The preceding two pages discuss symmetry in an intuitive fashion without actually defining it. Perhaps it is the best way. Symmetry simply means that one can draw many planes through the center of the globe. Then one hemisphere is a mirror image of the other. If you rotate the plane around the core, it will appear symmetric; the right hemisphere will be a mirror image of the left. In the case of a smooth, true globe, one can insert infinitely many planes through the center, and one hemisphere thus derived will be the mirror image of the other hemisphere, no matter what plane is chosen. A true globe would be smooth everywhere on the surface.
If there are 100 elements in a globe, its surface is not truly smooth and it cannot be a true globe; it is a "globe" in the sense of approximation. A pentagonal dodecahedron is an approximate globe. One can draw many planes through the center so that the dodecahedron is symmetric. For instance, in Picture 6, a vertical plane through the center can divide the dodecahedron equally. However, you can rotate this plane a little bit, say 5 degrees, and each half of the dodecahedron will not be a mirror image around this plane. Neither the pentagonal dodecahedron nor the soccer ball is truly symmetric; they are not true globes. However, a plane can be derived from each spoke so as to make the dodecahedrons and soccer balls symmetric.
This suggests that we need to confine our discussion to the realm of finite symmetry, in contrast to infinite symmetry, which is observed in a perfect globe. Three planes containing these spokes can be extended from the core, and if the structure is symmetric around these planes, we can say that it shows finite symmetry. As the number of planes of symmetry increases, the structure increasingly resembles the perfect globe.
For instance, in Picture 1, the red stick emanating from the center defines an axis, and one can insert a plane that separates the tetrahedron into two symmetric pieces. But if this plane were rotated a little around the imaginary center (the tip of the red stick), the tetrahedron will not be symmetric around the new plane.
Picture 1
Why is finte symmetry important?
If finite symmetry is tolerated, it is possible to put one ultimaton in the center. Three ultimatons equally spaced defines a flat plane. If we do not want the electron to be a flat disk, we need two other triangles, so that the planes defined by the three triangles are perpendicular to one another. These three triangles use up nine ultimatons, and nine spokes. Thus, a possible structure is:
(1) -- 1 -- 10
where as before, the center element ( ) refers to the core, and hence (1) means that the center has one ultimaton, and the first element (also 1) refers to the number of ultimatons on the first layer, and so on. Since nine spokes emanate from the center of this structure, the total number of ultimatons in the first and second layers is 9 x 11 = 99, and including the center, we have exactly 100 ultimatons.
A more likely structure with three layers is:
(1) -- 1 -- 3 -- 7
A typical spoke would be similar to Picture 3, if you can force an extra ultimaton on the last layer somewhere by zigzagging the sticks. This structure defines a flat surface on the last layer, but it is supposed to be spherical, and if the sticks are flexible, you might be able to squeeze an extra ultimaton there easily. Unfortunately, Douglas has not been able to construct a model due to inflexibility of our tools.
Picture 3
