Developers Club geek daily blog

1 year, 5 months ago
John Horton Conway claims that he did not work day in the life. This fragment from the biography "The Genius behind Game" shows what serious mathematical theories, like syurrealny numbers, can appear from entertainments and games.

image

Gnawing a forefinger of the left hand the old broken-off British teeth, with the bulked-up senile veins, with the eyebrow which is thoughtfully frowned under long ago unbarbered hair, the mathematician John Horton Conway without regret spends the time for thoughts and theoretical researches. Though he will claim that he is engaged, lazy nothing, and plays toys.

It works in Princeton though it found glory in Cambridge (being at first a student, and then professor from 1957 to 1987). Conway, 77, claims that he did not work day in the life. He means that he spends almost all the time for games. And at the same time, he is professor of Princeton of applied and calculus mathematics (already honourable). Member of Royal community. And recognized genius. "The title "genius" is often incorrectly used", - Percy Dyakonis, the mathematician from Stanford says. – John Conway is the genius. At the same time it can work in any area. And it has an intuition for any unusual things. It cannot be put in any mathematical framework".

The haughty atmosphere of Princeton as if not absolutely is suitable as base for such playful person. Buildings are built in Gothic style and tightened by an ivy. In this environment well well-groomed esthetics does not look old-fashioned. And by contrast, Conway - careless, something is eternal with a mysterious mine on the person, between Bilbo Baggins and Gandalf. Usually it can be found, loafing about in department of mathematics on the third floor in the general room. The department is in the 13-storey building Fayn Holla, the highest building of Princeton on which roof towers of the operators Sprint and AT&T; are located.; And inside the number of professors is approximately equal to the number of students. Usually in the company of the student, Conway settles or on one of couches in the general room, or in an alcove at a window in a corridor in one of two chairs looking at a board. From there Conway addresses the guest familiar to it, quoting Shakespeare with the Liverpool vivacity inherent to it:

Welcome! It’s a poor place but mine own!


(rephrases from the comedy "As to You It Will Be Pleasant": "Here the poor virgin, the duke, an ordinary-looking bagatelle, the duke, but actually to me belonging").

Conway's contribution to mathematics includes the uncountable number of games. Most of all it is known as the inventor of the game "Life" in the late sixties. Martin Gardner conducting a column in the Scientific American log called her "the most known child of reason of Conway". It is game about the cellular automaton – the machine with groups of cages which evolves on steps in discrete time. Cages are transformed, change a form and evolve in something else. "Life" is played on a grid where cages remind the microorganisms considered under a microscope.

image
Rules of the game

Strictly speaking, it is not absolutely game. Conway calls it "infinite game without player". The musician and the composer Brian Eno remembers how having seen demonstration of this game on the monitor of the museum Exploratorium in San Francisco once, felt "intuitive shock". "All system is so transparent that should not bring surprises, - Eno says. – But there are enough surprises there – complexity and harmony of evolution of point forms completely exceeds predictions". And, as it is told by the announcer in one of episodes of a TV show of Stephen Hawking of Grand Design: "It is quite possible to imagine that something like the game "Life" with several simple laws, can reproduce rather difficult things, and the intelligence can even. It can demand a grid in millions of small squares, but in it there will be nothing surprising. In our brains – hundreds of billions of cages".

"Life" became the first cellular automaton, and remains to the most known. Game led to what cellular automatons and similar simulations began to use in difficult sciences where they model behavior anything, from muravyyov to traffic jams, from clouds to galaxies. And it became classics for those who like to spend time aimlessly. The show of groups of the cages of game changing on the computer screen appeared, causes dangerous accustoming in the students studying mathematics, physics and computer sciences as well as in all people who had an access to computers. Affirmed as one of military reports of army of the USA that the hours spent behind supervision over the game "Life" cost to the government millions of dollars. Well the legend, at least, so says. Still claim that when in the mid-seventies game began to gain quickly popularity, it worked at a quarter of all computers of the world.

image
The overview of life forms which can be found in game / from the letter to Martin Gardner [clickable]

When rolls vanity (and it happens often) on Conway, he opens some new book on mathematics, finds index, and is irritated that his name is quoted most often only in connection with the game "Life". And except it a huge number of his ideas which expanded mathematics, very variously. For example, his first love – geometry, and as a result, symmetry. It was selected, having opened something under the name "Conway's constellation" – three sporadic groups in family of such groups in the ocean of mathematical symmetry. The largest group is founded on Leitch's grid representing the densest packing of spheres in 24-dimensional space where each sphere adjoins to 196560 other spheres. It also shed light on the largest sporadic group, "the monster Fischer — Gris", in a hypothesis of "Monstrous Moonshine". And its largest achievement, at least, in his own opinion – opening of new type of the numbers called "syurrealny numbers". It is the continuum from numbers including all real numbers (whole, fractional, irrational), and except them all infinity, infinitesimal values, etc. Syurrealny numbers, by its determination – "infinite classes of strange numbers, unprecedented the person hitherto". Perhaps, they will be able to describe everything, from infinite space to infinitesimal quantum values.

But how Conway came across them is the most surprising: playing and analyzing games. As in Asher's mosaic where birds turn into fishes. Concentrate on white, and you will see birds. Concentrate on red, and you will see fishes. Conway watched game like go, and saw in it something another – numbers. And when he saw these numbers, he was under impression several weeks.

image


In the best days in Cambridge in the 1970th years, Conway, it is eternal in sandals, at all seasons of the year, entering the general room of department of mathematics, usually declared the arrival by loud cotton a hand on one of beams in the center of the room. And this action was given by a ring. The ring meant the beginning of new day of games. One of games, fatbol was especially interesting (Phutball, AKA philosophical soccer).

Rules of a fatbol

Game begins with one counter ("ball") which locate on the central cross hairs of a square grid – for example, for game in go. Two players sit down from the opposite sides of a board and go in turn. On everyone to the course the player can or put one white counter ("person") on any free intersection, or execute sequence of jumps.

For a jump the ball has to be near one or several people. It moves on a straight line (to a vertical, a horizontal or diagonal) on the first free intersection for the person. The person over whom jumped leaves from a field. If the jump is carried out, the same player can jump further, all the time while the ball is near at least to one person – or can stop the course at any time. Jumps are not obligatory – it is possible to place instead the person in the field.

Game comes to an end when the sequence of jumps comes to an end on the edge or behind edge of a board, the next to the opponent ("goal"), and in this case the one who did jumps wins. A series of jumps can exactly pass across the goal line of the opponent, without coming for it. One of interesting properties of a fatbol is that each course can be played by any player.


This game was invented by Conway, but at the same time he plays it not really well.

Upon, rules of a fatbol allow the player who made very bad course to ask "it is possible, I will cry?", and if permit it, it withdraws the course and changes. But even in this case Conway does not manage to play well this game, and in general he not especially successfully plays any games – at least, not really often wins. Nevertheless, he participated in an infinite number of games in the general room, lifting them to level, worthy for serious scientific researches. But at the same time he sometimes dared to jump suddenly, grabbed a pipe under a ceiling and was shaken on it back and forth.

These representations did not make him the chief acrobat of department. Here it was bypassed by Frank Adams, the topologist who is fond of mountaineering which liked to climb under a table without concerning a floor. Conway found it frighteningly, it is impossible the serious mathematician. Professor of astronomy and geometry, Adams had reputation of the person who is difficult for pleasing which it is difficult for lecture to listen – and the person who was strictly treating himself. Colleagues considered that his ambition is based on its periodic nervous breakdowns. Adams worked as obsessed, as gave trouble to Conway. He was sure that Adams does not approve its passive weakened manner. And it forced Conway to feel guilty and to think that he has to be dismissed – and he thought of the wife and constantly increasing group of the daughters who needed to be supported.

He married Aileen Howie, the teacher of the French and Italian languages, in 1961. "He was an unusual young man, as attracted me, - she says. – We with John soon after acquaintance went to restaurant, and I stood, waited until it opens for me a door. And he told me – well pass that you cost! Most of young people opened doors, moved chairs, and so on. But it did not come to its mind. He thought in a different way. There is a door, you face it why not to open it? Likely, in it there is a logic".

After a marriage at them was born four daughters, with the periods in one, two and three years. Conway remembered their year of birth as the 1960th plus of number of Fibonacci: 1960 + 2, 3, 5, 8 = 1962, 1963, 1965, 1968.

And Conway not for nothing worried about an opportunity to lose work. By 1968 it a little what reached. Everything that it did – it played games in the general room, invented and pereizobretat rules of games which it found boring.

image
Conway plays "Life" in 1974

Conway liked games in which there are physical courses. It constantly played a backgammon on small rates (money, swept, just on interest). At the same time it did not reach heights in this game. He often risked, accepted doubles when it was not necessary to do it, and raised the stakes to 64 times from initial, just to look that he happens. And constantly spoke about mathematics. For example, "Conway's task about a piano" in which it is asked: what greatest object can be moved for a rectangular corner in a corridor of the fixed width?

It was not so interesting to it to win in a backgammon as it is interesting to investigate the opportunities which are available in game. It liked to lose specially, remaining in the tail with the worst players. His rivals often lost vigilance and began to lose. And then it did the course. Usually this strategy did not work. But sometimes he was lucky (randomness – a game element in a backgammon and therefore it does not give in to the strict scientific analysis), and then he did a dizzy victory.

While Conway sat down on a backgammon, other his colleagues were afraid to play them long, and others refused at all, being afraid that they will be involved in game and their researches will be thrown. Others claimed that Conway is a bad example for students. But it was also its plan.

Simon Norton, the child prodigy who was visiting the Eton college and received degree at the London university while he studied in the last class of high school was one of students. Having arrived to Cambridge, and being already experienced player in a backgammon, Norton easily fitted into the company. He was able to see off very quickly in mind of calculation, and became the protege Conway, solving for it problems which that could not solve. He monitored all tasks, deciding all and everyone, spied, overheard, interrupted any with shouts "a lie!", when noticed at it an error. It also had an impressive lexicon that was pleasant to Conway. It was known for capability to solve anagrams. For example, once someone cried out an anagram of "phoneboxes". Still before someone managed to raise the head and to understand what occurs, Norton proclaimed: "Xenophobes!".

Generally Conway played silly childish sports — Dots and Boxes, Fox and Geese, and sometimes it played them with children, generally with four daughters. And, of course, with the henchmen – it is frequent in games which those invented that it will be pleasant to the leader. Colin Vout thought up the game COL, and Simon Norton – SNORT, and both games consisted in coloring of areas. Also Norton thought up the game Tribulations, and Mike Guy parried, having issued Fibulations – both games similar on it, one based on triangular numbers, and another – on Fibonacci's numbers. Conway thought up Sylver Coinage in which two players in turn called different positive integral numbers, but they could not call the number which was the sum of any of the numbers called earlier. The first player calling unit lost.

Many games were included into the book "Advantageous Strategy in Mathematical Games" (Winning Ways for Your Mathematical Plays) written by Conway and two coauthors, Alvin Berlekamp, the mathematician from the Californian university and Richard Guy from University of Calgary.

image
The Trinity of pioneers of games theory agreed on the Computers in the Number Theory conference in 1969. Conway – in the row third on top, the second on the right (with a beard). Alvin Berlekamp – the fourth row on top, the sixth on the right (also with a beard). Richard Guy – the fourth row on top, the ninth at the left (with a striped tie). [clickable]

On writing of the book 15 years, partially because left that Conway and Guy liked to work nonsense, and spent Berlekamp's time. He called them "a couple of blockheads". And, nevertheless, the book became the best-seller in spite of the fact that need of printing in color and in unusual fonts took away so much money that on their advertizing almost did not remain. The book was the textbook how to win in games. Authors poured out in it theories, in abundance, and added a set of the new games suitable for the theoretical researches.

Conway wrote:

We invented new game in the morning in order that it served as the application of our theory. And after half an hour of research it was nonsense. Therefore we invented another. In the working day ten times on half an hour, roughly telling and therefore we invented 10 games a day. We investigated them, sifted, and about one of ten was rather good to be included in the book.


So they gathered games without names, and the name without games.

We had "a marriage problem". We invented game and if it was good, then there was a problem somehow it to call. If the name was not thought out, we placed it in the folder of "game without name". And then the scrupulous and loving accuracy Richad got one more folder, "names without games". All attempts to invent a name for game came to an end with the invention of a heap of different names which badly approached the necessary game, but in itself were quite good. Therefore they went to the folder of "the name without games". Each of folders grew, and we seldom managed "to marry" among themselves records from two folders.

I remember the best name without game. It sounds as "do not call us, we will call you" (Do not Ring Us, We’ll Ring You; ring – "to call", and also – "ring", or "to lead round a ring"). At us hands did not reach the invention of game, but its essence is absolutely clear: players would draw something on paper, and the purpose would be in leading round a ring of the opponent. For such game it would be the fine name. But we did not think up game.


image


Sometimes Conway visited Martin Gardner, and they exchanged materials on mathematical entertainments. It not necessarily were games – it could be puzzles, and other entertainments for nyord. Let's take, for example, Algorithm of the Doomsday which allowed to define a day of the week for any date. And though it showed this trick from teenage age, the algorithm was thought up during visit to Gardner. Conway arrived to New York and waited until his friend takes away it from the airport. He waited, and waited … And Gardner all did not appear.

At first I thought – all right, it will appear in five minutes. But I waited there very long – I do not know, the whole hour can. And I began to think: "And what if it does not appear?". I even had no its phone. And even if also was, I did not know how to call by telephone in America. Therefore I needed only to sit there and to hope.

Gardner was late more, than for two hours, ran in to the airport hall, without restraint swinging hands, with apologies and promises: "You will forgive me as soon as you learn that I found!". It was in public library of New York where found the note published in 1887 in the Nature log: "How to learn a day of the week for any date". Article was written by Lewis Carroll. He wrote: "Having come across the following method of calculation in mind of a day of the week for any selected date, I send it to you, hoping that it will interest any of your readers. I not really quickly consider, and usually at me leaves on this calculation of seconds on 20. But I think that it can take less than 15 seconds quickly considering person".

Gardner could not refuse to himself production of a photocopy of a note, but to the copy automatic machine there was long queue. He got up in it, but it very slowly moved. By then, when it became clear that it is late to take away Conway, it already staid 30 minutes, and decided that 15 more minutes will be enough for it. He considered that it is worth it, and knew that Conway will agree with it.

to be continued

This article is a translation of the original post at habrahabr.ru/post/274081/
If you have any questions regarding the material covered in the article above, please, contact the original author of the post.
If you have any complaints about this article or you want this article to be deleted, please, drop an email here: sysmagazine.com@gmail.com.

We believe that the knowledge, which is available at the most popular Russian IT blog habrahabr.ru, should be accessed by everyone, even though it is poorly translated.
Shared knowledge makes the world better.
Best wishes.

comments powered by Disqus