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Fun with microbiology: Virus, the Beauty of the Beast

Hamish Todd offers Virus, the Beauty of the Beast, an interactive documentary about viruses. Viruses have protein shells made of patterns which can be explored mathematically, and this link to tiling theory and geometric shapes provides a mathematical interest for the piece.

Hamish is a former game designer, former teacher turned PhD student in computational biology. He says:

While studying maths I had learned about viruses, and about their connection to Islamic art, which amazed me. I found it staggering that such beautiful things could surround us without most people being aware of it. I wanted to let people see it, and I knew that my game design skills could help me do that.

Apparently many viruses are arranged on what Hamish calls a ‘hexagons and pentagons’ structure (Caspar-Klug theory), and others have more exotic structures. Wikipedia says “most animal viruses are icosahedral or near-spherical with chiral icosahedral symmetry”, with other more complicated shapes also found.

As well as “interactive documentary”, Hamish calls the website an ‘explorable explanation’, which aims “to let laypeople play with the beautiful things that mathematicians and scientists spend their time with”. Overall, it seems like a nicely-produced series of interactive videos exploring an interesting topic. Give it a go!

More information

Virus, the Beauty of the Beast, the interactive documentary.

Virus, the Beauty of the Beast press pack.

A symmetry approach to viruses, an article at Plus.

2017 LMS prize winners announced

The London Mathematical Society has announced the winners of its various prizes and medals for this year.

Here’s a summary of the more senior prizes:

  • Alex Wilkie gets the Pólya prize for “his profound contributions to model theory and to its connections with real analytic geometry.”
  • Peter Cameron gets a Senior Whitehead prize for “his exceptional research contributions across combinatorics and group theory.” Peter has written a rare horn-tooting post on his excellent blog about winning the prize.
  • Alison Etheridge gets a Senior Anne Bennett prize “in recognition of her outstanding research on measure-valued stochastic processes and applications to population biology; and for her impressive leadership and service to the profession.”
  • John King gets a Naylor prize for “his profound contributions to the theory of nonlinear PDEs and applied mathematical modelling.”

The Berwick prize goes to Kevin Costello, and Whitehead prizes go to  Julia Gog, András Máthé, Ashley Montanaro, Oscar Randal-Williams, Jack Thorne, and Michael Wemyss.

Read the full announcement at the LMS website.

Laws of mathematics not as immutable as we thought, in Australia

Australian PM Malcolm Turnbull said, as part of a speech proposing a law to force tech companies to give the government access to encrypted messages,

“The laws of mathematics are very commendable, but the only law that applies in Australia is the law of Australia.”

The problem is that the end-to-end encryption schemes used by messaging apps make it practically impossible for the makers of the app to read messages, even if they really want to.

New Scientist writer Jacob Aron has seen the positive side of Turnbull’s comments:

Patterns and code to make your own cellular automaton scarves now online

If you remember our post about Fabienne Serrière’s amazing Cellular Automaton Scarves Kickstarter back in 2015, you’ll be pleased to hear Fabienne has now put the patterns, and all the code you need to make your own scarves, online on her Ravelry page.

If you have a knitting machine and are prepared to hack it to take code input (you can read Fabienne’s blog to find out how she’s done that), you can use JPG files to generate knitting patterns of your own, or use Fabienne’s code to create cellular automata from a seed row of pixels of your choice. She’s included the code for Rule 110, but I’m sure you could work out your own automata and knit those too. The patterns can also be knitted by hand, if you’re incredibly patient.

via KnitYak on Twitter.

Shaw Prize 2017 awarded to two algebraic geometers

The announcement of the Shaw Prize was posted on 23rd May, reading:

The Shaw Prize in Mathematical Sciences 2017 is awarded in equal shares to János Kollár and Claire Voisin for their remarkable results in many central areas of algebraic geometry, which have transformed the field and led to the solution of long-standing
problems that had appeared out of reach.

The prize is awarded annually to “individuals who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence”.

The two joint winners this year, Kollár and Voisin, are both professors of algebraic geometry, at Princeton and Collège de France respectively, and have made major contributions to the effort to characterise rational varieties – solution sets of polynomials which differ from a projective space only by a low-dimensional subset.

Kollár’s work relates to the Minimal Model Program, which concerns moduli of higher-dimensional varieties – spaces whose points represent equivalence classes of varieties. These spaces, which Kollár has extensively worked on and developed the field dramatically, have applications in topology, combinatorics and physics. Voisin’s achievements have included solving the Kodaira problem (on complex projective manifolds), developing a technique for showing that a variety is not rational, and even finding a counterexample to an extension of the Hodge conjecture (one of the Clay prize problems), which rules out several approaches to the main conjecture.

More information

Shaw Prize announcement, laureate biographies and press release

János Kollár’s homepage

Claire Voisin’s homepage

13532385396179 doesn’t climb to a prime

Someone called James Davis has found a counterexample to John H. Conway’s “Climb to a Prime” conjecture, for which Conway was offering \$1,000 for a solution.

The conjecture goes like this, as stated in Conway’s list of \$1,000 problems:

Let $n$ be a positive integer. Write the prime factorization in the usual way, e.g. $60 = 2^2 \cdot 3 \cdot 5$, in which the primes are written in increasing order, and exponents of $1$ are omitted. Then bring exponents down to the line and omit all multiplication signs, obtaining a number $f(n)$. Now repeat.

So, for example, $f(60) = f(2^2 \cdot 3 \cdot 5) = 2235$. Next, because $2235 = 3 \cdot 5 \cdot 149$, it maps, under $f$, to $35149$, and since $35149$ is prime, we stop there forever.

The conjecture, in which I seem to be the only believer, is that every number eventually climbs to a prime. The number 20 has not been verified to do so. Observe that $20 \to 225 \to 3252 \to 223271 \to \ldots$, eventually getting to more than one hundred digits without reaching a prime!

Well, James, who says he is “not a mathematician by any stretch”, had a hunch that a counterexample would be of the form $n = x \cdot p = f(x) \cdot 10^y+p$, where $p$ is the largest prime factor of $n$, which in turn motivates looking for $x$ of the form $x=m \cdot 10^y + 1$, and $m=1407$, $y=5$, $p=96179$ “fell out immediately”. It’s not at all obvious to me where that hunch came from, or why it worked.

The number James found was $13\,532\,385\,396\,179 = 13 \cdot 53^2 \cdot 3853 \cdot 96179$, which maps onto itself under Conway’s function $f$ – it’s a fixed point of the function. So, $f$ will never map this composite number onto a prime, disproving the conjecture. Finding such a simple counterexample against such stratospherically poor odds is like deciding to look for Lord Lucan and bumping into him on your doorstep as you leave the house.

A lovely bit of speculative maths spelunking!

via Hans Havermann, whom James originally contacted with his discovery.

Right answer for the wrong reason: cellular automaton on the new Cambridge North station

Cambridge North is a brand new train station, and the building’s got a fab bit of cladding with a design ‘derived from John Horton Conway’s “Game of Life” theories which he established while at Gonville and Caius College, Cambridge in 1970.’

One problem: that’s Wolfram’s Rule 135, not the Game of Life. You can tell because of the pixels.

Rule 135 is a 1-dimensional automaton: you start with a row of black or white pixels, and the rule tells you how the colour of each pixel changes based on the colours of the neighbouring pixels. The Cambridge North design shows the evolution of a rule 135 pattern as a distinct row of pixels for each time step. Conway’s Game of Life follows the same idea but in two dimensions – a pixel’s colour changes depending on the nearby pixels  in every compass direction.

Either way, it’s a lovely pattern. I suspect the designers went with Rule 135 instead of the Game of Life so that they’d get a roughly even mix of white and black pixels, which is hard to achieve under Conway’s rules.

Just in case gawping at train stations is your cup of tea, here’s a promotional video with lots of lovely panning shots of the design:

EDIT: James Grime has now also done a video, which can be seen here:

More information

Delayed £50m Cambridge North railway station opens on BBC News.

Cambridge North Station information from Atkins Group, the design consultancy responsible for the station building.

Press release from Greater Anglia trains.

The Game of Life: a beginner’s guide by Alex Bellos in the Guardian.

Brought to our attention by @Quendus on Twitter.