putting the spheres back into tetrahedra

This post has its roots in a discussion during the coffee-time, inspired by an almost empty jar of condensed milk. It seemed obvious that spherical jars do not pack as good as cylindrical jars. But how good do they pack? Some scribbling ensued.

Realising that solving the problem from the first principles might take much more than quarter an hour, I looked up the problem of close-packing of spheres, seemingly one of easier packing problems, in Wikipedia. I thought it was solved ages ago. I was wrong (or only partially right, if we are optimists.)

Sir Walter Raleigh posed so-called Cannonball Problem around 1587 to the English polymath Thomas Harriot, his scientific adviser on the expedition to the New World. Later, the correspondence of Harriot with Johannes Kepler influenced the Kepler conjecture (1611), which says that no arrangement of equally sized spheres filling space has a greater average density than that of the face-centered cubic (fcc) packing and hexagonal close packing (hcp) arrangements. Gauss proved in 1831 that the average density of close-packed spheres is π/√18 ≃ 74%. (Compare that with circle packing — or cylinder packing — density of π/√12 ≃ 90.7%.)

In 1998 Thomas Hales announced that the proof by exhaustion of Kepler conjecture was complete. It took four years for a twelve-strong panel of referees to agree with 99% certainty that Hales’s proof is correct. (How did they do that?) However a complete formal proof is still to be produced — Hales’s Flyspeck Project is said to last at least twenty more years.

What is the best container for close-packed spheres? At first, I naïvely thought that, if one can stack spheres as tetrahedral pyramids, maybe tetrahedral boxes? Provided, of course, that one knows what to do with tetrahedral boxes. Another discovery awaited me. Aristotle wrote in 350 BC in his treatise On the Heavens:

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.

It is somewhat reassuring that until recent I shared with great Aristotle the belief that one can fill space with regular tetrahedra. On the other hand, it is not. It was known at least since fifteenth century that these polyhedra do not tile space. Interestingly, the optimal packing for regular tetrahedra remains to be found. The best known packing of 85.63% was achieved as recently as 2010. Of course it is much better than 74% density of spheres, but maybe there is still room for improvement.

Even so, as this diagram shows, one can quite nicely stack the balls in more convenient cubic boxes, which do tile space for sure.

What, if any, is the moral of the story? Not every statement starting with “It is agreed that...” is true. The old problem does not mean it is a solved problem. You might as well have a go at it. Or, at least, read in Wikipedia about it.

Update (25.05.2023): The Flyspeck Project was completed in 2014 and a formal proof of the Kepler conjecture was published in 2015.

dead don't read emails

Over the last twenty or so years, I have changed several email addresses. Not too many — less than my postal addresses, in fact. It’s not a big deal. My old email addresses could be found in some papers that I authored, although I seriously doubt anyone will ever bother to check their validity. In any case, my readers had a few years to ask their questions. Now the shop is closed. I moved on. Emails happily bounce back.

But when a person dies and his or her email is still active — that by some reason freaks me out. That felt weird twenty years ago, that still feels weird today. Inbox full of unread emails. Creepy.

Now, with everyone connected by social networks, it feels weirder still. Yeah, let’s all write “R.I.P.” on the deceased’s Facebook wall, that will help.

Death of a friend. That emptiness left behind. Do we have to fill it with worthless words?

Hottabych principle

Two Young Pioneers, Volka and Zhenya, take the old genie Hassan Abdul-rahman ibn Khattab, aka “Hottabych”, to the football match. As the boys did not bother to explain the rules of the game to Hottabych, he had to figure them out himself.

The object of the game is not possession of the ball but to score more goals at the end of the game than the opposing team — something that is not immediately obvious in a match that can go for ninety minutes without scoring a single goal. So Hottabych decides to give each player a ball of his own. As I just learned, in Russian-language cyberspace this approach to football (and its extensions) is referred to as Принцип Хоттабыча (“Hottabych principle”).

— Не сочтешь ли ты, о Волька, возможным объяснить твоему недостойному слуге, что будут делать с мячом эти двадцать два столь симпатичных мне молодых человека! — почтительно осведомился Хоттабыч.
Но Волька в ответ только нетерпеливо отмахнулся:
— Сейчас все сам поймешь.
Как раз в этот момент игрок “Зубила” звонко ударил носком бутса по мячу — и состязание началось.
— Неужели этим двадцати двум приятным молодым людям придется бегать по столь обширному полю, терять силы, падать и толкать друг друга только для того, чтобы иметь возможность несколько мгновений погонять невзрачный кожаный мячик? И все это лишь потому, что на всех нашелся для игры только один мяч? — недовольно спросил Хоттабыч через несколько минут.
Но Волька, увлеченный игрой, снова ничего старику не ответил. Было не до Хоттабыча: нападение “Шайбы” завладело мячом и приближалось к воротам “Зубила”.
— Знаешь что, Волька? — шепнул своему приятелю Женя. — Мне кажется, просто счастье, что Хоттабыч ничего не понимает в футболе. А то бы он тут таких дров наколол, что ой-ой-ой!
— И мне так кажется, — согласился с ним Волька и вдруг, ахнув, вскочил со своего места.
Одновременно с ним вскочили на ноги и взволнованно загудели и все остальные восемьдесят тысяч зрителей.
Пронзительно прозвучал свисток судьи, но игроки и без того замерли на месте.
Случилось нечто неслыханное в истории футбола и совершенно необъяснимое с точки зрения законов природы: откуда-то сверху, с неба, упали и покатились по полю двадцать два ярко раскрашенных мяча. Все они были изготовлены из превосходного сафьяна.

Лазарь Лагин, Старик Хоттабыч

“Will you, O Volka, consider it possible to explain to your unworthy servant what these twenty-two pleasant young men are going to do with the ball?” Hottabych asked respectfully.
Volka waved his hand impatiently and said, “You’ll see for yourself in a minute.”
At that very moment a Zubilo player kicked the ball smartly and the game was on.
“Do you mean that these twenty-two nice young men will have to run about such a great field, get tired, fall and shove each other, only to have a chance to kick this plain-looking leather ball around for a few seconds? And all because they gave them just this one ball for all twenty-two of them?” Hottabych asked in a very displeased voice a few minutes later.
Volka was completely engrossed in the game and did not reply. He could not be bothered with Hottabych at a time when the Shaiba’s forwards had got possession of the ball and were approaching the Zubilo goal.
“You know what, Volka?” Zhenya whispered. “It’s real luck Hottabych doesn’t know a thing about football, because he’d surely stick his finger in the pie!”
“I know,” Volka agreed. Suddenly, he gasped and jumped to his feet.
At that very moment, the other hundred thousand fans also jumped to their feet and began to shout. The umpire’s whistle pierced the air, but the players had already come to a standstill.
Something unheard-of in the history of football had happened, something that could not be explained by any law of nature: twenty-two brightly coloured balls dropped from somewhere above in the sky and rolled down the field. They were all made of top-grain morocco leather.

Lazar Lagin, The Old Genie Hottabych

At this point, the referee stops the game, depriving us of the opportunity to see what could happen next. On one hand, the players are saturated with footballs. So passing the ball is possible but not really needed. Ditto tackling the opponent, unless you know what to do with two or more footballs. On the other hand, the object of the game remains the same. One still can score, and one still can defend. Of course, that requires an adjustment of the rules and some extra skills. I wonder how many footballers would be up to challenge.

As it happens, most of the magic performed by well-meaning Hottabych is not welcome. This has nothing to do with the fact that the old genie finds himself in the 1930s USSR, even though both the book and the film have their fair share of Soviet propaganda.

Much of the comedy is based on old Hottabych failing to understand what people really mean when they express their wishes. Or is it people who cannot formulate their wishes clearly and frankly? I think that the latter is true, and genie is brought into the picture simply to demonstrate that.

When a wish is granted, especially by a genie, you may come to realise that it is not what you actually want or need. For example, to possess the ball in the game of footie is an opportunity, not an end in itself. Twenty-two balls falling from the sky serve to prove it.

I suspect there is no scientist alive who never dreamed of getting a Nobel Prize. Imagine that Hottabych, in form of the Nobel Committee, awards you a Nobel Prize, sparing you years of tedious experiments, paper writing and other activities that scientists habitually torture themselves with. Will you be happy? I guess not. To quote Russell W. Belk, “the gift of an award may be a Trojan horse that destroys our initiative for further achievements”. Or, as Mikhail Zhvanetsky put it: «Процесс — жизнь, результат — смерть» (“Life is the process; death is the result”).