I have nothing interesting to contribute to the area of geometric packing. These are just some things about it that I think are interesting and some things that I want to note to look into in the future.
I think the thing that really got me thinking about geometric packing was Alone With You in the Ether by Olivie Blake. I read it a couple months ago, and in it there is a character who kind of uses math to distract himself from other things. That is not the point of the book, it is actually more of a love story (and I had read it because I had thought it would be a story about how somebody does NOT need a relationship, but then the whole thing was about a relationship, but I actually loved it, but it did not help me get over the person I was trying to get over, but even if it was about not needing relationships, it's not like a book is gonna fix me, but none of this is the point). Something that was periodically mentioned was the importance of the number six, and also bees and their honeycombs. So I fell down a rabbit hole about why honeycombs are hexagons.
I learned then that it was not proved until the late 1990s by Thomas Hales that the densest packing of equally sized spheres gives an efficiency of approximately 74%, and one of the ways to pack the spheres is hexagonally. Shortly after, he published a proof that hexagonal packing is the most efficient way to pack equal areas in a 2D space. I have not yet read these proofs. The first is over a hundred pages long and relied on a computer, and for the honeycomb conjecture, I have only been able to understand the introduction so far; therefore, this little summary of his work may not be wholly correct. But it is very interesting to me that the honeycomb conjecture was first known to be written about over two millennia ago, but was not proved until fairly recently.
Among other things, this has pointed out to me that I should learn about mathematical proofs. The only experience I have with them is from high school geometry. Luckily, there are a variety of decent courses and textbooks online, such as those published by MIT or approved by AIM. So theoretically I have access to the information needed to learn about proofs, although I suppose I do not have the checks given by graded homeworks and exams that would be a part of a typical upper-level math class about proofs. This is neverthelesss something which I think is worth looking into.
Once I have some understanding of upper level math concepts and notations, I would like to read Thomas Hales's Honeycomb Conjecture. Then I would like to look deeper into 3D packing. This relates interestingly to chemistry in two ways: soap bubbles and crystal structures. Crystal structures and the packing of atoms is a big part of inorganic chemistry. Soap bubbles relate less professionally, but if you wash a volumetric flask, you may notice gorgeous and fascinating shapes. A volumetric flask has a large round bulb with a long narrow neck, designed for precise measurement of one specific volume, and being made of glass, you can look into the bulb at the bubbles filling the flask from a variety of angles. You can look through the bubbles against the surface of the flask to the bubbles deeper in the flask, appearing to be planar with no curved surfaces, because they only touch other bubbles. It is very cool.
There are also interesting things about truncated octahedrons and less appealing but better packing shapes discovered later. This is, I suppose, part of the reason for writing proofs, rather than just coming up with a solution and it being the best one yet. Not that that is meaningless, but I can see why people would want to know if it is the best one yet or the best one possible. So once again I am thinking that I should learn more about formal proofs and upper level mathematics.
I do not expect to solve one of Hilbert's problems or win one of the CLI Millenium Prizes, although a part of me does certainly dream about how interesting that would be. Rather, I like having something interesting to think about. I was thinking and writing about triangles the other day at work, and one of my coworkers said how it's funny and easy to forget that some people just have math as a hobby, and another of my coworkers said of course they do, that's why we know as much as we do now. I thought that was a cool thing of him to say. Idk. I am really enjoying having something to work on and think about. Maybe I'll jinx it and go numb again now that I've said that. But this stuff is really cool.