dodecahedron

The dodecahedron has 12 faces...one for each note in the chromatic scale. Has the shape ever been used to create music. For example rolling it as a dice? Or even something more structural like using its symmetry somehow to generate lines of music?

My new book is written as 12 chapters, one chapter for each month of the year. I used a spinning dodecahedron at the top of each chapter to keep track of the chapter number.

Repeating decimals

Here's something that I find rather amusing about fractions.

Write each fraction as a decimal:
1/1 =1.000000... Repeats in 1
1/2 =0.500000... Repeats in 1
1/3 =0.333333... Repeats in 1
1/4 =0.2500000... Repeats in 1
1/5 =0.2000000... Repeats in 1
1/6 =0.1666666... Repeats in 1
1/7 =0.142857142857142857... Repeats in 6
1/8 =0.1250000000... Repeats in 1
1/9 =0.111111111... Repeats in 1
1/10=0.1000000... Repeats in 1
1/11=0.09090909090909.... Repeats in 2
1/12=0.0833333333333... Repeats in 1
1/13=0.076923076923076923... Repeats in 6
1/14=0.0714285714285714285... Repeats in 6
1/15=0.0666666666666... Repeats in 1
1/16=0.0625000000000... Repeats in 1
1/17=0.05882352941176470588235294117647... Repeats in 16
1/18=0.055555555555... Repeats in 1
1/19=0.052631578947368421052631578947368421... Repeats in 18

The point is that if p is a prime then
1/p expressed as a decimal will have a repeating pattern of
length N where N divides p-1.

[More generally if a number n=p(1)xp(2)...xp(k) then 1/n written as a decimal will have a repeating
pattern of length N where N divides (p(1)-1)(p(2)-1)...(p(k)-1). See for example 1/14.]

The interesting question is: which primes p do you get expansion where the repeating pattern of length N is equal to (p-1), the maximum length of a repeating pattern?

In the examples above this happens for 7, 17 and our prime 19. One reason I like the 19th step. Some of the primes after this for which you get this maximal repeating pattern are: 23, 29, 47, 59, 61, 97, 109.

Gauss made the conjecture that there are infinitely many primes for which this is true.

This is still unknown. We know that if the extended Riemann Hypothesis is true, then there are infinitely many such primes.

It relates to something called Artin's conjecture. Artin was a mathematician who did his mathematics in the 20s and 30s. I rather liked the following quote I found from Artin:
"We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt, he must always fail. Mathematics is logical to be sure, each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that, its perception should be instantaneous. We have all experienced on some rare occasions the feeling of elation in realising that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications."

I was quite struck by this statement that mathematics was non-linear. This sense of suddenly seeing what was going on is quite true. One might proceed through a linear, logical argument. But then there is this moment when you suddenly see what's going on. Like someone switching on a light and revealing the architecture of the room. Does music have that moment? A non-linear experience amongst the very linear nature of listening to a piece of music?

possible publicity pic (from Carols' archive)

pages and infinty quotients

With the flute material I have started thinking in terms of 'pages' rather than books. The page torn from a book has a high 'infinity quotient' as there could be any number of other pages. A page taken at random could have any kind or quantity or arrangement of information. There are of course many precedents for this in music, starting with Cage. At this stage for me it is, as much as anything a device to keep the palette 'open' and keep the sense of exploration and responsiveness to the fore.

This opens up the idea that there could be different kinds of material generated in numerous different ways within a piece. I am currently working on pre-composed material but I want to leave space for devising material with performers within the development period, in collaboration with Carol and also with Marcus.

There is an inherent openness then to the possible order of events. The 'generic' and multiple identity of the piece becomes stronger too. Some pages are instrument-specific, some are open to being played on any instrument, others might be transcribed between instruments.

On Friday 14th I will be working with Katie and exploring the 'spaces' made by moving between different fingerings outside the normal 12-note chromatic for which the flute is designed. The timbre becomes unstable and you discover 'half dimensions' in the sound-world. Working with Katie, we will find ways of fingering and refining each gesture to reap full colouristic value from it, a process of mapping the abstract musical gestures onto the body of the flute.

the nineteenth step

I'm really enjoying having this image on my desktop. It is the view down the stairwell from the 18th floor of NZ House. The eighteenth floor is the penthouse, where the CD launch was, and so the 'nineteenth step' is perhaps the step you'd take to see all the other floors, as in this image, or perhaps it's the balcony around the penthouse from where you can see the whole of 'the universe' of the west end of London...

Anyway, when I see this now I am thinking about the Library - it feels infinite and vertiginous.

Infinity Dances

I've been thinking about your provocation Dorothy regarding infinities in other forms. I took this question to Rose and we came up with revolutions, spinning and a maze. It was interesting to compare two different modes of turning. For Rose, with a classical ballet organisation of space, the pirouette and the fouette are about 'flipping' front to back, back to front, whilst keeping a sense of frontality using 'spotting' (keeping your eyes looking to the front until the last moment and swinging them around again). Interestingly, a New Zealander, Rowena Jackson holds the world record for 152 consecutive fouettes. In performing Odette or Odeile the expectation is 32 fouettes. My own experience of revolutions is in spinning within the post-tanztheatre version of this which is a kind of ecstatic turning, taking your gaze around with you until the world swings into horizontal bands of fluid colour, losing the solidity of forms to become liquescent matter, keeping the momentum of spinning going and sustaining it for as long as possible whilst feet step a constant pattern. Perhaps this distinction - between a classical and a contemporary version of revolutions - can be compared to turning the page or leafing through the book, between wakefulness and dreaming. Borges: "Arthur Schopenhauer wrote that dreaming and wakefulness are the pages of single book, and that to read them in order is to live, and to leaf through them at random to dream." Merging these modes of turning as a way to explore infinity we came up with a skimming turn that travels the perimeter of a triangle and resolves itself in infinite steps in a maze like pattern. The idea is that this material is constant and continuous, the turning suspending a sense of vertical and temporal divisions.

What shape is our universe

For me one of the central themes of The Library of Babel and Borges work in general is an attempt to understand the shape of the space that we live in. One of the sentences in The Library of Babel that stuck out for me: The Library is a sphere whose exact centre is any one of the hexagons and whose circumference is inaccessible. This, Borges declares, is a classic dictum about the library. It occurs at the end of the second paragraph. What I discovered yesterday whilst reading around for my History of Maths series is that this actually picks up on a quote from Pascal:Nature is an infinite sphere, whose centre is everywhere and whose circumference is nowhere. It turns out that Pascal's original version had the extra word FEARFUL sphere.

In the essays in my addition of Labyrinths there is an essay by Borges dedicated to The Fearful Sphere of Pascal. It traces back how the sphere has been an important symbol for many philosophers. Parmenides: The divine being is like the mass of a well-rounded sphere whose force is constant from the centre in any direction. The music of the spheres. Kepler's model of the universe as interconnected spheres.

But what is an infinite sphere? A sphere is a surface whose points are equidistant from one fixed point. If that distance is infinite then indeed one can take any internal point as a centre. But the interesting thing is that there is something rather unsatisfying about an infinite sphere because that surface - the circumference - is unattainble, unreachable. Even if you did reach it, the curvature of the sphere is such that it is an essentially flat surface. Think of a circle that gets bigger and bigger. The curvature of the circle gets flatter and flatter till at infinity it is really a straight line.

I think most people's perception of the shape of the universe is something like a huge infinite ball with us at the centre.

Interestingly Plato in his text Timaeus associates the dodecahedron rather than a sphere with the shape of the cosmos. The Platonic solids are the building blocks of his atomic theory:
the tetrahedron- fire
the cube - earth
the octahedron - air
the icosahedron (the most spherical of all the Platonic solids)- water.
and finally the dodecahedron is left as the shape which captures the universe.

These shapes are made from triangles, squares and pentagons. Once one gets to a hexagon, you find that the only way to put them together makes an infinite flat surface, like a honeycomb. This is if you are in a Euclidean universe. I'm intrigued to see whether you can get interesting shapes with hexagonal faces in non-Euclidean space. I'll try to chase this up.

But both the universe and the library are ultimately not believed to be infinite but rather finite but unbounded. For example the surface of a two dimensional sphere (like the surface of our earth) is finite but unbounded. Perhaps we live on the three dimensional surface of a four-dimensional sphere. Then we would actually be living on the circumference and the centre would be unattainable.

For me, the exploration of the library leads ultimately to a way to discover what sort of surface we are a living on - a 3-dimensional surface of a 4-dimensional shape. The building block of this surface is the hexagon (or rather a hexagonal box) and the way these are put together, the way they are connected gives us clues to the overall shape, a shape we can never see in its entirety. I'll post later my thoughts on what this shape might be.

This has been the fun part for me of reading the Library of Babel. It's been like a puzzle. Taking the clues that Borges gives us and then trying to find a shape that fits the clues. A lot of geometry recently has been about finding ways to describe a shape from its internal structure. How can I tell whether I'm on a sphere or a torus (a ball or a bagel) just from internal features of the surface.

By the way, I love the idea of the floor being a chalk surface that we can draw on. Also love our name: The Nineteenth Step.

Marcus.