I am a high school student doing a mathematics research project and I am in dire need of help! My math project is "determining the crystals needed to bedazzle a top" and the topic might seem extremely silly to most, but when choosing what to write about, I simply could not think of anything else.
So far, I have calculated the total area of the pattern piece, which is about 1144,758 cm^2 and I divided this result by the area of a single crystal with a diameter of 8.4 mm, which gave me the absolute maximum amount of the crystals needed, which is about 2067 pieces. Afterwards, I divided the total area of the pattern piece by the square in which the crystal fits, which is 0.7056 cm^2, which gave me 1623 pieces which is the minimal amount of the crystals needed. This all due to the fact that I will be placing the crystals in the way that minimizes the space unused as seen in the second picture that I attached.
As you can see, the produced interval is quite huge and I would like to make it more precise. I have been trying to read many articles and solutions to problems like these and I know that @RobPratt specializes in space optimization, however, I am a high school student and these scientific papers are quite...well...scientific..., which I find a bit confusing.
I attached the photo of the pattern pieces and its measurements are 43.2 X 42.0 cm. I will be extremely helpful for any help:) pattern piece
You must find an appropriate polygon for the circle and (perhaps) the container. True, "circles" (for simplicty) behave like a squares but a hexagon is a better approximation for a circle than a square and I can see one in the center of your diagram. It's very much like how Archimedes caclulate $\pi$ using polygons. Better your approximation, the more efficient the utilization of space. It's also a tiling problem from the looks of it.
Both size and shape matter as far as I can tell.