Conway’s Glass of Coke

A Conway Life animation by Manfredas Zabarauskas

Last year, I posted here about Conway’s Game of Life, a cellular automaton that simulates certain complex processes.

A few minutes ago, I poured a glass of Coke and noticed that the bubbles were moving in patterns that vaguely reminded me of Conway Life.  So I took a video of it with my cellphone and posted it on YouTube:

Conway’s Game of Life

A celebrated Conway Life pattern, the "Gosper Glider"

Years ago, I read a piece in The Atlantic about something demographer Thomas Schelling had figured out:  

In the 1960s he grew interested in segregated neighborhoods. It was easy in America, he noticed, to find neighborhoods that were mostly or entirely black or white, and correspondingly difficult to find neighborhoods where neither race made up more than, say, three fourths of the total. “The distribution,” he wrote in 1971, “is so U-shaped that it is virtually a choice of two extremes.” That might, of course, have been a result of widespread racism, but Schelling suspected otherwise. “I had an intuition,” he told me, “that you could get a lot more segregation than would be expected if you put people together and just let them interact.”

One day in the late 1960s, on a flight from Chicago to Boston, he found himself with nothing to read and began doodling with pencil and paper. He drew a straight line and then “populated” it with Xs and Os. Then he decreed that each X and O wanted at least two of its six nearest neighbors to be of its own kind, and he began moving them around in ways that would make more of them content with their neighborhood. “It was slow going,” he told me, “but by the time I got off the plane in Boston, I knew the results were interesting.” When he got home, he and his eldest son, a coin collector, set out copper and zinc pennies (the latter were wartime relics) on a grid that resembled a checkerboard. “We’d look around and find a penny that wanted to move and figure out where it wanted to move to,” he said. “I kept getting results that I found quite striking.”

Programming computers to play this game, Schelling found that strong residential segregation arose even if he assumed that each member of the set would stay put with only a single neighbor of the same category.  This provided evidence, not only that Schelling might be right about residential segregation, but also that social order in general can arise in ways that do not directly reflect the intentions of any particular member of that society.  All of Schelling’s virtual people wanted to live in integrated neighborhoods, yet it was precisely the actions they took to pursue that goal that inexorably led to the creation of segregated neghborhoods. 

Schelling’s tests reminded me of Conway’s Game of Life, a cellular automaton that mathematician John Conway invented in 1970.  The procedure of Conway is very similar to Schelling’s.  An indefinite number of square cells are arranged in a square grid.  Each cell is in one of two conditions, live or dead.  Each cell is in contact with eight other cells: one directly above, one directly below, one directly to the right, one directly to the left, and one on each of the four corners.  If a cell is alive, it remains alive if and only if it is in contact with two or three other live cells.  If a cell is dead, it remains dead unless it is in contact with exactly three dead cells.    Some very simple initial patterns take a surprisingly long time to stabilize in Conway Life: for example, this fellow (which Conway called the R-pentomino, though others call it the F-pentomino) goes on generating new forms for 1103 generations, and along the way produces a number of spectacular structures:

You can easily test out patterns here; some especially famous patterns are collected here and here.

Conway’s Game of Life came back to mind a couple of weeks ago, when this xkcd strip appeared:

Someone came up with a cellular automaton that could qualify as “Strip Conway’s Game of Life”:

Various commenters tried to put humans in the role of the automated cells, and tried to devise rules based on what the people around each human are wearing that would determine which clothes the human was required to remove.  It occurred to me that a more promising approach would be to have one person start by wearing a great many articles of clothing, leaving those clothes on that were touching either two or three other articles of clothing, removing those that were touching fewer than two or more than three articles of clothing, and putting clothes on bare spots that were bordered by exactly three articles of clothing.  Eventually, somebody might get naked. 

Then yesterday, Alison Bechdel announced on her blog that she’d drawn a comic for McSweeney’s magazine.  The comic represents a modified version of Milton Bradley’s board game called The Game of Life

Bechdel's Life

In the comments on that post, I brought up Conway’s Game of Life.  So, I decided the time had come to post about it here.

The Atlantic Monthly, December 2008



In this issue, Virginia Postrel reports on the rising discipline of “experimental economics.”  The experiments are similar to those to which psychologists routinely subject their undergraduate students.  A group of test subjects plays a game that is supposed to simulate a market phenomenon.  The experimenters then analyze the results.  (As it happens, I recently posted a link to a discussion of the theoretical limitations of this sort of attempt to translate one game into another.) 

The studies Postrel discusses deal with the origin and nature of speculative bubbles.  Even games in which players are given perfect information from the outset regularly generate bubbles.  Experience matters; repeat players generate smaller bubbles.  One point particularly arrested my attention.  When teams of players have gone through a trading game often enough that they no longer generate ruinous bubbles, experimenters sometimes rearrange the players into new teams.  These new teams, even though they are composed of experienced players, then proceed to behave just as wildly as the teams had at the beginning of the game.  Postrel quotes one of the founders of experimental economics, Caltech professor Charles Plott,  to the effect that the experience that matters is not at the level of the individual trader, but at the level of the organization through which that trader operates.  So the key thing about experience in particular and information in general may be how the organizational principles of a given group allow that information to be deployed.  

Disgraced stock analyst Henry Blodget gives a first person account of the way the organization of his former employer, Merrill Lynch, guided his deployment of information.


Translating Games

Grocery Checkers, by Scott Moore

Grocery Checkers, by Scott Moore

Is it possible to translate games as we translate language?  That is, can a particular instance of game- one round of table tennis, say-  be said to represent a particular instance of another game- say, one swim meet- in the same way that a particular sentence of English can be said to represent a particular sentence of Latin?  It would seem obvious that the answer is no, and it probably is.  But here’s an argument that the answer might not be so obvious.  Follow the argument to the end, and you begin to suspect that if games can’t be translated into each other, then metaphors in general might be trickier than they at first seem.

A strangely fascinating website

genealogy_skeletonThe Mathematics Genealogy Project is a vast family tree connecting mathematicians to their dissertation advisors, going back in some lines to the 15th century.   It can be a compelling toy- after I mentioned Georg Christoph Lichtenberg in a post Thursday, I looked up a math professor who works across the street from me and traced his lineage back to Lichtenberg.  That’s pretty easy to do- of about 130,000 mathematicians indexed, 23,522 are descendants of one or the other of Lichtenberg’s two advisees, Heinrich Brandes and Bernhard Thibault.  So you have about a 1/5 chance that any living mathematician you choose will be a descendant of Lichtenberg.  

I don’t know anything about how mathematics works as a field, but I do know enough of certain other fields to say that a reference tool like this would be of great value to them.  For example, the research careers of most classical scholars are largely defined by their dissertations, so it would be natural to sort classicists into families defined by dissertation advisor.  Efforts have been made to copy the Mathematics Genealogy Project in some other fields; here for example is “The Philosophy Family Tree.”

Altered Chess

Thanks to for pointing to this Russian site that illustrates alternative versions of chess.  A few examples: