
One of the coolest (pure math) facts in Conway’s book ONAG is the explicit construction of the algebraic closure $\overline{\mathbb{F}_2}$ of the field with two elements as the set of all ordinal numbers smaller than $(\omega^{\omega})^{\omega}$ equipped with nimber addition and multiplication. Some time ago we did run a couple of posts on this. In… Read more »

Today, we will expand the game of Nimbers to higher dimensions and do some transfinite Nimber hacking. In our identification between $\mathbb{F}_{16}^* $ and 15th roots of unity, the number 8 corresponds to $\mu^6 $, whence $\sqrt{8}=\mu^3=14 $. So, if we add a stone at the diagonal position (14,14) to the Nimbersposition of last time… Read more »

Nimbers is a 2person game, winnable only if you understand the arithmetic of the finite fields $\mathbb{F}_{2^{2^n}} $ associated to Fermat 2powers. It is played on a rectangular array (say a portion of a Goboard, for practical purposes) having a finite number of stones at distinct intersections. Here’s a typical position The players alternate making… Read more »

The odd Knight of the round table problem asks for a consistent placement of the nth Knight in the row at an odd root of unity, compatible with the two different realizations of the algebraic closure of the field with two elements. The first identifies the multiplicative group of its nonzero elements with the group… Read more »

For the better part of the 30ties, Ernst Witt (1) did hang out with the rest of the ‘Noetherknaben’, the group of young mathematicians around Emmy Noether (3) in Gottingen. In 1934 Witt became Helmut Hasse‘s assistent in Gottingen, where he qualified as a university lecturer in 1936. By 1938 he has made enough of… Read more »

The Leech lattice was, according to wikipedia, ‘originally discovered by Ernst Witt in 1940, but he did not publish his discovery’ and it ‘was later rediscovered in 1965 by John Leech’. However, there is very little evidence to support this claim. The facts What is certain is that John Leech discovered in 1965 an amazingly… Read more »

The Knightseating problems asks for a consistent placing of nth Knight at an odd root of unity, compatible with the two different realizations of the algebraic closure of the field with two elements.

Here’s a tiny problem illustrating our limited knowledge of finite fields : “Imagine an infinite queue of Knights ${ K_1,K_2,K_3,\ldots } $, waiting to be seated at the unitcircular table. The master of ceremony (that is, you) must give Knights $K_a $ and $K_b $ a place at an odd root of unity, say $\omega_a… Read more »

To mark the end of 2009 and 6 years of blogging, two musical compositions with a mathematical touch to them. I wish you all a better 2010! Remember from last time that we identified Olivier Messiaen as the ‘Monsieur Modulo’ playing the musical organ at the Bourbaki wedding. This was based on the fact that… Read more »

Are the valencies of the 171 moonshine groups are compatible, that is, can one construct a (disconnected) graph on the 171 vertices such that in every vertex (determined by a moonshine group G) the vertexvalency coincides with the valency of the corresponding group? Duncan describes a subset of 9 moonshine groups for which the valencies… Read more »

We have seen that Conway’s big picture helps us to determine all arithmetic subgroups of $PSL_2(\mathbb{R}) $ commensurable with the modular group $PSL_2(\mathbb{Z}) $, including all groups of monstrous moonshine. As there are exactly 171 such moonshine groups, they are determined by a finite subgraph of Conway’s picture and we call the minimal such subgraph… Read more »

Conway and Norton showed that there are exactly 171 moonshine functions and associated two arithmetic subgroups to them. We want a tool to describe these and here’s where Conway’s big picture comes in very handy. All moonshine groups are arithmetic groups, that is, they are commensurable with the modular group. Conway’s idea is to view… Read more »

While the verdict on a neolithic Scottish icosahedron is still open, let us recall Kostant’s grouptheoretic construction of the icosahedron from its rotationsymmetry group $A_5 $. The alternating group $A_5 $ has two conjugacy classes of order 5 elements, both consisting of exactly 12 elements. Fix one of these conjugacy classes, say $C $ and… Read more »

Hendrik Lenstra found an effective procedure to compute the mysterious elements alpha(p) needed to do actual calculations with infinite nimarithmetic.

Conway’s nimarithmetic on ordinal numbers leads to many surprising identities, for example who would have thought that the third power of the first infinite ordinal equals 2…

Surely Georg Cantor’s transfinite ordinal numbers do not have a reallife importance? Well, think again.

About a year ago I did a series of posts on games associated to the Mathieu sporadic group $M_{12} $, starting with a post on Conway’s puzzle M(13), and, continuing with a discussion of mathematical blackjack. The idea at the time was to write a book for a general audience, as discussed at the start… Read more »

Arnold has written a followup to the paper mentioned last time called “Polymathematics : is mathematics a single science or a set of arts?” (or here for a (huge) PDFconversion). On page 8 of that paper is a nice summary of his 25 trinities : I learned of this newer paper from a comment by… Read more »

Referring to the triple of exceptional Galois groups $L_2(5),L_2(7),L_2(11) $ and its connection to the Platonic solids I wrote : “It sure seems that surprises often come in triples…”. Briefly I considered replacing triples by trinities, but then, I didnt want to sound too mystic… David Corfield of the ncategory cafe and a dialogue on… Read more »

Exactly one year ago this blog was briefly renamed MoonshineMath. The concept being that it would focus on the mathematics surrounding the monster group & moonshine. Well, I got as far as the Mathieu groups… After a couple of months, I changed the name back to neverendingbooks because I needed the freedom to post on… Read more »

Most chess programs are able to give a numerical evaluation of a position. For example, the position below is considered to be worth +8.7 with white to move, and, 0.7 with black to move (by a certain program). But, if one applies combinatorial game theory as in John Conway’s ONAG and the BerlekampConwayGuy masterpiece Winning… Read more »

Here’s a sweet Easter egg for you to crack : a mysterious message from none other than the discoverer of Monstrous Moonshine himself… From: mckayj@Math.Princeton.EDU Date: Mon 10 Mar 2008 07:51:16 GMT+01:00 To: lieven.lebruyn@ua.ac.be The secret of Monstrous Moonshine and the universe. Let j(q) = 1/q + 744 + sum( c[k]*q^k,k>=1) be the Fourier expansion… Read more »

Monstrous moonshine was born (sometime in 1978) the moment John McKay realized that the linear term in the jfunction $j(q) = \frac{1}{q} + 744 + 196884 q + 21493760 q^2 + 864229970 q^3 + \ldots $ is surprisingly close to the dimension of the smallest nontrivial irreducible representation of the monster group, which is 196883…. Read more »

John Conway once wrote : There are almost as many different constructions of $M_{24} $ as there have been mathematicians interested in that most remarkable of all finite groups. In the inguanodon post Ive added yet another construction of the Mathieu groups $M_{12} $ and $M_{24} $ starting from (half of) the Farey sequences and… Read more »

On friday, I did spot in my regular Antwerpbookshop Finding Moonshine by Marcus du Sautoy and must have uttered a tiny curse because, at once, everyone near me was staring at me… To make matters worse, I took the book from the shelf, quickly glanced through it and began shaking my head more and more,… Read more »
Close