A Crystal Is a Repeated Unit Cell
A crystal is one small box copied over and over in all three directions. That smallest repeating box is called the unit cell. Even a single grain of salt stacks this box many millions of times in a regular pattern. That is why crystals show flat faces and sharp corners, and why they produce crisp diffraction patterns under X-rays. In the end, the whole solid you can see is nothing more than one unit cell repeated.
The single gold box in the middle is the unit cell. Drag to rotate it and see its three-dimensional shape, then raise the tile count from 1 to 2x2x2 to 3x3x3. The same box copies with no gaps, and a regular lattice appears. The key idea is that once you fix one box, the whole crystal is fixed automatically.
Atoms sitting on corners and faces are shared with the neighboring boxes. A corner atom is shared by eight boxes, so its share for one box is 1/8; a face atom is split between two boxes, so 1/2; and only an atom right at the center of the box counts as a full 1. Toggle corners, faces, and body on and off to count how many atoms truly belong to one box. Corners alone give 1, adding the body gives 2, and adding the faces gives 4.
The edge length of the unit cell is called the lattice constant a. Sliding a makes the whole box grow or shrink. Even though the number of atoms n in one box and the molar mass M stay the same, the volume changes as a to the third power, so the density = n·M / (NA·a3) is recomputed at once. Shrink a and watch the same atoms get squeezed into a smaller volume, raising the density.
Crystal planes carry name tags called Miller indices. You read where the plane meets the three axes in units of the lattice constant, then take the reciprocals of those values. The (100) plane cuts the a-axis at 1 and runs parallel to the other two axes, so it never meets them. The (110) plane cuts the a and b axes, and the (111) plane cuts all three axes, forming a triangle. Press each plane to see how the slice and its intercepts change.
The three cubic lattices differ only in where the extra atoms sit. Simple cubic (SC) has atoms only on the corners; body-centered cubic (BCC) adds one at the center of the box; and face-centered cubic (FCC) adds one at the center of each of the six faces. So one box holds 1, 2, and 4 atoms respectively. Press each type to watch the atoms rearrange, which leads naturally into the close packing of the next lesson.