From black friday toys

With magnetic beads and rods, the whole world's symmetry is under the children's finger tips.

Magnetic beads model the structure of metal, collection of beads are flexible, fluid, like metal. Beads hit each other, spread movement (heat) to each other, like metal. We can mix two different sized beads to get the model of copper and iron alloy.

The rod and beads model crystals. We notice the 3 edge, 4 edge, 5 edge and 6 edge space structure ends there. 

If we ask "(AI) expert" how many shapes we can build with rod and beads.

There are 230 different space groups that are recognized in crystallography. A space group is a set of symmetry operations that includes all of the symmetry operations of a crystal lattice. The specific symmetry operations that make up a space group depend on the symmetry of the crystal lattice it describes. The 230 space groups are organized into 32 crystal classes, which are groups of space groups that have the same symmetries. The crystal classes are further organized into seven crystal systems, which are groups of crystal classes that have the same basic symmetries. The seven crystal systems are the triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, rhombohedral, and cubic systems.

The 230 space groups recognized in crystallography are not all distinct groups with unique symmetries. Some of the space groups are derived from one another by applying certain symmetry operations, such as translations or screw axes. These derived space groups can have the same symmetries as the original group, but they are considered to be different space groups because they have different sets of symmetry operations.

For example, consider the space group P4/mbm, which has a fourfold rotational symmetry (P4), a glide plane perpendicular to the fourfold axis (m), and a mirror plane perpendicular to the glide plane (bm). This space group can be derived from the space group P4/mmm by applying a screw axis along the direction of the glide plane. The resulting space group, P4/mbm, has the same symmetries as P4/mmm, but it is considered to be a different space group because it has a different set of symmetry operations. In this way, two different space groups can have the same symmetries and still be considered distinct groups.

The 32 crystal classes are groups of space groups that have the same symmetries. Each crystal class is represented by a symbol that describes its symmetries. The symbols for the 32 crystal classes are as follows:

1. Triclinic crystal system group: P1
   
2. Monoclinic crystal system group: C2, Cs, C2h
   
3. Orthorhombic crystal system group: D2, D2h, C2v, C2h, D2d
   
4. Tetragonal crystal system group: C4, C4h, S4, C4v
   
5. Rhombohedral crystal system group: C3, C3i
   
6. Hexagonal crystal system group: C6, C6h, D6
   
7. Cubic crystal system group: T, Th, Td, O, Oh

These are the symbols used to represent the 32 crystal classes in crystallography. The specific symbol used to represent a crystal class depends on the symmetries of the class. For example, the crystal class C4 represents a tetragonal crystal lattice with a fourfold rotational symmetry about an axis passing through the center of the lattice. The symbol C4h represents a tetragonal crystal lattice with a fourfold rotational symmetry about an axis passing through the center of the lattice and a horizontal mirror plane perpendicular to the fourfold axis.

the list of symbols includes 23 different symbols, not 32. The 32 crystal classes are represented by a combination of these 23 symbols, as well as other symbols not listed above. For example, the symbol D2 is used to represent two different crystal classes: the orthorhombic class D2, and the hexagonal class D6. The symbol C4 is also used to represent two different crystal classes: the tetragonal class C4, and the cubic class T. In this way, the 23 symbols can be combined to represent the 32 different crystal classes.

Notice the above are are just some examples of the symbols used to represent the seven crystal system groups.

The seven crystal system groups are based on the type of symmetry and the number of axes of symmetry in the crystal lattices belonging to each group. These seven groups were chosen because they represent the seven possible combinations of symmetry and the number of axes of symmetry that can occur in a crystal lattice. They provide a comprehensive classification of the symmetries that can occur in a crystal lattice, and they allow for a systematic study of the properties of crystal lattices and their symmetry operations.

In two-dimensional (2D) space, there are 17 space groups, which are groups of symmetry operations that include all of the symmetry operations of a crystal lattice in 2D space. The 17 space groups in 2D space are organized into the three crystal system groups I mentioned earlier: monogonal, orthogonal, and circular. The specific symbols used to represent each space group depend on the type of symmetry and the number of axes of symmetry in the crystal lattices belonging to the group. Some examples of the symbols used to represent the 17 space groups in 2D space are as follows:

1. Monogonal crystal system group: Pm, Pmn, Pmc
   
2. Orthogonal crystal system group: Pmm, Pmc, Pm2, Pm3
   
3. Circular crystal system group: P4, P4m, P4g
   

These are just some examples of the symbols used to represent the 17 space groups in 2D space. The specific symbols used to represent a given space group can vary depending on the properties of the crystal lattices belonging to the group.

In one-dimensional (1D) space, there is only one crystal system group, which is the linear crystal system group. This group includes crystal classes with one axis of symmetry, which is the only type of symmetry that can occur in a crystal lattice in 1D space.

The specific symmetries and symmetry operations of the crystal classes in the linear crystal system group in 1D space can be represented using different symbols, depending on the properties of the crystal lattices being considered. For example, the symbol P1 can be used to represent a crystal class with no symmetry, while the symbol C1 can be used to represent a crystal class with a single mirror plane.

In addition, there are only two space groups in 1D space, which are groups of symmetry operations that include all of the symmetry operations of a crystal lattice in 1D space. These two space groups are the P1 space group, which includes all of the symmetry operations of a crystal lattice with no symmetry, and the C1 space group, which includes all of the symmetry operations of a crystal lattice with a single mirror plane. These two space groups are the counterparts of the 17 space groups in 2D space and the 230 space groups in 3D space.