How Are the Platonic Solids Related to Temari Divisions?

Because of the focus on congruent spaces, the combination temari divisions (C4, C6, C8 and C10) are closely related to the Platonic solids. It is possible to derive all of the Combination temari divisions from the projection of the Platonic Solids on the sphere.

The first step in relating the temari divisions to platonic solids is to know what a platonic solid is. Platonic solids are a special set of polyhedra (3-D shapes) where each face is a congruent regular polygon. That is, all of the faces are exactly the same size and same shape, having the same length edges and the same size angles. It turns out that there are only five platonic solids: the icosahedron made up of 20 equilateral triangles, the octahedron made up of 8 equilateral triangles, the tetrahedron made up of 4 equilateral triangles, the cube or hexahedron made up of six squares, and the dodecahedron made up of 12 regular pentagons. There is an excellent picture of them on the mathworld site that you can view from different angles. You'll find some history and background information there as well. Their mathematical explanation is a bit more involved than needed for this discussion so feel free to just look at the pictures.

Placing the Platonic Solids on Temari

For each of the Platonic Solids, the set of their verticies lie on the surface of a circumscribed sphere, which is a complicated way of saying that you can put a sphere around them and all of the points are exactly on the surface. So what? Well, that means that the Platonic Solids translate nicely to the sphere and divide the surface into congruent spaces. Here they are stitched on temari:

A finished temari stitched with 4 triangles
A tetrahedron, 4 triangles A finished temari stitched with 6 squares
A cube, 6 squares A finished temari stitched with 8 triangles
An octahedron, 8 triangles A finished temari stitched with 12 pentagons
A dodecahedron, 12 pentagons A finished temari stitched with 20 triangles
An icosahedron, 20 triangles

Platonic Solids with Temari Divisions C4, C6 and C10

Since one goal of marking a temari is to create congruent spaces to use as the basis of the design, it is not surprising to find some similarities between the Platonic Solids with their congruent faces and the standard temari divisions. A temari division divides the surface of the sphere into congruent spaces using great circles. If we project a Platonic solid onto the surface of a sphere, it also divides the surface of the sphere into congruent spaces, but the edges of those shapes are not necessarily along complete great circles. So can you just extend the edges into great circles to make a division? Well yes, you can.

A tetrahedron temari with the triangle edges extended to great circles
A tetrahedron: great circles along the edges make a C6 division. A cube temari with the squares edges extended to great circles
A cube: great circles along the edges make a C6 division. An octahedron temari with the triangles edged extended to great circles
An octahedron: great circles along the edges make a C4 (S4 w/ obi) division. A dodecahedron temari with the pentagon edges extended to great circles
A dodecahedron: great circles along the edges make a C10 division. An icosahedron temari with the triangle edges extended to great circles
An icosahedron: great circles along the edges make a C10 division.

You get the C6 from the tetrahedron and the cube. You get the C10 from the icosahedron and the dodecahedron. The edges of the triangles on an octahedron already make complete great circles, forming the C4 (or S4 with obi). So what about the C8? It is suspiciously absent. Hmmmm...

The Octahedron and the C8

An equilateral triangle with red lines drawn from the verticies to the centerConsider this. If you want to stitch a shape on a temari you will need the lines that extend from the corners of the shape to the middle of the shape to place your stitches, like the red lines in the triangle to the left. Merely having the outline of the shape doesn’t give you the guidelines you need. Notice that for the tetrahedron, cube, dodecahedron and icosahedron we do get those lines from extending the edges into great circles. But, we do not have those lines for the octahedron.

So, if we actually wanted to stitch the triangles of an octahedron we will need the lines through the center of the triangles. Conveniently we can add those lines with great circles. When we apply them to the octahedron together with the great circles for the edges, we get the C8 temari division.

An octahedron temari with great circles added through the triangle centers, making a C8 division
An octahedron with great circles through the triangle centers

Comparing the C6 and the C8

You might notice that the lines we just added to the octahedron are the same ones that were on the cube and tetrahedron in our first exercise. Alternately we could add the lines of the C4 (edges of the octahedron) to the tetrahedron and cube and also arrive at a C8 division.

A tetrahedron temari with gold lines added as described above.
A tetrahedron; the gold lines are C4 lines or edges of the octahedron completing the C8 division. A cube temari with gold lines added as described above.
A cube; the gold lines are C4 lines or edges of the octahedron completing the C8 division.

You can see that the C6 is a subset of the lines on a C8. It is not very common to find published temari designs that use a C6. More often they use a C8 with inconspicuous marking lines. That makes it easier on the temari student. Learning the techniques for marking a C8 and C10 are sufficient and, strictly speaking, you don't need to know how to mark a C6.

I've established a close relationship between the Platonic Solids and the combination temari divisions. In particular, the tetrahedron, cube and octahedron are matched to the C8 division, while the dodecahedron and icosahedron are matched to the C10 division. The octahedron is also related to the C4. And finally, the tetrahedron and cube are also related to the C6. Since the lines of the C6 are present in the C8, it is not as commonly used in temari as the C8.

Now that we have established the relationship between the divisions and the Platonic Solids, we can expand our study to other properties of the Platonics in order to find some insight into temari designs. That is the topic of the next study.

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