They each give slightly different results in terms of the geometry of the ball and in terms of the number of facets that can be created. When shown in the Japanese books they are often represented by a simple diagram of the geometry. I have seen and experimented with several different stitching paths to create these. Ultimately it does not matter what stitching path you choose as long as you are comfortable enough with it to reproduce the lines accurately. This discussion is not the place to go into specific stitching methods but I will either add a pattern investigation about it or will put together a page of links later.

Common Method 1: Subdividing the 6-part triangles

The Geometry

figure 1: 6-part triangle method

In this method, the sides of the highlighted triangle are subdivided into sections. Notice that this splits the triangle into a number of smaller triangles that are more or less equilateral. Most of those smaller triangles will be combined to create the hexagons in the multi-facet marking. The ones at the tips of the triangle will be combined to form the pentagons. The number of sections you divide the triangle side into will determine the number of facets or poles the final marking will have. In order to get the pentagons and hexagons commonly used in multi-faceted designs, you need to divide the triangle side with a multiple of three. If you use other numbers, you do not get the tiling of hexagons and pentagons that is usually expected on a multi-faceted design; you will also get some small triangles. (There's more about that later in the formula discussion.) When you use this method on a C10 you will always get 12 pentagons and a number of hexagons. The following table shows the number of facets you will get from the different numbers.

How many facets do you get?

Number to divide side by	Number of facets
3	32
6	122
9	272
12	482
15	752
18	1082

What does it look like?

On the picture the original C10 lines are done with blue and the additional marking lines are done with green. You can recognize a marking done with the triangle method by locating three of the 10-way intersections at the center of the pentagons. Those are the corners of the triangle (highlighted on the picture). Once located you can see that the additional marking lines are all parallel to the sides of the triangle (within that region).

On this picture the pentagons and hexagons are outlined so you can see them more easily. There will always be 12 pentagons and a number of hexagons. The small pentagons will be oriented so that their points are towards each other. Depending on the number used to split the triangle, there are up to three different types of hexagons: ones with 6 lines radiating from the center, ones with 8 lines radiating from the center and ones with 12 lines radiating from the center.

Common Method 2: Subdividing the 4-part diamonds

The Geometry

figure 2: 4-part diamond method

I have illustrated this method with a diamond because it allows for easier naming of the method. In some sources it is described using a smaller triangle that is the top half of the diamond but it makes no difference in terms of the geometry of the finished marking. In this method, the sides of the highlighted diamond are subdivided into sections. Like the previous method, you end up creating lots of small triangles that will be combined into hexagons and pentagons. The small triangles at the upper and lower tips of the diamond are combined to form pentagons. The number you use to divide the side by will determine the number of facets or poles the final marking will have. This one is not as commonly done as the triangle method above. It is similar to the triangle method because you will get 12 pentagons and a number of hexagons. But, unlike the triangle method, you can use any number to split the side of the diamond and still get a marking with nicely tiled hexagons and pentagons.

How many facets do you get?

Number to divide side by	Number of facets
2	42
3	92
4	162
5	252
6	362
7	492

What does it look like?

For this example, the original C10 lines are marked in red and the additional lines are marked in purple. It is probably easiest to recognize a marking done with the diamond method by again locating the triangle with the pentagon centers as its points (highlighted with yellow on picture). When a marking has been done with the diamond method the additional lines will not be parallel to the sides of the triangle. Alternately, you can locate the diamond; it has pentagon centers at its top and bottom (highlighted with green in picture) and you can see that the lines are parallel to its sides in the region.

The 12 pentagons are oriented so that their sides are facing each other. There are up to two different types of hexagons: ones with 6 lines radiating from the center, and ones with 8 lines radiating from the center.

'New' Method: Multiple of three

This method is detailed in the Japanese book Edo Temari pg. 70 and 71 (ISBN4-8377-0394-1). The author (Sensei Toshiko Ozaki) calls it the multiple of three method since it results in approximately three times the number of facets than the marking started with.

Why do we need a third method?

We've known about the first two methods for a long time and they are commonly in use. They work quite well once you get used to stitching them. So why would we even need a third method? According to the article in Edo Temari the method of three was devised to achieve a perfect split for the hexagons of the marking. But there are other benefits as well. First let's look at the idea of a perfect split.

figure 6: an imperfectly split hexagon
figure 7: a perfectly split hexagon

Both the diamond and the triangle method have hexagons that are not evenly split (see figure 6). They have an extra line that runs through the center to the middle of a side. Now, if you consider the pentagons and triangles on a C10 division you see that they are divided into 10 parts and 6 parts respectively. Mathematically those lines are all of the lines of reflection symmetry for the shapes. If the same were true of hexagons, they would be split into 12 parts (figure 7).

The multiple of three method creates those 12 parts for all of the hexagonal facets on the ball so it is more complete, perfect or symmetrical than the other markings. (Pick your favorite adjective...the translation from Edo Temari uses the word perfect.)

It may seem like a small thing but when stitching a design on these hexagons you either have to ignore the extra line or you have to add lines in to get an even split to stitch with. And that extra line becomes quite obvious when you look at the overall marking. Since those hexagons with the extra line are aligned along the original C10 marking, it can make the C10 marking show through the multi-facet lines. That is a problem if the actual marking is going to be your design since it disrupts the evenness of the look. This will be more apparent with a large number of facets.

In the image to the left all color has been removed to better demonstrate the 'ghosting' of the C10 lines. The large ball on the left is marked with the diamond method; the ball on the right is marked with the triangle method. Notice that you can clearly see the C10 lines on both balls with the lines being more obvious on the triangle method ball.

When you make sure to split all of the hexagons perfectly, it effectively hides the C10 lines in the marking, making it look more even. The ball in this picture was marked using the multiple of three method. It has the added benefit of making it easier to see the individual hexagons and how they tile together.

As in the other markings, you can use any stitching path you wish to apply the extra lines needed. The multiple of three method is one particular way to streamline the addition of those lines. You can start with a C10 marking, but unlike the other methods you can start with a different number of facets (maybe a 42) and still apply the method. Because of this you can use the method recursively to build up to a large number of facets. It does result in about three times the number of facets than you started with so you have to choose your starting number carefully if you are interested in a particular number of facets for the final marking. It is fairly easy to establish smooth stitching lines around the ball making it ideal for temari where the marking will be the design.

So, how do you do it?

figure 3: multiple of three method

I will describe the technique starting from a C10 here, but this method can be used from any multi-faceted marking starting place to achieve different numbers of facets. Notice the pentagon highlighted on the diagram and the red lines separating it into triangles. You will be placing lines to subdivide each of those triangles into 6-part triangles as shown with the blue lines. As in the other methods, you can do one triangle at a time or you can work out a more continuous stitching path. You will get a smoother look to the marking if you use as continuous a path as possible. Starting from the C10 you will get a 32 facet marking with 12 pentagons and 20 hexagons.

For the next step, locate the triangles in the pentagons and subdivide them as before into 6-part triangles. Also locate the triangles in the hexagons and subdivide those into 6-part triangles. When you have done all of them you will have a 92 faceted marking with 12 pentagons and 80 hexagons.

figure 4: multiple of three method on pentagons
figure 5: multiple of three method on hexagons

You can continue to subdivide triangles in this way to achieve ever larger numbers of facets. For example: Start with a C10, apply the method to get to a 32, apply the method again to get to a 92, then apply once more to get to a 272. You don't have to start with a C10, you can use any multi marking as a starting point and thus get to different numbers of facets for your final marking. If you want a really large number of facets, you can start small and work your way up to it with successive iterations. There is a technique of applying colors in each of the layers to create designs that only consist of the marking lines. It is detailed in the Multiple of Three Pattern Investigation. You'll find detailed pictures of how to do the multiple of three method there as well. The table below shows how many facets you will get with different numbers as a starting point. Notice that you could start with a 42, go to a 122, and then to a 362 and so on.

How many facets do you get?

Number of starting facets	Number of facets after subdividing
12	32
32	92
42	122
92	272
122	362
162	482

A more complete mathematical look

In the triangle and diamond methods you are splitting up the triangle or diamond into many smaller triangles that get recombined into hexagons and pentagons. It takes 5 of the small triangles for a pentagon. They will be the ones at the points of the triangles or at the top and bottom points of the diamond. It takes 6 of the small triangles to make up a hexagon. You can combine the small triangles in other ways if you choose, but the formulas and typical naming of the multis are based on counting pentagons and hexagons.

The multiple of three method is a bit different mathematically. In this case you are taking every intersection (or vertex) on the ball and making sure that it is evenly split into 12 (unless it is one of the original 10-way intersections). At the same time you are adding a 6-way intersection in the middle of each of the small triangles. Those new 6-way intersections become the points or vertices of the hexagons in the final marking. The vertices you started with become the centers of the hexagons and pentagons in the final marking thus, creating approximately triple the number of facets from where you started.

The formulas

The formulas used to calculate the number of facets in all of the tables so far are based upon Euler's formula for the number of edges, faces and vertices of a polyhedron: V+F-E=2. With a small bit of algebra and an understanding of what we are counting you can derive them in a reasonably simple manner. I won't do it here but if you are curious I've included a pdf with the full derivation. (It is a pdf rather than a webpage because of all the math formatting.) Here is a table summarizing the formulas for a C10 starting point:

Formulas for a C10 based multi-facet marking
6-part triangle method				4-part diamond method
Number to subdivide triangle by	facets	perfect facets after multiple of three method		Number to subdivide diamond by	facets	perfect facets after multiple of three method
X	(1/3)x10X² + 2	10X² + 2		Y	10Y² + 2	30Y² + 2

Notice the 1/3 in the formula for the number of facets for the triangle method. If you use a number other than a multiple of 3 for X you will get a fraction for an answer to the formula. That fraction amounts to small triangles that are 'left over' and not included in hexagons. Those leftovers will give you a marking with tiled pentagons, hexagons and triangles rather than just pentagons and hexagons. While it is not a commonly used geometry on temari, it might be interesting to play with and could be used to artistic advantage.

If we expand the table with numbers instead of formulas we can see some interesting patterns. Notice that the numbers in the 3rd column (the facets after applying the multiple of three method to a triangle marking) are the same as the numbers in the diamond method column. This works in reverse too, the numbers in the last column (facets after applying multiple of three method to the diamond markings) are the same as the triangle method numbers. If you look closely at where the lines are being placed in each step of the multiple of three process you can see that they are essentially placing diamond method lines on top of triangle method lines or vice versa.

Those gaps in the triangle method column are useful too. Although you don't typically use those numbers with the triangle method, you can combine them with a multiple of three method to get a marking with a specific number of perfectly split faces. For example, if I want a perfect 162 marking, I can start with a triangle method using a split of 4 and then apply the multiple of three method to that.

6-part triangle method				4-part diamond method
Number to subdivide triangle by	facets	perfect facets after multiple of three method		Number to subdivide diamond by	facets	perfect facets after multiple of three method
1		12		1	12	32
2		42		2	42	122
3	32	92		3	92	272
4		162		4	162	482
5		252		5	252	752
6	122	362		6	362	1082
7		492		7	492	1472
8		642		8	642	1922
9	272	812		9	812	2432

Want more numbers? I've created a chart with 30 rows giving up to 27002 facets (after applying the multiple of 3 method). It is available as a pdf so that you can print it and use it for reference.

Does all of this only work with a C10 starting point?

No! These processes and formulas are based on the geometry of the triangular faces of the icosahedron and a related solid, the RhombicTriacontahedron (mathworld.com link). Both of those are represented in the C10 division. The C8 division can be related to the octahedron and the RhombicDodecahedron (mathworld.com link). (There's more information about the divisions and the icosahedron and octahedron in the Platonic Solid Relationship study elsewhere on this site.) So, the diamond, triangle, and multiple of three methods can all be applied with a C8 division starting point. The resulting geometry will include squares rather than pentagons. The article in Edo Temari included the formulas for calculating the facets on a C8 that are similar to the ones above. (See next table.) They can be derived in a manner similar to the others. Although I have not stitched any examples to share yet, they are on my to do list and will probably show up in a pattern investigation at some point. (In fact, I just discovered that one of my favorite patterns in my 'I've got to try this sometime' list is done on a C8 multi using the triangle method.) I have created pdf of a table for the number of facets on a C8 multi-facet that you can access if you are interested in pursuing it.

Formulas for a C8 based multi-facet marking
6-part triangle method				4-part diamond method
Number to subdivide triangle by	facets	perfect facets after multiple of three method		Number to subdivide diamond by	facets	perfect facets after multiple of three method
X	(1/3)x4X² + 2	4X² + 2		Y	4Y² + 2	12Y² + 2

Besides using these methods on a C8, the multiple of three method is fairly generic. Potentially it could be used on any marking that starts with shapes having more than 3 sides. That will need more research and some sample temari before I can draw any definite conclusions about it.

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