Tag: Conway

Mark
Ronan
has written a beautiful book intended for the general public
on Symmetry and the Monster. The
book’s main theme is the classification of the finite simple groups. It
starts off with the introduction of groups by Galois, gives the
classifivcation of the finite Lie groups, the Feit-Thompson theorem and
the construction of several of the sporadic groups (including the
Mathieu groups, the Fischer and Conway groups and clearly the
(Baby)Monster), explains the Leech lattice and the Monstrous Moonshine
conjectures and ends with Richard Borcherds proof of them using vertex
operator algebras. As in the case of Music of the
Primes
it is (too) easy to be critical about notation. For example,
whereas groups are just called symmetry groups, I don’t see the point of
calling simple groups ‘atoms of symmetry’. But, unlike du Sautoy,
Mark Ronan stays close to mathematical notation, lattices are just
lattices, characer-tables are just that, j-function is what it is etc.
And even when he simplifies established teminology, for example
‘cyclic arithmetic’ for modular arithmetic, ‘cross-section’
for involution centralizer, ‘mini j-functions’ for Hauptmoduln
etc. there are footnotes (as well as a glossary) mentioning the genuine
terms. Group theory is a topic with several colourful people
including the three Johns John Leech, John
McKay
and John Conway
and several of the historical accounts in the book are a good read. For
example, I’ve never known that the three Conway groups were essentially
discovered in just one afternoon and a few telephone exchanges between
Thompson and Conway. This year I’ve tried to explain some of
monstrous moonshine to an exceptionally good second year of
undergraduates but failed miserably. Whereas I somehow managed to give
the construction and proof of simplicity of Mathieu 24, elliptic and
modular functions were way too difficult for them. Perhaps I’ll give it
another (downkeyed) try using ‘Symmetry and the Monster’ as
reading material. Let’s hope Oxford University Press will soon release a
paperback (and cheaper) version.

If you prefer
Neal Stephenson’s Snow crash to his
bestseller Cryptonomicon
you may have a fun time reading through Jeff Noon’s
Nymphomation. In a
‘parallel’ 1999’s Manchester, blurbflies (blurb stands for
their slogans, especially for _DominoBones_, the new lottery game
which is on a year trial run in Manchester before going National.
A group of mathematics students are searching for the hidden
mysteries behind the game. Their promotor is Prof. Max(imus) Hackle who
has written a series of psychedelic sixties papers in a `journal’
called _Number Gumbo: A Mathemagical Grimoire_ with titles like
“The No-Win Labyrinth: A solution to any such Hackle maze”,
“Maze Dynamics and DNA Codings, a special theory of
Nymphomation” and “Fourth-Dimensional Orgasms and the
Casanova Effect.” He is also keen on using fun-terminology
defining processes happening in the ‘Hackle maze’ and as
such is a bit like John Conway. In fact, Conway’s Game of Life is a lot
like Hackle’s maze. There is some statistics and game
theory in the book but the plot and ending are that of a good Postcyberpunk
novel, that is rather chaotic depending on possible future technological
advanves rather than the logical and unescapable ending of a good
whodunnit. After reading Nymphomation, a fly or a game of dominoes will
never seem quite the same again. Another nice feature are the
non-sensical beginning sentences of every ‘chapter’. To some they seem like the rantings of a mad
mathematician. To me they sound like a tribute to Finnegan’s Wake

A _quintomino_ is a regular pentagon having all its sides
colored by five different colours. Two quintominoes are the same if they
can be transformed into each other by a symmetry of the pentagon (that
is, a cyclic rotation or a flip of the two faces). It is easy to see
that there are exactly 12 different quintominoes. On the other hand,
there are also exactly 12 pentagonal faces of a dodecahedron
whence the puzzling question whether the 12 quintominoes can be joined
together (colours mathching) into a dodecahedron.
According to
the Dictionnaire de
mathematiques recreatives
this can be done and John Conway found 3
basic solutions in 1959. These 3 solutions can be found from the
diagrams below, taken from page 921 of the reprinted Winning Ways for your Mathematical
Plays (volume 4)
where they are called _the_ three
quintominal dodecahedra giving the impression that there are just 3
solutions to the puzzle (up to symmetries of the dodecahedron). Here are
the 3 Conway solutions

One projects the dodecahedron down from the top face which is
supposed to be the quintomino where the five colours red (1), blue (2),
yellow (3), green (4) and black(5) are ordered to form the quintomino of
type A=12345. Using the other quintomino-codes it is then easy to work
out how the quintominoes fit together to form a coloured dodecahedron.

In preparing to explain this puzzle for my geometry-101 course I
spend a couple of hours working out a possible method to prove that
these are indeed the only three solutions. The method is simple : take
one or two of the bottom pentagons and fill then with mathching
quintominoes, then these more or less fix all the other sides and
usually one quickly runs into a contradiction.
However, along the
way I found one case (see top picture) which seems to be a _new_
quintominal dodecahedron. It can't be one of the three Conway-types
as the central quintomino is of type F. Possibly I did something wrong
(but what?) or there are just more solutions and Conway stopped after
finding the first three of them…
Update (with help from
Michel Van den Bergh
) Here is an elegant way to construct
'new' solutions from existing ones, take a permutation $\\sigma \\in S_5$ permuting the five colours and look on the resulting colored
dodecahedron (which again is a solution) for the (new) face of type A
and project from it to get a new diagram. Probably the correct statement
of the quintominal-dodecahedron-problem is : find all solutions up to
symmetries of the dodecahedron _and_ permutations of the colours.
Likely, the 3 Conway solutions represent the different orbits under this
larger group action. Remains the problem : to which orbit belongs the
top picture??

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

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.

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…