devoted to the study of temari and mathematics
In the art of temari there are very few traditional ways of initially dividing up the sphere to create the design. An interesting question to consider is: What are all of the divisions possible? It turns out that the traditional Simple, Simple with Obi, C8 and C10 are almost all of the divisions possible mathematically. The addition of the seldom used C6 completes the set.
The first step to answering this question is to be precise about what constitutes a division. There are two critical attributes to all traditional divisions. First, the traditional divisions are all done with great circles. This makes sense from a practical viewpoint. It is difficult to accurately mark angles, line segments or points on the surface of a sphere without reference points, but by placing great circles and using distances along them, individual angles and points can be placed with greater accuracy. Great circles can be placed accurately using only the traditional method of paper tape and pins. The second piece to a traditional division is that the surface of the ball is divided into congruent spaces. Since one of the goals of design in temari is the symmetry and spacing of the design, this also makes sense to have in our definition. So, for the purposes of this discussion, a temari division is defined as a collection of great circles applied to the sphere so as to divide it into congruent spaces. Our problem is to list all of the temari divisions mathematically possible using the constraints of the above definition.
The solution is done in a number of small logical steps. Within each step I’ll provide a statement of the result, a link to the proof of why (if needed) and a discussion of the implications.
The simplest way to apply the great circles is to have them all meet at exactly two points, the N and S pole. This will create sections of the sphere or lunes. In order for those lunes to be congruent, the angles around the vertices must be equal. On a sphere this is equivalent to having the lines be equidistant along the equator (even if the equator is not applied).
Implications: You can create any even number of lunes by applying anywhere from zero to infinity great circles; in practice there is an upper limit based on the thickness of the thread involved. This configuration is known as the Simple without Obi division in temari terms. They are designated as S0, S2, S4 etc. The S0 is a special case where there are no great circles. See Step 3 below for a discussion of odd Simple divisions.
The congruent shapes created by the great circles must be either lunes or triangles.
If all of the great circles intersect at the N and S pole then the shapes created will be lunes as in step 1. If the great circles intersect in other places as well, triangles will always be created no matter what other shapes are present. (Why?) So, if we are to have congruent shapes they must all be lunes or triangles.
Implications: Since we have already exhausted the case of congruent lunes, we need only consider the case of congruent triangles for the rest of this study. In the paper ‘Classification of tilling of the 2-dimensional sphere by congruent triangles’ by Ueno and Agaoka, all of the possible ways to tile the sphere with triangles placed edge to edge are listed along with their number of vertices, edges, faces and the number and types of angles at each vertex. Ueno and Agaoka also provide diagrams of the tilings. There are ten that are made up of a specific triangle arranged in a specific way. And there are 10 that are infinite classes where the arrangement of the triangles is set but the number of the triangles (and hence their size) can vary, similar to the lunes in step 1. By determining which of these twenty tilings can be done with great circles we can obtain a list of all tilings possible that meet our original definition of a temari division.
There must be an even number of angles at each vertex.
This is because we are using great circles. They can’t ‘stop’ at a vertex or they would not be circles. They must go all the way through the vertex and thus will always create angles in pairs at an intersection. (Try it! Get a ball and some rubber bands and see what happens.)
Implications:
A tiling created by great circles must have 4-way intersections and the angles at those intersections must be equal.
Implications:
The angles must all be equal at each intersection (vertex).
Implications: We can eliminate 3 more tilings from Ueno and Agaoka's list since they have intersections with unequal angles. We can limit the cases of the G4n (n>=2) tiling to those where α = β and the cases of the I8n tiling to I24. (Why?)
The possible temari divisions correspond to the Ueno and Agaoka tilings of F48, F120, G4n (n >= 2 and α = β), and I24. They are related as follows: F48 is the C8, F120 is the C10, G4n is the Simple with Obi, and I24 is the C6. Add in the Simple without Obi that was discussed in Step 1 and you have enumerated all possible temari divisions. Both simple classes are infinite, since you can put any number of great circles on the sphere that intersect in only the N and S pole.
Traditionally the C6 is seldom used because the lines are a subset of the C8, but it does have some interesting properties of its own. While these five divisions are the only possibilities when we use a rigorous definition for temari division, they allow for an infinite number of design possibilities. Interestingly, they are closely related to the Platonic Solids. That will be discussed in another topic.
Other temari markings are possible but they either use only segments of great circles or they do not create congruent spaces on the sphere. Few of these are named; rather, they are detailed in the instructions and diagrams for any given design. There is a special named class called Multi-poles that could easily be the focus of an investigation all by itself. Some of the more common support line configurations are represented in the Ueno and Agaoka tilings and will most likely be studied on this site at a later date.
I would like to extend a special thank-you to Dr. Robert Dawson at St. Mary's University in Nova Scotia for his assistance in locating resources and his guidance. His research on the non-edge-to-edge tilings of triangles on a sphere provides for great temari design possibilities and will probably be investigated on this site at a later time.