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Category: books

human-, computer- and fairy-chess

It was fun following the second game last night in real time. Carlsen got a winning endgame with two bishops against a rook, but blundered with 62. Bg4?? (winning was Kf7), resulting in stalemate.

There was this hilarious message around move 60:

“The computer has just announced that white mates in 31 moves. Of course, the only two people in the building who don’t benefit from that knowledge are behind the pieces.”

[section_title text=”Alice’s game from ‘Through the Looking-Glass'”]

The position below comes from the preface of Lewis Carroll’s Through the Looking-Glass



The old notation for files is used:

a = QR (queen’s side rook)
b = QKt (queen’s side knight)
c = QB (queen’s side bishop)
d = Q (queen)
e = K (king)
f = KB (king’s side bishop)
g = KKt (king’s side knight)
h = KR (king’s side rook)

Further, the row-number depends on whose playing (they both count starting from their own side). Here’s an animated version of the game:



And a very strange game it is.

White makes consecutive moves, which is allowed in some versions of fairy chess.

And, as the late Martin Gardner explains in his book The Annotated Alice:

“The most serious violation of chess rules occurs near the end of the
problem, when the White King is placed in check by the Red Queen without
either side taking account of the fact. “Hardly a move has a sane purpose,
from the point of view of chess,” writes Mr. Madan. It is true that both sides
play an exceedingly careless game, but what else could one expect from the
mad creatures behind the mirror? At two points the White Queen passes up
a chance to checkmate and on another occasion she flees from the Red
Knight when she could have captured him. Both oversights, however, are in
keeping with her absent-mindedness.”

In fact, the whole game reflects the book’s story (Alice is the white pawn travelling to the other side of the board), with book-pages associated to the positions listed on the left. Martin Gardner on this:

“Considering the staggering difficulties involved in dovetailing a chess
game with an amusing nonsense fantasy, Carroll does a remarkable job. At
no time, for example, does Alice exchange words with a piece that is not
then on a square alongside her own. Queens bustle about doing things while
their husbands remain relatively fixed and impotent, just as in actual chess
games. The White Knight’s eccentricities fit admirably the eccentric way in
which Knights move; even the tendency of the Knights to fall off their
horses, on one side or the other, suggests the knight’s move, which is two
squares in one direction followed by one square to the right or left. In order
to assist the reader in integrating the chess moves with the story, each move
will be noted in the text at the precise point where it occurs.”

The starting position is in itself an easy chess-problem: white mates in 3, as explained by Gardner:

” It is amusing to note that it is the Red Queen who persuades Alice to advance along her file to the eighth square. The Queen is protecting herself with this advice, for white has at the outset an easy, though inelegant, checkmate in three moves.
The White Knight first checks at KKt.3. If the Red King moves to either Q6
or Q5, white can mate with the Queen at QB3. The only alternative is for
the Red King to move to K4. The White Queen then checks on QB5,
forcing the Red King to K3. The Queen then mates on Q6. This calls, of
course, for an alertness of mind not possessed by either the Knight or
Queen. ”

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NaNoWriMo (3)

In 2001, Eugenia Cheng gave an interesting after-dinner talk Mathematics and Lego: the untold story. In it she compared math research to fooling around with lego. A quote:

“Lego: the universal toy. Enjoyed by people of all ages all over the place. The idea is simple and brilliant. Start with some basic blocks that can be joined together. Add creativity, imagination and a bit of ingenuity. Build anything.

Mathematics is exactly the same. We start with some basic building blocks and ways of joining them together. And then we use creativity, and, yes, imagination and certainly ingenuity, and try to build anything.”

She then goes on to explain category theory, higher dimensional topology, and the process of generalisation in mathematics, whole the time using lego as an analogy. But, she doesn’t get into the mathematics of lego, perhaps because the talk was aimed at students and researchers of all levels and all disciplines.

There are plenty of sites promoting lego in the teaching of elementary mathematics, here’s just one link-list-page: “27 Fantastic LEGO Math Learning Activities for All Ages”. I’m afraid ‘all ages’ here means: under 10…

lego-math-teaching-children-alycia-zimmerman-fb__700-png

Can one do better?

Everyone knows how to play with lego, which shapes you can build, and which shapes are simply impossible.

Can one tap into this subconscious geometric understanding to explain more advanced ideas such as symmetry, topological spaces, sheaves, categories, perhaps even topos theory… ?

Let’s continue our

[section_title text=”imaginary iterview”]

Question: What will be the opening scene of your book?

Alice posts a question on Lego-stackexchenge. She wants help to get hold of all imaginary lego shapes, including shapes impossible to construct in three-dimensional space, such as gluing two shapes over some internal common sub-shape, or Escher like constructions, and so on.

escherlego

Question: And does she get help?

At first she only gets snide remarks, style: “brush off your French and wade through SGA4”.

Then, she’s advised to buy a large notebook and jot down whatever she can tell about shapes that one can construct.

If you think about this, you’ll soon figure out that you can only add new bricks along the upper or lower bricks of the shape. You may call these the boundary of the shape, and soon you’ll be doing topology, and forming coproducts.

These ‘legal’ lego shapes form what some of us would call a category, with a morphism from $A$ to $B$ for each different way one can embed shape $A$ into $B$.

Of course, one shouldn’t use this terminology, but rather speak of different instruction-manuals to get $B$ out of $A$ (the morphisms), stapling two sets of instructions together (the compositions), and the empty instruction-sheet (the identity morphism).

Question: But can one get to the essence of categorical results in this way?

Take Yoneda’s lemma. In the case of lego shapes it says that you know a shape once you know all morphisms into it from whatever shape.

For any coloured brick you’re given the number of ways this brick sits in that shape, so you know all the shape’s bricks. Then you may try for combination of two bricks, and so on. It sure looks like you’re going to be able to reconstruct the shape from all this info, but this quickly get rather messy.

But then, someone tells you the key argument in Yoneda’s proof: you only have to look for the shape to which the identity morphism is assigned. Bingo!

Question: Wasn’t your Alice interested in the ‘illegal’ or imaginary shapes?

Once you get to Yoneda, the rest follows routinely. You define presheaves on this category, figure out that you get a whole bunch of undesirable things, bring in Grothendieck topologies to be the policing agency weeding out that mess, and keep only the sheaves, which are exactly the desired imaginary shapes.

Question: Your book’s title is ‘Primes and other imaginary shapes’. How do you get from Lego shapes to prime numbers?

By the standard Gödelian trick: assign a prime number to each primitive coloured brick, and to a shape the product of the brick-primes.

That number is a sort of code of the shape. Shapes sharing the same code are made up from the same set of bricks.

Take the set of all strictly positive natural numbers partially ordered by divisibility, then this code is a functor from Lego shapes to numbers. If we extend this to imaginary shapes, we’ll rapidly end up at Connes’ arithmetic site, supernatural numbers, adeles and the recent realisation that the set of all prime numbers does have a geometric shape, but one with infinitely many dimensions.

primenumbers

Not sure yet how to include all of this, but hey, early days.

Question: So, shall we continue this interview at a later date?

No way, I’d better start writing.

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Ulysses and LaTeX

If you’re a mathematician chances are that your text-editor of choice will be TeXShop, the perfect environment for writing papers. Even when writing a massive textbook, most of us stick to this or a similar LaTeX-frontend. The order of chapters in such a book is usually self-evident, and it is enough to use one TeX-file per chapter.

If you’re a blogger, chances are you spend a lot of writing time within the WordPress-editor. If you have a math-blog, there’s no longer the issue of including TeX-output images in some laborious way, thanks to MathJax. Even for a longer series of blog-posts there’s no problem staying within the WordPress-environment.

However, if you’re reckless enough to want to write a novel, or a math-book for a larger audience, you may need different equipment.

You will have to be able to follow story-lines, to follow your main characters throughout the plot, get word counts on scenes and chapters, jot down ideas and results from research, but most of all: you will have to be able to remain focussed just on your writing, as far away as possible from all bells and whistles and thrills of internet and preview-on-the-go editors.

In short, you may consider moving all of your writing to Ulysses.

I’ve been an early adopter from the days their iPad-app was called Daedalus, which I found cute, being a pathetic Joyce-fan. However, the app’s iCloud syncing sucked, but it is now replaced by the Ulysses.app which works like magic, syncing every keystroke between iPad, iPhone and whatever Mac you use as your workhorse.

But, what if you want to write about math and are unwilling to ban all LaTeX-formulas from your text.

Well, I’ve tried everything, including the approach below (in a faulty way), and figured it was impossible due to the fact that Ulysses is a MarkDown editor in which underscores are entirely different from indices.

Fortunately, yesterday Eline Steffens posted “Writing Mathematical Equations in Ulysses” showing me what I did wrong.

If you want MathJax to parse your text you need to include the standard code in your header. What I missed was that you have to include it as a ‘Raw source block’ (under ‘Markup’ in Ulysses).

Further, I forgot to prepend dollar-signs with a tilde, which works as an escape character in Ulysses so that all underscores are safe within the LaTeX-boundaries.

But now it works like a charm.

Ulysses is able to export your text in a variety of ways. You can preview it as HTML, including all rendered LaTex, and you can export directly either to Medium (on which I should begin to cross-post stuff asap) or your own WordPress-site.

In fact, I wrote this in Ulysses, then clicked the export-icon, choose ‘Publishing’ and NeverEndingBooks, and bingo I was able to post it as a draft. I can even fill in categories and tags, even add the featured image appearing at the top of this post, check everything in WordPress-admin and hit: “Publish”.

I guess I’ll be doing all my non-paper writing from now on entirely in Ulysses.

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