OLD Tetrahedron Packings
TETRAHEDRON PACKINGS
A story of the time when a bunch of undergraduates students discovered that by putting dice in a fishbowl and shaking them that they were able to destroy a leading (Princeton) mathematician’s published results!
To really appreciate what happened here though we need to take a few steps back and go through a bit of history and background info
A QUICK(ISH) HISTORY
One of the things that Geometers (mathematicians who study Geometry) used to do back in the day is tilings. The simplest form of this is using regular polygons and see which one can fill up a plane completely.
Here you can see that the triangle, hexagon, and square will all tile a 2-dimensional plane indefinitely (assuming you have enough of them I suppose). By the way, they really go down the rabbit hole with these tiling things, formally called Tessellations (if you want to research the topic yourself, M.C. Escher has some incredible Tessellations!).
So around 360BC the big new thing in math/philosophy were the Platonic solids
Which are just like a 3D version of the polygons (the triangle/hexagon/squares above). So of course it was not too long after that folks tried to do a 3D version of tilings which would come to be known as packings. This of course was much harder (since it was in 3D) and was harder to visualize.
So back in 325BC Aristotle is writing a book On Heavenly Bodies and discusses the elements earth, air, fire, water, etc., and the regular solids. He states:
“It is agreed that there are only three plane figures which can fill a space, the triangle, the square and the hexagon, and only two solids, the pyramid and the cube.”
In this context pyramid=regular tetrahedron. This may be taken to mean that Aristotle asserted: regular tetrahedrons completely pack space… If so, Aristotle made a mistake! Amazingly this mistake went unnoticed for close to 1700 years until around 1470 AD, Regiomontanus showed that Aristotle was wrong. Tetrahedrons cannot 100% pack space. Since he was able to prove that tetrahedrons could not completely pack space the question had to be asked (in 1896 by Minkowski)… how well can tetrahedrons pack space? That is, even if there are gaps what is the best way to arrange tetrahedrons (imagine you are putting them in a box) so you can fit as many as possible? Formally this is called a packing density. So, even if tetrahedrons can’t 100% fill up space can you get them to fill it 90%? 80%? What is the best packing density that tetrahedrons can do?
So now you know the history and some terms. Now let’s jump forward a bit and get to the fishbowl filled with dice.
RESULTS FOR TETRAHEDRON PACKINGS
A few mathematicians tried to solve this Tetrahedron packing density problem when Minkowski first proposed it and interest was renewed in the 1960’s and 1970’s but there were not any great results until 2006. John Horton Conway (a well-known mathematician at Princeton), and Salvatore Torquato is a chemist, materials scientist (and much more) at Princeton tackled the problem together. They found by packing 20 tetrahedrons together in a group they could nearly form an icosahedron. Using these icosahedron groups they were able to show that they could achieve a total packing density of 71.66%.
But in 2007 (the very next year) Paul Chaikin, a physics professor at New York University, working with undergraduate students, experimented with filling fishbowls and other containers with tetrahedral dice. With these experiments, they were able to get a packing density of about 75%. Of course, because the dice are not perfect tetrahedra this was not a formal proof, but even when accounting for possible error they were above the 71.66% that Conway had proposed the last year… out-mathing a Princeton professor with fishbowls and dice!
WHERE WE ARE TODAY
From 2006 to 2010 there was significant progress made on this problem. In 2010 when a research group out of the University of Michigan was able to achieve a packing density of 85.63%. But there is no reason to think that this is the best possible packing density. This is an open problem that nobody knows the answer to! Maybe one day a clever person will put those tetrahedrons together in a way that breaks a 90% packing density! As a side note, many of the models used now to achieve these high packing densities have a “shake the tetrahedrons” feature built in to make sure everything has settled as close as they possibly can be. I have to imagine this is thanks to work done by the New York University undergraduates shaking fishbowls filled with dice to try to fit more!
MORAL OF THE STORY
Many students who we have talked to have the idea that
- Math is a complete subject that was discovered hundreds of years ago and there is nothing more we can research or discover today.
- Math is always neat and tidy with variables dancing across an equals sign until the answer is all that is left.
- Math research is ungodly complicated and there is no way I could ever understand it.
So what we should all leave here with is that pretty much the exact opposite is true. Math research is dirty and messy and filled with mistakes (Aristotle made the initial mistake thinking that tetrahedrons 100% packed space, Minkowski made a big mistake too that I didn’t mention, and Conway published results that were worse than dice in a fishbowl). There are tons and tons of problems that we don’t know the answers to (some much more simply stated than the Tetrahedron packing problem displayed here) and you can absolutely understand them and can even contribute to solving them if you are so inclined.
We also believe that
- Math is interesting — It may not feel like it when you have 20 homework problems due tomorrow but if you start looking into some math topics on your own it is amazing how interesting it is.
- Math is for everyone — You don’t have to be perfect to be a mathematician or to do math. We will 100% make mistakes along the way and that is completely okay. It is part of the learning process… and who knows, maybe no one will catch our mistakes for 1700 years.
Thank you so much for taking the time to read this far! We hope you enjoyed.
REFERENCES
https://dept.math.lsa.umich.edu/~lagarias/TALK-SLIDES/icerm-clay2015apr.pdf
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