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Tag: Conway

simple groups

I
found an old copy (Vol 2 Number 4 1980) of the The Mathematical Intelligencer with on its front
cover the list of the 26 _known_ sporadic groups together with a
starred added in proof saying

  • added in
    proof … the classification of finite simple groups is complete.
    there are no other sporadic groups.

(click on the left picture to see a larger scanned image). In it is a
beautiful paper by John Conway “Monsters and moonshine” on the
classification project. Along the way he describes the simplest
non-trivial simple group $A_5 $ as the icosahedral group. as well as
other interpretations as Lie groups over finite fields. He also gives a
nice introduction to representation theory and the properties of the
character table allowing to reconstruct $A_5 $ only knowing that there
must be a simple group of order 60.
A more technical account
of the classification project (sketching the main steps in precise
formulations) can be found online in the paper by Ron Solomon On finite simple
groups and their classification
. In addition to the posts by John Baez mentioned
in this
post
he has a few more columns on Platonic solids and their relation to Lie
algebras
, continued here.

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capita selecta


Rather than going to the NOG
III Workshop
I think it is more fun to give a talk for the Capita
Selecta
-course for 2nd year students on “Monstrous Moonshine”. If
I manage to explain to them at least something, I think I am in good
shape for next year\’s Baby Geometry (first year) course. Besides,
afterwards I may decide to give some details of Borcherds\’ solution next year in my 3rd year
Geometry-course…(but this may just be a little bit
over-optimistic).
Anyway, this is what I plan to do in my
lecture : explain both sides of the McKay-observation
that

196 884 = 196 883 + 1

that is, I\’ll give
the action of the modular group on the upper-half plane and prove that
its fundamental domain is just C using the modular j-function (left hand
side) and sketch the importance of the Monster group and its
representation theory (right hand side). Then I\’ll mention Ogg\’s
observation that the only subgroups Gamma(0,p)+ of SL(2,Z)
for which the fundamental domain has genus zero are the prime divisors
p of teh order of the Monster and I\’ll come to moonshine
conjecture of Conway and Norton (for those students who did hear my talk
on Antwerp sprouts, yes both Conway and Simon Norton (via his
SNORT-go) did appear there too…) and if time allows it, I\’ll sketch
the main idea of the proof. Fortunately, Richard Borcherds has written
some excellent expository papers I can use (see his papers-page and I also discovered a beautiful
moonshine-page by Helena Verrill which will make my job a lot
easier.
Btw. yesterday\’s Monster was taken from her other monster story…

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Fox & Geese


The game of Fox and Geese is usually played on a cross-like
board. I learned about it from the second volume of the first edition of
Winning Ways
for your Mathematical Plays
which is now reprinted as number 3 of
the series. In the first edition, Elwyn Berlekamp,
John Conway and
Richard Guy claimed that the value of their
starting position (they play it on an 8×8 chess board with the Geese on
places a1,c1,e1 and g1 and the Fox at place e8) has exact value

1 +
1/on

where on is the class of all ordinal numbers so
1/on is by far the smallest infinitesimal number you can think
of. In this second edition which I bought a week ago, they write about
this :

We remained steadfast in that belief until we heard
objections from John Tromp. We then also received correspondence
from Jonathan Weldon, who seemed to prove to somewhat higher standards
of rigor that
“The value of Fox-and-Geese is 2 +
1/on”

Oops! But of course they try to talk themselves out
of it

Who was right? As often happens when good folks
disagree, the answer is “both!” because it turns out that the parties
are thinking of different things. The Winning Ways argument
supposed an indefinitely long board, while Welton more reasonably
considered the standard 8×8 checkerboard.

Anyway, let us be
happy that the matter is settled now and even more because they add an
enormous amount of new material on the game to this second edition (in
chapter 20; btw. if after yesterday you are still interested in the game of sprouts you might be interested in
chapter 17 of the same volume). Most of the calculations were done with
the combinatorial game suite program of Aaron
Siegel.

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COLgo

COL is a map-coloring game invented by Colin Vout.
Two players Left (bLack) and Right (white) take turns in coloring the
map subject to the rule that no two neighboring regions may be colored
the same. The last player to be able to move wins the game. For my talk
on combinatorial game theory in two weeks, I choose for a simplified
version of COL, namely COLgo which is played with go-stoned on a
(partial) go-board. Each spot has 4 neighbors (North, East, South and
West). For example, the picture on the left is a legal COLgo-position on
a 5×5 board. COL is a simple game to illustrate some of the key features
of game theory. In sharp contrast to other games, one has a general
result on the possible values of a COL-position : each position has
value $z$ or $z+\\bigstar$ where $z$ is a (Conway)-number (that is, a
dyadic integer) and where $\\bigstar$ is the fuzzy game {0|0}. In
the talk I will give a proof of this result (there are not so many
results in combinatorial game theory one can prove from scratch in 50
minutes but this is one of them). Of course, to illustrate the result I
had to find positions which have counter-intuitive values such as 1/2.
The picture on the left is an example of such a position on a 5×5 board
but surely one must be able to find 1/2-positions on a 4×4 board
(perhaps even on a 3×3?). If you have an example, please tell me.

On a slightly different matter : I used the psgo.sty package in LaTeX to print the (partial)
go-boards and positions. If I ever write out the notes I’ll post them
here but they will be in Dutch.

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antwerp sprouts

The
game of sprouts is a two-person game invented by John Conway and Michael Paterson in 1967 (for some
historical comments visit the encyclopedia). You just need pen and paper to
play it. Here are the rules : Two players, Left and Right, alternate
moves until no more moves are possible. In the normal game, the last
person to move is the winner. In misere play, the last person to move is
the loser. The starting position is some number of small circles called
“spots”. A move consists of drawing a new spot g and then drawing two
lines, in the loose sense, each terminating at one end at spot g and at
the other end at some other spot. (The two lines can go to different
spots or the same spot, subject to the following conditions.) The lines
drawn cannot touch or cross any line or spot along the way. Also, no
more than three lines can terminate at any spot. A spot with three lines
attached is said to be “dead”, since it cannot facilitate any further
action.

You can play sprouts online using this Java applet.
There is also an ongoing discussion about sprouts on the geometry math forum. Probably the most complete
information can be found at the world game
of sprouts association
. The analysis of the game involves some nice
topology (the Euler number) and as the options for Left and Right are
the same at each position it is an impartial game and the outcome
depends on counting arguments. There is also a (joke) variation on the
game called Brussels sprouts (although some people seem to miss the point
entirely).

Some years ago I invented some variations
on sprouts making it into a partizan game (that is, at a given
position, Left and Right have different legal moves). Here are the rules
:

Cold Antwerp Sprouts : We start with n White
dots. Left is allowed to connect two White dots or a White and bLue dot
or two bLue dots and must draw an additional Red dot on the connecting
line. Right is allowed to connect two White dots, a Red and a White dot
or two Red dots and must draw an additional bLue dot on the connecting
line.

Hot Antwerp Sprouts : We start with n
White dots. Left is allowed to connect two White dots or a White and
bLue dot or two bLue dots and must draw an additional bLue dot on the
connecting line. Right is allowed to connect two White dots, a Red and a
White dot or two Red dots and must draw an additional Red dot on the
connecting line.

Although the rules look pretty
similar, the analysis of these two games in entirely different. On
february 11th I’ll give a talk on this as an example in
Combinatorial Game Theory. I will show that Cold Antwerp Sprouts
is very similar to the game of COL, whereas Hot Antwerp Sprouts resembles SNORT.

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