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How Can Platonic Solids be Used for Temari Designs?

Relating the Platonic Solids to the Temari Divisions

There are five Platonic solids but only four combination temari divisions. In the first Platonic Solid study I demonstrated that the Platonics are closely related to the combination temari divisions as follows:

• C4 - octahedron. The C4 is also the same as the S4 with obi division and mathematically is slightly better placed in the Simple group. By itself it is not useful in this discussion, but we'll consider it as a subset of the lines on the C8 and C10.
• C6 - tetrahedron and cube. The C6 is seldom used because it's lines are a subset of the more common C8. But I think that in some cases it gives a better way of viewing the tetrahedron.
• C8 - tetrahedron, cube and octahedron.
• C10 - dodecahedron and icosahedron.

The relationships between platonic solids are often shown by comparing the number of edges, vertices and faces that each one has. This information is summarized in the following table.

	Edges	vertices	Faces	Edges per face	Faces per Vertex
Tetrahedron	6	4	4	3 (triangles)	3
Cube	12	8	6	4 (squares)	3
Octahedron	12	6	8	3 (triangles)	4
Dodecahedron	30	20	12	5 (pentagons)	3
Icosahedron	30	12	20	3 (triangles)	5

Look at the entries for the cube and octahedron. They each have the same number of edges, but the number of faces and vertices are switched with each other as are the numbers of the edges per face and faces per vertex. You'll notice that the same is true of the dodecahedron and icosahedron. The reason for this relationship is the duality property of the platonic solids that is explained later in this page. This is also why the cube and octahedron are associated with the C8. And it is why the dodecahedron and the icosahedron are associated with the C10.

There is a similar table for the combination temari divisions. In this case the headings are the types of intersections created on the ball.

	4-way	6-way	8-way	10-way
C6	6	8	0	0
C8	12	8	6	0
C10	30	20	0	12

Notice the similarities between the two tables. The number of edges in the Platonic solids corresponds to the number of 4-way vertices on the temari divisions. The numbers for the vertices and faces are related to the number of 6-way, 8-way and 10-way intersections. The C6 and tetrahedron relationship is different because the number of faces plus the number of vertices is equal to the 6-way intersections. This difference has to do with the duality property of the tetrahedron.

The information in these tables can be used in two ways: to help determine the division used for any given temari pattern and to help devise a method for embedding the platonic solids on the divisions.

Identifying the Division Used for a Temari Pattern

It is often the case when creating temari that we see a design we would like to recreate but unfortunately there are no diagrams or instructions written in our own language. You can actually glean a fair bit of information just by examining the picture closely. Designs stitched on Simple divisions are fairly easy to spot; they have very clear north and south poles and you can usually count the number of lines needed to create the sections. Combination divisions can be more challenging to determine.

For starters, notice the entries for the dodecahedron and icosahedron in the first table above. Those are the only solids with the number 5 in them. They correspond closely to the C10 temari division. That means if you see five sided shapes or vertices with five shapes coming together you can deduce that the design is worked on a C10 division. You can extend this even further with multiples of five. The C6 and C8 divisions just do not have the structure to easily create shapes in multiples of five.

Now, examine the entries for the cube and octahedron. The number 4 is unique to them with the cube having four sided faces and the octahedron having vertices with four faces. In this case if you are examining a temari that has four sided shapes or four shapes coming together at a vertex (and not any five sided ones) then you are most likely looking at a design worked on a C8 division.

Looking at the number of design centers on a temari can also help in identifying the division used. Design centers often correspond to the faces of the Platonic Solids. A design with 4 centers generally matches the geometry of a tetrahedron and might be worked on a C6 although a C8 is more commonly used. A design with 6 centers usually matches a cube and a design with 8 centers often matches an octahedron. Both would most likely be done on a C8. Lastly, a design with 12 centers corresponds to a dodecahedron and a design with 20 centers corresponds to an icosahedron; both of which would be done on a C10.

Understanding the geometry of the divisions and their relationship to the Platonic Solids is a very useful way to start deciphering unknown patterns. However, these ideas are just guidelines to use; they are not necessarily foolproof.

Placing Platonic Solids within the Temari Divisions

An interesting property of the platonic solids is that of embedding. Any platonic solid can be embedded into any other platonic solid. Similarly any platonic solid can be circumscribed around any other platonic solid. For temari you are working entirely on the surface of the sphere so you don't really care whether you are embedding or circumscribing. All that matters is that you can match the set of vertices, edges and faces of one platonic solid with those of another in a logical way. The combination divisions are a projection of the platonic solids onto the surface of a sphere. That means you should be able to place any of the platonic solids on any of the temari divisions.

Some of these pairings are more obvious than others. The tables in the first section give clues of how you might embed the solids. Consider the cube and the C8. The cube has twelve edges and the C8 has twelve 4-way vertices. Those 4-way vertices are nicely matched to the edges of the cube. The cube has six faces and the C8 has six 8-way intersections. So, the center of the faces of the cube are at the 8-way intersections. Lastly, the cube has 8 vertices and the C8 has eight 6-way intersections. So, the vertices of the cube are at the 6-way intersections of the C8. The lines of the C8 naturally create 6 square areas that correspond to the face of the cube.

Although all of the embeddings are possible, they are not all as obvious as the cube on the C8. And, just because an embedding is possible does not mean it is useful for the creation of temari designs. In some cases it makes a lot more sense to use one division rather than another for embedding a particular solid because the division has better marking lines for placing the vertices of the embedded solid without having to do additional measuring. In some cases it is possible to embed the solid in more than one way giving arrangements that correspond to compound polyhedra. The following table summarizes how to place each of the five Platonic Solids onto the C6, C8 and C10 temari divisions.

	C6	C8	C10
Tetrahedron	Connect four of the 6-way intersections leaving the other four 6-way intersections in the center of the triangles formed. All edges needed are already part of the division lines. There are two different tetrahedra that can be placed on a C6. See notebook page #060304 for diagram.	The tetrahedron is placed in the same way as on a C6 with the vertices on the 6-way intersections. The edges of the tetrahedron are the diagonals of the squares. The faces of the tetrahedron are commonly used on '4-center' temari designs. See notebook page #060304 for diagram. There are two different tetrahedra that can be placed on a C8.	The vertices for the tetrahedron are on a set of four 6-way intersections. There is another set of four 6-way intersections at the center of the faces. The edges cut across two pentagonal sections and there are three 10-way intersections in the interior of each face. There are ten different ways to place the tetrahedron on the C10 division. See notebook page #060604 for diagram and more details.
Cube	The 4-way intersections are at the center of the square faces of the cube. The 6-way intersections are the vertices of the cube. All edges needed are already part of the division lines. See notebook page #060305 for diagram.	The 8-way intersections are at the center of the square faces of the cube. The 6-way intersections are the vertices of the cube and the 4-way intersections are in the middle of the edges of the cube. All edges needed are already part of the division lines. See notebook page #060305 for diagram.	The vertices of the cube are on a set of eight 6-way intersections. The edges of the cube cut off the top of a pentagon shape on the C10 and are not already lines in the division. The center of a face is at a 4-way intersection. There are five different ways to place the cube on the C10. See notebook page #060602 for diagram and more details.
Octahedron	Although you can place an octahedron on a C6, the necessary lines for the stitches are not there. You would need to take the stitches in between division lines. For that reason it is better placed on a C8.	The vertices of the octahedron are the 8-way intersections. The 4-way intersections are in the center of the edges and the 6-way intersections are at the center of the faces. The edges correspond to the perpendicular lines through the squares, not the diagonal lines. See notebook page #060306 for diagram.	The vertices of the octahedron are a set of eight of the 4-way intersections. The center of a face is a 6-way intersection. The edges go completely across one pentagon and along the edge of an adjacent pentagon following the division lines. There are 5 different ways to place an octahedron on a C10. See notebook page #060603 for diagram and more details.
Dodecahedron	To place the dodecahedron on the C6 we will need either the 12 points for the center of the faces or the 20 points for the vertices. If you refer to the table in the first section, you can see that the C6 does not have enough points marked so we would have to add more to place the dodecahedron. Essentially this involves measuring and marking a C10 on top of the C6. In this case it would be better to start with the C10 to place the dodecahedron and then use the tetrahedron embedding to place the lines of the C6 if needed.	Similar to the C6, the C8 does not inherently have the points necessary for the dodecahedron and we would need to add some to place it. (Although there are twelve 4-way intersections on a C8 they are not placed correctly for the twelve faces of a dodecahedron.) In this case it is better to use the C10. You can use the cube embedding on a C10 to add the C8 lines if needed.	The 10-way intersections are at the center of the faces. The faces are the pentagons outlined by the lines of the division. The 4-way intersections are at the midpoint of the edges and the 6-way intersections are the vertices. All edges needed are already part of the division lines. See notebook page #060307 for diagram.
Icosahedron	Like the dodecahedron, the icosahedron has more points needed than are in place on a C6 division so the C10 is a better place to start. You can use the tetrahedron embedding on the C10 to place the C6 lines if you need them.	This case is also similar to the dodecahedron. You would need to measure and place more points on the C8 to stitch the icosahedron. It is better to use the C10 for the icosahedron, using a cube embedding to add the lines of a C8 if necessary.	The 10-way intersections are the vertices of the icosahedron. The 6-way intersections are at the center of the faces and the 4-way intersections are in the middle of the edges. All edges needed are already part of the division lines. The triangular faces are created by taking the connecting lines between three of the 10-way intersections. See notebook page #060308 for diagram.

Some of these configurations are commonly used in published temari designs. In particular the cube faces on the C8, tetrahedron faces on a C6 or C8, the octahedron faces on a C8, the dodecahedron faces on a C10 and the Icosahedron faces on a C10 are all well represented in published sources. Others such as the tetrahedron, cube or octahedron on a C10 are much less intuitive so they are rare. Nonetheless, they provide opportunities for creating interesting and beautiful designs.

Combining Platonic Solids

Since you can put any of the platonic solids on any of the divisions you can then combine their geometries to create designs, perhaps stitching a cube and a dodecahedron on the same ball. Some of the combinations are easier to stitch and more common than others. In particular there is a special relationship between some pairs of solids called duality.

Duality

Each polyhedron can be paired with another special polyhedron called it's dual. To find the dual of a polyhedron you place a new vertex at the center of each face, then connect these vertices with new edges. This new set of vertices, edges and faces is the dual of the original polyhedron. Platonic solids are special in that the dual of each Platonic solid is also a Platonic solid. Recall the tables at the top of this page and how the numbers for the faces and vertices were switched for some of the solids. Those solids are duals of each other as shown below.

This temari has been stitched with orange squares as the faces of a cube, and purple triangles as the faces of an octahedron. Notice how the points of the triangles come to the center of the squares, and the points of the squares come to the center of the triangles. The cube (square faces) and the octahedron (triangular faces) are duals. Because you need the C8 lines to stitch an octahedron, this temari and any that use the cube/octahedron dual relationship are stitched on a C8 division.

This temari has been stitched with green pentagons as the faces of a dodecahedron, and orange triangles as the faces of an icosahedron. Notice how the points of the pentagons come to the center of the triangles, and the points of the triangles come to the center of the pentagons. The dodecahedron (pentagonal faces) and the icosahedron (triangular faces) are duals. Temari using the dodecahedron/icosahedron dual relationship are stitched on a C10 division.

Lastly, this temari has been stitched with green triangles as the faces of a tetrahedron, and purple triangles as the faces of another tetrahedron. Notice how the points of the green triangles come to the center of the purple triangles, and the points of the purple triangles come to the center of the green triangles. The tetrahedron is called self-dual because the dual of the tetrahedron is another tetrahedron. Dual tetrahedra designs can be stitched on a C6 division or C8 division. I find it slightly easier to see the tetrahedron in the lines of the C6, but few published sources use it.

The property of duality is used in many design variations. The overlapping areas of the duals' faces can easily be stitched so that there are no gaps left making it an ideal geometry for all-over designs. By staggering the starting places for the shapes or changing how they overlap you can create many different effects including some interesting negative space designs. Duality of the Platonic Solids is well represented in published sources because it is so versatile.

Beyond Duality - other platonic combinations

If you allow no repeats, there are 26 different ways to combine the Platonic solids into pairs, triplets, foursomes and fivesomes including the three dualities listed in the previous section. (What are they? How do you know that is all?) Since some of the solids can be embedded in more than one way on a division (like the 5 cubes on a C10) you can create even more possibilities. Mathematically speaking, these combinations correspond to the compound polyhedra. Compound polyhedra are a large enough topic to warrant a study all to themselves, so I won't delve into them here other than to state that once again, some combinations are more common in published sources than others. It may be that the rare ones are much more difficult to describe and stitch so do not get included, or it could be that they just have not been fully explored yet.

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