Packing Efficiency
Question
Packing efficiency is defined as the percentage of space that is filled by an object as a percentage of the total available space. What is the packing efficiency of a box full of cylinders? A box full of spheres?
Hints
To find the packing efficiency for an object, consider a small tile that can be infinitely repeated. An example of this might be a circle in the middle of a square. This has a tiling efficiency of $\frac{A_{circle}}{A_{square}}=\frac{\pi r^2}{4r^2}=\frac{\pi}{4}$. However, this isn't the most efficient packing of circles in a grid.
For cylinders, simplify to a 2D case - what is the packing efficiency of a circle?
For spheres, look up "Face Centred Cubic" to have an idea of what the packing would look like
Answer
Cylinder packing: Consider triangle made up of joining centres of 3 touching circles $$A_{triangle}=\frac{1}{2} (2R)^2 \sin(60)$$ $$A_{circle}=\frac{180}{360} \pi R^2$$ $$A_{circle}/A_{triangle} =\pi/(2\sqrt{3})=90.7 \% $$
Sphere packing: Consider FCC packing. In each corner, there is $\frac{1}{8}$ of a sphere. At each face, there is $\frac{1}{2}$ of a sphere. Therefore in total, there are $\frac{8}{8}+\frac{6}{2}=4$ spheres in the unit cube. Let the radius of the spheres be $R$, and the side-length of the cube be $a$. $$V_{sphere}=4∗\frac{4}{3} \pi R^3$$ Pythagoras: $$a=2\sqrt{2} R$$ $$V_{cube}=16\sqrt{2} R$$ $$V_{sphere}/V_{cube} = 74.1 \% $$