The 19th Step Development Day Six and Seven, Equations, Sequencing, Patterns and Yellow Paper

Dorothy works with Richard and Scott to make music to accompany Marcus as he creates an equation through dance. Marcus showing the performers his moves which show an equation through his body.

These two pics show Marcus teaching Dylan the body equation moves.

Below are two photos showing the performers practicing their 12 body positions.


The clip below shows ideas developing the opening of the piece.

The 19th Step Development Day Four, developing each strand

Today we are all working on various parts of the project separately.

Below are some videos beginning to collect material to create an idea of the Aleph?
Just been listening to this programme,
http://www.bbc.co.uk/radio3/discoveringmusic/pip/5056m/
you need to listen to it by Sunday 30th as the link will run out. Its about Tango and one of the contributors just said "Borges describes Tango as a heart dancing on four legs" this made me think his short story the Circular Ruins.




Marcus explains irrational numbers.



Here are some pages from Marcus's sketch book a lot of the diagrams show him trying to workout how to navigate through the hexagonal rooms of the Library of Babel.

The 19th Step Development Day Three, Patterns, Shapes and Tango

"Musically, the tango is not important; its only importance is what we give it. This reflection is correct, but perhaps applies to everything."
"The tango can be debated, and we have debates over it, but it still guards, as does all that is truthful a secret"
Jorge Luis Borges, A History of Tango.

Today Marcus our mathmatician had his first ever tango lesson. Using the shape of the circle the dancers and Marcus improvised moves drawing circles on the floor inspired by tango.

Marcus drew the diagram of the dance moves he and Rose had developed. Inside the circle the 6 triangles make a hexagon, the shape of the rooms in Borges story the Library of Babel.

Making a hexagon

One thought is to use the giant set of compasses to create a perfect hexagon on the floor. This is a construction that the Ancient Greeks discovered.

There is an animation on Wiki which shows how the construction is done. The point is that you can't measure anything. You can only draw a straight line or a circle with the compass.

It is possible to draw a pentagon but it is impossible to draw a 7-sided figure with this equipment. In fact if the number of sides of the shape is N then the shape can be constructed if and only if the odd primes dividing N are Fermat primes, that is primes of the form 2^2^n+1. The only Fermat primes known are 3, 5, 17, 257, 65537.

As a 19 year old, Gauss discovered a construction of the 17-sided figure. His discovery prompted him to begin a mathematical diary, one of the most important documents in the history of mathematics. Here is his construction of the 17-gon.

Maths Moments II

Shapes and the nature of space
(a) A collection of triangles, squares, pentagons, hexagons, septagons and octagons. How can you put them together to cover the floor? If all the shapes are the same then only triangles, squares or hexagons will work. But what if you mix them up?
(b) What 3 dimensional shapes can you build with the flat shapes? How many are there if the faces are all the same? Plato proved there are five shapes. Their symmetry makes them perfect as dice. Plato associated the shapes with fire – tetrahedron; earth – cube; air – octahedron; water – icosahedron; and finally the shape of the universe corresponded to the dodecahedron. If you mix shapes then Archimedes proved there are another 13 shapes you can make.
(c) What are the shapes of the universe? What is the shape of the library? If the library only has one floor, then it is like a flat universe. The library could extend without limit in all directions, the hexagons just tiling an infinite flat expanse. But more interestingly it might fold up on itself. It could be like the surface of a sphere or the surface of a torus. But how could the librarian tell which shape he lives in, stuck as he is on the surface of this universe? On a sphere if you draw a closed path, a loop, on the surface, then it is possible to continuously morph the loop until it vanishes to a point. On a torus there are loops that can’t be shrunk like this. These loops are like closed journeys on these surfaces. The way the hexagons are arranged and the doors going from one to another, it looks like the floors of the library are actually like torus shaped universes. As you travel through the hexagons you find yourself returning to the original hexagon from which you started. But what are the other possible shapes that you can use as plans for wrapping up the floors of the library? Poincare proved at the beginning of the twentieth century that any library can be morphed until it either looks like a sphere, or a torus with one hole, or a torus with two holes and so on. Perhaps you could see the librarian going through the proof of this projected up onto the walls.
(d) But the library is many layered, it is a 3 dimensional universe where you can go from one floor to the next…a third dimension. So how can this universe be wrapped up? Now we are having to wrap up a three dimensional universe in 4 dimensions. If we head up to the higher layers of the library we suddenly find ourselves returning to the layer we started at then the library is wrapped up into a higher dimensional bagel or torus. But what other shapes are possible? This is what Perelman answered in his solution to the Poincare conjecture, possible one of the greatest achievements in mathematics in the last century.
(e) There is also the interesting issue of whether the library is a Euclidean geometry or possible a non-Euclidean geometry. In non-Euclidean geometry, triangles have angles that don’t add up to 180 degrees. Consider triangles drawn on the surface of a sphere. See this link for an interesting article: plus article
(f) And what about fractals? Could the library reflect in some way some fractal characteristics?

Maths Moments

My intention was to carry on the little narrative I began with the librarian faced with the two piles of books and to tell you what happens in the next room. But perhaps in preparation for our meeting at The Hope I'll record some of the things I was thinking of including.

I think the maths moments I hope we might include will focus on two topics
(1) number and infinity
(2) shapes and the nature of space

This post will deal with the first one.

(1) Number and infinity
(a) I quite like the fact that you can calculate exactly how many books there are in the library given that the number of pages is limited. It's a large number but one can also calculate from this the number of hexagons.
(b) But it's still possible to make infinite books by combining the books in the library. So for example given ten books numbered 0 to 9, any number could represent a new book where I read the ten books in the order given by the number. For example 134115 means the book got by reading Book 1 followed by Book 3 followed by Book 4 followed by Book 1 again then Book 1 again then Book 5. It’s almost like using the first library to build a second library with infinitely many books.
(c) But then perhaps you come across a third library where the books are all labeled with fractions. Does this library have more books in than the second library where all the books are labeled with whole numbers? This was Cantor’s great discovery that actually these two infinities have the same size because there is a way to pair up all the books in each library. It’s called Cantor’s diagonal slash. There is a very graphic way to show how to pair these numbers up which I’ve done in one of our workshops and I’ve also done with the workshops I’ve created with Complicite. I wonder if one could explore also using the audience as books. The Cantor argument depends on arranging the fractions in a big infinite two dimensional grid. Here is the script I’ve written yesterday for the TV programme I’m making:

Cantor needed to count all the fractions in a systematic way. To do this he started by arranging all the fractions in an infinite grid. The first row contained all the fractions with 1 on the bottom. [See these fractions being laid out] In the second row came all the fractions with 2 on the bottom. [Next see these fractions] Carrying on like this, the 6th row for example would contain all the fractions with 6 on the bottom.

Every fraction appears somewhere in this grid. Where’s 5/7…go to the 5th column of the 7th row. [See this being lit up] But how can we count these fractions. How can we pair up the whole numbers with this infinite grid of fractions. [Perhaps see the whole numbers appear along the top of the screen]

Cantor’s snake is the key. Imagine a line snaking back and forward diagonally through the fractions [see the line running through the fractions] then by pulling this line straight we can match up every fraction with one of the whole numbers. [Now see the fractions being pulled up by the line and aligning with the whole numbers. We can even have the whole numbers running off the screen to the left so that we gradually see more and more fractions pairing up.]

So the fractions are the same sort of infinity as the whole numbers.
(d) But then there comes a library with books labeled with infinite decimal numbers, like pi=3.14159… or e=2.71828… Each library could even have its own librarian. Almost like the librarians showing off to each other about whose library has the most books. It turns out that this librarians library has an infinity which is much bigger than the fractional library or the first infinite library. There is a nice way to show this physically as well, an exercise which I’ve done before with us and Complicite.
(e) One of the big mysteries of the twentieth century was: Is there an infinite library with strictly fewer books than the infinite decimal library but strictly larger than the infinite fractional library. The problem to sort this out was called the Continuum Hypothesis. In the middle of the twentieth century a rather surprising answer was reached: Both answers could be true. There was an entirely consistent mathematical world where there was such a library. And an equally consistent mathematical world where no such library existed. It was a discovery that was almost as shocking as the discovery of non-Euclidean geometries.

The first book

Dorothy asked me some time ago what I thought was the first book in the library. A strange question. I found it very difficult to answer. I wasn't sure why. Then I realised that it was because my image of the library was very continuous. It was a shape without a beginning or an end. This is a universe without a boundary. You can't hit up against a deadend. I can keep on journeying through one door into the hexagon and out the opposite door. Or climb up and down between the layers of the library. But the library isn't infinite, just without a boundary. So if I keep on journeying I will eventually come back to where I started.

But as I thought a bit more about the first book and the beginning, I realised that the story of the Library Babel mirrors the two first steps we make in navigating the world around us: to count and to explore the space around us. I believe that animals are almost programmed by evolution to be mathematicians. Counting is essential for survival. To assess whether there are more of you than there are of the enemy will inform the decision to fight or fly. To assess the the nature of the space around us allows us to judge whether we are out of range of our prey or our predator. Those that can count survive. As Plato wrote across the top of the Academy: Let no one ignorant of geometry enter here.

So maybe the first book contains the numbers 1, 2, 3,... and pictures of triangles (although the books have no pictures...or do they?)

Galileo's famous quote: The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

Saw this today in a copy of the Western Daily Press! Not sure I totally understand how you can inadvertantly nudge the cosmos by measuring...??

Some thoughts about Borges, Tango......

Been reading Borges A History of Tango which is in The Total Library Non Fiction 1922-1986 came across these quotes and thoughts not sure what they have to do with 19step yet but here they are anyway

"Schopenhaur (Welt als Wille und Vorstelling I, 52) has written that music is as near to us as the world itself; without the world, without a common stock of memories summoned by language, there would be no literature, but music does not need, could exist, without the world. Music is will and passion; the old tango, as music, immediately transmits that joy of combat which Greek and German poets, long ago tried to express in words."

“We read in one of Oscar Wilde’s dialogues that music reveals a personal past which, until then, each of us was unaware of, moving us to lament misadventures we have never suffered and wrongs we did not commit.”

Was discussing with Dorothy idea of Tango as an example of passion and humanness being controlled and tamed through the structure of language, learning steps and routines and nostalgia rather than its original passion Borges describes it originally as The Fighting Tango, which now can only lament for what is lost…………..

Suppose been thinking about the differences between learning ballroom tango and argentine tango, sequences and patterns as opposed to intuition and improvisation and the relationship between mathematics, music and art……..

Dorothy keeps mentioning a primordial state…….do we make sense of this through patterns and geometry?

I keep thinking of the drawing exercise Carol led were we linked the hexagon to parts of the body with letters, which creates a dance sequence, the letters and geometry/numbers take over the body movements.... does it in Dorothy’s score?

Is Borges hexagonal library built to control all our physical and emotional needs?

“ To the left and right of the entrance way are two miniature rooms. One allows standing room for sleeping; the other, the satisfaction of faecal necessities”