Geometrical proof of number of lattice points in 3D lattice

Question

It is well known that number of lattice points in three-dimensional (3D) objects of simple cubic lattice, body-centered cubic lattice, and face-centered cubic lattice are 1, 2, and 4, respectively (source). What I don't get is how to prove that the value of each lattice point in these lattices, that's in corner, center of cube, and center of surface (face) are $\frac{1}{8}$, $1$, and $\frac{1}{2}$.

I hypothesized that the values were made due to its definition of lattice and primitive cell itself, that

Lattice is an infinite array of points in space that each point has identical surroundings to each other (that is the distance from each other should be same). A lattice may be viewed as a regular tiling of a space by a primitive cell.

Primitive cell is a minimum-volume cell (a unit cell) corresponding to a single lattice point of a structure with discrete translational symmetry.

(The first sentence in definition of lattice is from lecture note, the second sentence in definition of lattice is from Wikipedia)

Thus, I thought that these lattice points in a 3D lattice corresponds to lattice points in one 3D unit cell (because the primitive cell corresponds to a structure that has minimum-volume cell that is 3D unit cell) and with the definition of lattice, all points should have same distances to each other, so the 8 corners in simple cubic lattice should represent single lattice point only. But, I don't think it makes sense because the corners can have distance of $l \sqrt{2}$ and $l \sqrt{3}$ ($l$ is length of edge of cube) to other corners and these points are not identical (and can't represent as single lattice point).

1. What kind of lattice that makes the lattice point values $\frac{1}{8}$, $1$, and $\frac{1}{2}$? Many sources said that the lattices are hard spheres (atoms) but I don't think these lattices were made for that situation because I think Bravais lattices were not made for hard spheres (the primitive cell page doesn't mention any criteria for the lattice). Are the values correct for simple dot (not hard sphere) of Bravais lattices?

2. What would be the value of lattice point in middle-edge of cube?

3. Does the value of lattice point in corner work ($\frac{1}{8}$) in hexagonal 3D unit cell? Wikipedia said that the unit cell is not primitive (hence non-Bravais lattice) because of two nonequivalent sets of lattice points. I can't determine the value of each lattice point in the cell and prove it doesn't correspond to a single lattice point (because it is not primitive cell).

Is there any geometrical proof to obtain the value of lattice point in these three-dimensional lattices?

Answer 1

Just count how many unit cells share a given point.

In a cubic lattice, each point is part of 8 cubes (or 8 unit cells), hence the value 1/8. This also means that each unit cells contains 1 atom (8 vertexes, each with a value of 1/8).

The vertexes of a FCC still have a value of 1/8 as they are shared between 8 cubes. However, the atom on the face is shared by only two cubes, hence 1/2. The total number of atom in the unit cell is then 8 vertexes times 1/8 and 6 faces times 1/2, i.e. $$8 {1\over8}+6{1\over2}=4$$ atoms per unit cell.

Try to figure out the BCC on your own. Anyways, you can find all you need here to..