# Tag: symmetry

A natural question these days might be: “what are the rotational symmetries of the Covid-19 virus?”

Most illustrations show a highly symmetric object, suggesting it might have icosahedral symmetry. In fact, many viruses do have icosahedral symmetry as a result of the ‘genetic economy principle’ proposed by Watson and Crick in 1956, resulting in the Caspar-Klug classification of viral capsids.

But then, perhaps this icosahedral illusion is a result of design decisions illustrators made turning scientific data into pictures. Veronica Falconieri Hays wrote a beautiful article describing the effort going into this: How I built a 3d-model of the coronavirus for Scientific American. Here’s her final picture

And yes, icosahedral symmetry was one of her design decisions:

The M proteins form pairs, and it is estimated that there are 16–25 M proteins per spike on the surface of the virus. I ended up modeling 10 M protein pairs (so 20 M proteins) per spike in my model. Some researchers hypothesize that the M proteins form a lattice within the envelope (interacting with an underlying lattice of N proteins; see below). I decided to use an icosahedral sphere to create a regular distribution of the M protein dimers to hint at this hypothesis.

The spikes (or S-proteins) are the tentacles in these pictures, and one of the few hard figures on Corona is that ‘on average’ there are 74 of them.

This fact is enough to rule out icosahedral symmetry.

If the icosahedral rotation group (of order $60$, isomorphic to $A_5$) acts on the $74$ spikes, then each orbit consists of $60$ spikes unless that spike lies on a twofold, threefold or fivefold rotation axis, in which cases the number of spikes in its orbit are respectively $30$, $20$ or $12$. So, we can’t get a total number of $74$ spikes!

However, just looking at the number of spikes we cannot rule out octahedral symmetry!

The octahedral rotation group (of order $24$, isomorphic to $S_4$) will have orbits of size $24$ unless the spike lies on a twofold, threefold or fourfold rotation axis, giving orbits of size $12$, $8$ and $6$ respectively (the midpoints of edges, the vertices and the midpoints of faces of the octahedron), and

$74 = 24+24+12+8+6$

The most symmetric arrangement of spikes would be to subdivide each of the $8$ triangular faces of the octahedron into $6$ triangles with vertices the midpoint of the face, a vertex and a midpoint of an edge, and then to position the spikes on the axis through the vertices and midpoints of these smaller triangles.

Googling around I found very few references to symmetries of Covid-19, probably because it has an helical RNA-coil, which seems not to go well with Caspar-Klug type polyhedral viral capsids.

Here’s an exception: A structural model for the Coronavirus nucleocapsid by Federico Coscio, Alejandro D. Nadra, and Diego U. Ferreiro.

They propose a truncated octahedron as capsid (in transparent brown) with interior a continuous coil packing of blue and cyan helices. The virus membrane with the spikes and M proteins is drawn in blue.

Last time we’ve seen that on June 3rd 1939, the very day of the Bourbaki wedding, Malraux’ movie ‘L’espoir’ had its first (private) viewing, and we mused whether Weil’s wedding card was a coded invitation to that event.

But, there’s another plausible explanation why the Bourbaki wedding might have been scheduled for June 3rd : it was intended to be a copy-cat Royal Wedding…

The media-hype surrounding the wedding of Prince William to Pippa’s sister led to a hausse in newspaper articles on iconic royal weddings of the past.

One of these, the marriage of Edward VIII, Duke of Windsor and Wallis Warfield Spencer Simpson, was held on June 3rd 1937 : “This was the scandal of the century, as far as royal weddings go. Edward VIII had just abdicated six months before in order to marry an American twice-divorced commoner. The British Establishment at the time would not allow Edward VIII to stay on the throne and marry this woman (the British Monarch is also the head of the Church of England), so Edward chose love over duty and fled to France to await the finalization of his beloved’s divorce. They were married in a private, civil ceremony, which the Royal Family boycotted.”

But, what does this wedding have to do with Bourbaki?

For starters, remember that the wedding-card-canular was concocted in the spring of 1939 in Cambridge, England. So, if Weil and his Anglo-American associates needed a common wedding-example, the Edward-Wallis case surely would spring to mind. One might even wonder about the transposed symmetry : a Royal (Betti, whose father is from the Royal Poldavian Academy), marrying an American (Stanislas Pondiczery).

Even Andre Weil must have watched this wedding with interest (perhaps even sympathy). He too had to wait a considerable amount of time for Eveline’s divorce (see this post) to finalize, so that they could marry on october 30th 1937, just a few months after Edward & Wallis.

But, there’s more. The royal wedding took place at the Chateau de Cande, just south of Tours (the A on the google-map below). Now, remember that the 2nd Bourbaki congress was held at the Chevalley family-property in Chancay (see the Escorial post) a bit to the north-east of Tours (the marker on the map). As this conference took place only a month after the Royal Wedding (from 10th till 20th of July 1937), the event surely must have been the talk of the town.

Early on, we concluded that the Bourbaki-Petard wedding took place at 12 o’clock (‘a l’heure habituelle’). So did the Edward-Wallis wedding. More precisely, the civil ceremony began at 11.47 and the local mayor had to come to the castle for the occasion, and, afterwards the couple went into the music-room, which was converted into an Anglican chapel for the day, at precisely 12 o’clock.

The emphasis on the musical organ in the Bourbaki wedding-invitation allowed us to identify the identity of ‘Monsieur Modulo’ to be Olivier Messiaen as well as that of the wedding church. Now, the Chateau de Cande also houses an impressive organ, the Skinner opus 718 organ.

For the wedding ceremony, Edward and Wallis hired the services of one of the most renowned French organists at the time : Marcel Dupre who was since 1906 Widor’s assistent, and, from 1934 resident organist in the Saint-Sulpice church in Paris. Perhaps more telling for our story is that Dupre was, apart from Paul Dukas, the most influential teacher of Olivier Messiaen.

On June 3rd, 1937 Dupre performed the following pieces. During the civil ceremony, an extract from the 29e Bach cantate, canon in re-minor by Schumann and the prelude of the fugue in do-minor of himself. When the couple entered the music room he played the march of the Judas Macchabee oratorium of Handel and the cortege by himself. During the religious ceremony he performed his own choral, adagium in mi-minor by Cesar Franck, the traditional ‘Oh Perfect Love’, the Jesus-choral by Bach and the toccata of the 5th symphony of Widor. Compare this level of detail to the minimal musical hint given in the Bourbaki wedding-invitation

“Assistent Simplexe de la Grassmannienne (lemmas chantees par la Scholia Cartanorum)”

This is one of the easier riddles to solve. The ‘simplicial assistent of the Grassmannian’ is of course Hermann Schubert (Schubert cell-decomposition of Grassmannians). But, the composer Franz Schubert only left us one organ-composition : the Fugue in E-minor.

I have tried hard to get hold of a copy of the official invitation for the Edward-Wallis wedding, but failed miserably. There must be quite a few of them still out there, of the 300 invited people only 16 showed up… You can watch a video newsreel film of the wedding.

As Claude Chevalley’s father had an impressive diplomatic career behind him and lived in the neighborhood, he might have been invited, and, perhaps the (unused) invitation was lying around at the time of the second Bourbaki-congress in Chancay,just one month after the Edward-Wallis wedding…

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 a name for himself to be offered a lecturer position in Hamburg and soon became an associate professor, the down-graded position held by Emil Artin (2) until he was forced to emigrate in 1937.

A former fellow student of him in Gottingen, Erna Bannow (4), had gone earlier to Hamburg to work with Artin. She continued her studies with Witt and finished her Ph.D. in 1939. In 1940 Erna Bannow and Witt married.

So, life was smiling on Ernst Witt that sunday january 28th 1940, both professionally and personally. There was just one cloud on the horizon, and a rather menacing one. He was called up by the Wehrmacht and knew he had to enter service in february. For all he knew, he was spending the last week-end with his future wife… (later in february 1940, Blaschke helped him to defer his military service by one year).

Still, he desperately wanted to finish his paper before entering the army, so he spend most of that week-end going through the final version and submitted it on monday, as the published paper shows.

In the 70ties, Witt suddenly claimed he did discover the Leech lattice ${\Lambda}$ that sunday. Last time we have seen that the only written evidence for Witt’s claim is one sentence in his 1941-paper Eine Identität zwischen Modulformen zweiten Grades. “Bei dem Versuch, eine Form aus einer solchen Klassen wirklich anzugeben, fand ich mehr als 10 verschiedene Klassen in ${\Gamma_{24}}$.”

But then, why didn’t Witt include more details of this sensational lattice in his paper?

Ina Kersten recalls on page 328 of Witt’s collected papers : “In his colloquium talk “Gitter und Mathieu-Gruppen” in Hamburg on January 27, 1970, Witt said that in 1938, he had found nine lattices in ${\Gamma_{24}}$ and that later on January 28, 1940, while studying the Steiner system ${S(5,8,24)}$, he had found two additional lattices ${M}$ and ${\Lambda}$ in ${\Gamma_{24}}$. He continued saying that he had then given up the tedious investigation of ${\Gamma_{24}}$ because of the surprisingly low contribution

$\displaystyle | Aut(\Lambda) |^{-1} < 10^{-18}$

to the Minkowski density and that he had consented himself with a short note on page 324 in his 1941 paper.”

In the last sentence he refers to the fact that the sum of the inverse orders of the automorphism groups of all even unimodular lattices of a given dimension is a fixed rational number, the Minkowski-Siegel mass constant. In dimension 24 this constant is

$\displaystyle \sum_{L} \frac{1}{| Aut(L) |} = \frac {1027637932586061520960267}{129477933340026851560636148613120000000} \approx 7.937 \times 10^{-15}$

That is, Witt was disappointed by the low contribution of the Leech lattice to the total constant and concluded that there might be thousands of new even 24-dimensional unimodular lattices out there, and dropped the problem.

If true, the story gets even better : not only claims Witt to have found the lattices ${A_1^{24}=M}$ and ${\Lambda}$, but also enough information on the Leech lattice in order to compute the order of its automorphism group ${Aut(\Lambda)}$, aka the Conway group ${Co_0 = .0}$ the dotto-group!

Is this possible? Well fortunately, the difficulties one encounters when trying to compute the order of the automorphism group of the Leech lattice from scratch, is one of the better documented mathematical stories around.

The books From Error-Correcting Codes through Sphere Packings to Simple Groups by Thomas Thompson, Symmetry and the monster by Mark Ronan, and Finding moonshine by Marcus du Sautoy tell the story in minute detail.

It took John Conway 12 hours on a 1968 saturday in Cambridge to compute the order of the dotto group, using the knowledge of Leech and McKay on the properties of the Leech lattice and with considerable help offered by John Thompson via telephone.

But then, John Conway is one of the fastest mathematicians the world has known. The prologue of his book On numbers and games begins with : “Just over a quarter of a century ago, for seven consecutive days I sat down and typed from 8:30 am until midnight, with just an hour for lunch, and ever since have described this book as “having been written in a week”.”

Conway may have written a book in one week, Ernst Witt did complete his entire Ph.D. in just one week! In a letter of August 1933, his sister told her parents : “He did not have a thesis topic until July 1, and the thesis was to be submitted by July 7. He did not want to have a topic assigned to him, and when he finally had the idea, he started working day and night, and eventually managed to finish in time.”

So, if someone might have beaten John Conway in fast-computing the dottos order, it may very well have been Witt. Sadly enough, there is a lot of circumstantial evidence to make Witt’s claim highly unlikely.

For starters, psychology. Would you spend your last week-end together with your wife to be before going to war performing an horrendous calculation?

Secondly, mathematical breakthroughs often arise from newly found insight. At that time, Witt was also working on his paper on root lattices “Spiegelungsgrupen and Aufzähling halbeinfacher Liescher Ringe” which he eventually submitted in january 1941. Contained in that paper is what we know as Witt’s lemma which tells us that for any integral lattice the sublattice generated by vectors of norms 1 and 2 is a direct sum of root lattices.

This leads to the trick of trying to construct unimodular lattices by starting with a direct sum of root lattices and ‘adding glue’. Although this gluing-method was introduced by Kneser as late as 1967, Witt must have been aware of it as his 16-dimensional lattice ${D_{16}^+}$ is constructed this way.

If Witt wanted to construct new 24-dimensional even unimodular lattices in 1940, it would be natural for him to start off with direct sums of root lattices and trying to add vectors to them until he got what he was after. Now, all of the Niemeier-lattices are constructed this way, except for the Leech lattice!

I’m far from an expert on the Niemeier lattices but I would say that Witt definitely knew of the existence of ${D_{24}^+}$, ${E_8^3}$ and ${A_{24}^+}$ and that it is quite likely he also constructed ${(D_{16}E_8)^+, (D_{12}^2)^+, (A_{12}^2)^+, (D_8^3)^+}$ and possibly ${(A_{17}E_7)^+}$ and ${(A_{15}D_9)^+}$. I’d rate it far more likely Witt constructed another two such lattices on sunday january 28th 1940, rather than discovering the Leech lattice.

Finally, wouldn’t it be natural for him to include a remark, in his 1941 paper on root lattices, that not every even unimodular lattices can be obtained from sums of root lattices by adding glue, the Leech lattice being the minimal counter-example?

If it is true he was playing around with the Steiner systems that sunday, it would still be a pretty good story he discovered the lattices ${(A_2^{12})^+}$ and ${(A_1^{24})^+}$, for this would mean he discovered the Golay codes in the process!

Which brings us to our next question : who discovered the Golay code?