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.

# Tag: Conway

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

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.