How many 1/2-inch diameter marbles could I fit in a 1-gallon bucket? What is the formula for determining this? Can you walk me through figuring it out for myself?
To determine how many 1/2-inch diameter marbles you can fit in a 1-gallon bucket, you'll need to calculate the volume of a single marble and the volume of the bucket, then divide the bucket's volume by the marble's volume. Marbles won't perfectly fill the entire space, so we'll also need to account for the packing efficiency.

Here is a step-by-step guide:

1. **Calculate the volume of a single marble:**
The formula for the volume of a sphere is V = (4/3)πr^3, where V is the volume and r is the radius. Since the diameter of the marble is 1/2 inch, the radius is 1/4 inch.
V = (4/3)π(1/4)^3
V ≈ 0.06545 cubic inches

2. **Convert the volume of the 1-gallon bucket to cubic inches:**
1 gallon = 231 cubic inches

3. **Calculate the packing efficiency:**
The most efficient way to pack spheres is in a hexagonal close-packed or face-centered cubic arrangement, which has a packing efficiency of about 74% (0.74).

4. **Calculate the effective volume of the bucket:**
Effective volume = Total volume × Packing efficiency
Effective volume = 231 cubic inches × 0.74
Effective volume ≈ 170.94 cubic inches

5. **Divide the effective volume of the bucket by the volume of a single marble:**
Number of marbles = Effective volume / Volume of a single marble
Number of marbles ≈ 170.94 / 0.06545
Number of marbles ≈ 2611

So, you can fit approximately 2611 marbles with a 1/2-inch diameter in a 1-gallon bucket, considering the packing efficiency.
How much space, in marbles, is in-between the marbles?
Since the hexagonal close-packed marbles have a packing efficiency of about 74%, approximately 2611 marbles make up 74% of the bucket, and the other 26% is the space in-between the marbles. The space in-between the marbles in marbles can then be calculated as:
2611 * 0.26/0.74 ≈ 917 marbles.
Well now I understand that the packing efficiency is what's needed to answer my question. Can you give me the packing efficiency for more shapes like cubes and pyramids?
Sure! This is highly relevant to the atomic packing factor (APF) in crystallography, which can be simplified as If we fill a space with identical rigid spheres in highly ordered a structure, what is the fraction of volume that is occupied the spheres. To calculate APF, we usually use a unit cell to represent the space to be filled, which is the smallest portion of the space that can make up it by being repeated over and over in three dimensions. Note that spheres in a unit cell can be shared by multiple adjacent unit cells. For example, a sphere on the face of a unit cell is shared by two adjacent cells having this same face, and the space it occupies in one unit cell should be half of its volume. Here are the APFs, or packing efficiencies of some common structures:

- Simple cubic. It is a cubic unit cell with 8 adjacent spheres on its 8 vertices, and each sphere is shared among 8 cells having this same vertex. So if the radius of the sphere is r, its side length is 2r, leading to an APF of about 0.52.

- Face centered cubic. It is a cubic unit cell with 8 spheres on its 8 vertices, as well as 6 spheres on its 6 faces. You can also think of this as placing a layer of spheres on the cross points of a square grid with a side length of 2r, and placing the spheres of the second layer in the center points of the grid of the first layer, then repeating this process to fill up a space. Itis has an APF of about 0.74.

- Body-centered cubic.  It is a cubic unit cell with 8 spheres on its 8 vertices and one in its center. It has an APF of about 0.68.

- Hexagonal close-packed. The unit cell is a hexagonal prism. There are 12 spheres on its vertices and 2 on the centers of its top and bottom faces. There are another 3 sphere inside it. If we vertically divide it into 6 identical triangular prisms, these 3 spheres are in the center of 3 of the 6 prisms separated by one. You can also think of this as placing a layer of spheres on the cross points of a triangle grid with a side length of 2r, and placing the spheres of the second layer in the center points of the grid of the first layer, then repeating this process to fill up a space. It has an APF of about 0.74.

- Diamond cubic. It is also a cubic unit cell, but the locations of the spheres is hard to describe with words. An easier way is to explain the basic structure as a tetrahedron with 4 spheres on its 4 vertices and one in the center, and this structure spreads by using the 3 bottom vertices of one tetrahedron as the top vertices of another 3. It has an APF of about 0.34.