The Mathematics of 'Circle Limit III'

M.C.Escher's series Circle Limit I-IV series was inspired by his friendship with the great geometer H.M.S. Coxeter, and Escher's Wikipedia biography mentions this explicitly:

Around 1956, Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's works Circle Limit I–IV demonstrate this concept. In 1995, Coxeter verified that Escher had achieved mathematical perfection in his etchings in a published paper. Coxeter wrote, "Escher got it absolutely right to the millimeter."

The paper in question by Coxeter is here, and demonstrates that Escher was a research mathematician in his own right:

Abstract. In M. C. Escher’s circular woodcuts, replicas of a fish (or cross, or angel, or devil), diminishing in size as they recede from the centre, fit together so as to fill and cover a disc. Circle Limits I, II, and IV are based on Poincare's circular model of the hyperbolic plane, whose lines appear as arcs of circles orthogonal to the circular boundary (representing the points at infinity). Suit-able sets of such arcs decompose the disc into a theoretically infinite number of similar “triangles,” representing congruent triangles filling the hyperbolic plane. Escher replaced these triangles by recognizable shapes. Circle Limit III is likewise based on circular arcs, but in this case, instead of being orthogonal to the boundary circle, they meet it at equal angles of almost precisely 80◦.(Instead of a straight line of the hyperbolic plane, each arc represents one of the two branches of an “equidistant curve.”) Consequently, his construction required an even more impressive display of his intuitive feeling for geometric perfection. The present article analyzes the structure, using the elements of trigonometry and the arithmetic of the biquadratic field, subjects of which he steadfastly claimed to be entirely ignorant.

Here's a paper by Douglas Dunham(pdf) that generalizes the fishy model a little further.

Ragas for ETLFs: Music for Aliens

Music composed for extra-terrestrial life-forms is certainly much harder to come by than music composed BY certain earth-dwelling aliens themselves.

Feel no shame, fellow pirates, as you download this free music CD from the rock stars of the internet. The tracks were apparently arranged on this album by Carl Sagan, and feature an astonishing variety:

102 Greeting From The Secretary General Of The UN
103 Greetings In 55 Languages
104 UN Greetings & Whale Greetings
105 The Sounds Of Earth
106 J. S. Bach -- Brandenburg Concerto No. 2 In F, First Movement
107 Java, court gamelan -- Kinds Of Flowers
108 Senegal, percussion -- Tchenhoukoumen
109 Zaire -- Pygmy Girls' Initiation Song
110 Australian Aborigine songs -- Morning Star And Devil Bird
111 Mexico -- El Cascabel (performed by Lorenzo Barcelata)
112 Chuck Berry -- Johnny B. Goode
113 Papua New Guinea -- Men's House Song
114 Japan, shakuhachi -- Cranes In Their Nest (performed by Coro Yamaguchi)
115 J. S. Bach -- Gavotte En Rondeaux, from the Partitia No. 3 In E Minor For Violin
116 Mozart -- The Magic Flute_ Queen Of The Night Aria, No. 14
201 Georgia, chorus -- Tchakrulo
202 Peru -- Panpipes And Drum Song
203 Louis Armstrong & His Hot Seven -- Melancholy Blues
204 Azerbaijan Bagpipes -- Ugam
205 Stravinsky -- Rite Of Spring, Sacrificial Dance
206 J. S. Bach -- The Well-Tempered Clavier, Book 2, Prelude And Fugue In C, No. 1
207 Beethoven -- Symphony No. 5 In C Minor, First Movement
208 Bulgaria -- Izlel Je Delyo Hagdutin (sung by Valya Balkanska)
209 United States -- Navajo Night Chant
210 Holborne -- Fairie Round, from Paueans, Gaillards, Almains, And Other Short Aeirs
211 Solomon Islands -- Melanesian Panpipes
212 Peru -- Wedding Song
213 China, Ch'in -- Flowing Streams (performed by Kuan P'ing-hu)
214 India, Raga -- Jaat Kahan Ho (sung by Surshri Kesar Bai Kerkar)
215 Blind Willie Johnson -- Dark Was The Night
216 Beethoven -- String Quartet No. 13 In B Flat, Opus 130, Cavatina

The Sunya Machina - Part III

The central theme in this series of notes (previously:Part I & Part II) is that the beginning of the number line (0,1,2,3) is a machine. It is shown as a generator that spits out the rest of the numbers until infinity. I have used the metaphor of cellular automatons and finite state machines, from my limited knowledge. As I learn more everyday, I will try and describe the developments here.

The Fundamental Theorem of Arithmetic states very simply that:

Every integer n ≥ 2 either is a prime or can be expressed as a product of primes. The factorization into primes is unique except for the order of the factors.
- from Elementary Number Theory with Applications 2e, Thomas Koshy

Such a factorization of any number is called the 'canonical decomposition' of the number, for instance 2520 = 2^3 x 3^2 x 5 x 7 where (2,3,5,7) are primes that make up 2520's internal structure. So perhaps large integers are not unlike organic molecules, made up of simpler components arranged in unique compositions, which gives them their particular properties. Such as, 2520 has a trailing zero because two of it's prime factors are 2 and 5 (2x5=10). The very existence of composite numbers appears to be just a classification tool, since prime numbers are the bricks that make up ALL numbers, big and small. How are molecules held together? By nuclear forces.

So shouldn't there be a bridge that links number theory to particle physics?

In this regard, eminent Japanese scientist Akio Sugamoto (he has co-written several papers with Nobel Laureate Makoto Kobayashi) submitted a paper on 24 Oct, 2008 titled "Factorization of number into prime numbers viewed as decay of particle into elementary particles conserving energy":

In number theory, how a number factorizes into prime numbers is a key issue, while in particle physics how a particle decays into elementary particles is also a key issue. These two key issues are intimately related, if we identify the energy E(n) of a particle labeled by a positive integer n = 1, 2, 3, . . . is proportional to ln(n). Then, factorization of a number into prime numbers can be viewed as the energy conservation law.

While I'm slowly becoming conversant in number theory, the mathematics Sugamoto-san uses in the paper is beyond me at this stage, and I'll leave it to better mathematics geeks. However, it does re-assert my idea that the generation of the number line is connected to physical systems and the mathematical models we have created to understand them.

The Dreaming Sea - Part III: "The Cellaphopod & the Mobtar"

The Hindu idea of avatar is connected with collective action, and has become emblematic of the way people are cooperating across borders. An avatar is perhaps not a single person, but a configuration of people, a mob acting together in unison, perhaps towards a common goal. Lets call it a Mob-tar.

At this moment millions of people are hooked onto the Internet, gazing into their liquid crystal displays; people at bus stops, in the trains, peering into their fluid mobile phone gooey (GUI) menus. The global telecommunication system is not a sea, but it feels like one because of this - the digital screen is a porthole; beyond the porthole we can see liquid, and some kind of lucid dream unfolding. It's a sea that is almost like a cephalopod with millions of pods (iPods are Steve Jobs' tentacles, people). It's a cell-a-phopod out there!

Cephalopods communicate with their environment in a very different manner than human beings. They have the ability to morph into various shapes and colours, they have the ability to physically simulate something else - like a rock or entirely different creature. There's a brilliant column by Jaron Lanier called "What cephalopods can teach us about language", in which he describes something called 'postsymbolic communication':

Suppose we had the ability to morph at will: What sort of language might that make possible? Would it be the same old conversation, or would we be able to "say" new things to one another?For instance, instead of saying, "I'm hungry; let's go crab hunting," you might simulate your own transparency so your friends could see your empty stomach, or you might turn into a video game about crab hunting so you and your compatriots could get in a little practice before the actual hunt. I call this postsymbolic communication. Some people think that the ability to morph would just give you a new dictionary mapping to the same old set of ideas, with avatars in place of words, while others, including me, think there would be fundamental differences.

While Lanier is a visionary in his own right, this gem of an idea above would be better explored outside the context of 'virtual reality' and avatars in arenas like Second Life. Instead, lets us imagine the descent of mob-tars in the context of locative media and satellite imagery.

The fact that now we can see human configurations on such a large scale from a bird's viewpoint, or from the Moon even, begins to convert the Earth itself into a cephalopod, watching itself via digital media. It's a feedback loop that allows us to re-configure ourselves as a mobtar. The new linguistic context here is collective expression, millions of people arranging themselves as human pixels arranged on a screen. If executed as it was in the 2008 Beijing Olympic Games opening ceremony, it appears as an expression of totalistic governance, and perfect order. But when it is the people themselves who self-organize this shared meaning, it becomes an example of emergence, an example of mobtaric behaviour.

The cellphone is the new astrolabe, but instead of telling a single user where he is, it can be used to tell all users (the mobtar) where everybody is located. The mobtar can see how it's doing by watching it's own satellite imagery on the cellphone screens. The mobtar can morph! This heralds the arrival of a new language of mobtars, as language zooms out from the scale of your desk, to that of the entire planet.

Of course, the question is not whatthe mobtar is going to say, but what the mobtar is going to do to be able to say it.

The Dreaming Sea- Part II

One of the great spectacles of the modern world is it's simultaneity, or the way certain events seem to happen in a synchronized manner. The great invention of terrorism today is 'serial bombings', where a number of small bombs go off in multiple locations - almost together, as if a conductor were performing a symphony, like an orchestra. It is not entirely accidental that serialism has a rich history in Western classical music.

Simultaneity was difficult to observe for the ancients, because there was no system of communication that could move information faster than a human being, or perhaps a pigeon. If mathematicians in Ancient Greece and the Indus Valley's Vedic civilisation discovered a prime factorization algorithm simultaneously, the fact of this independent discovery would not be known until today. The ancients had no way of observing patterns on a global scale, but we do - and the Economic Crisis of 2008 is one such event.

We are not surprised by the fact that financial markets are falling, as it were, but by the fact that the same thing is happening, at the same time, all over the planet. Also, as observers of these phenomenon through our media technologies, we as a specie are instrumental in generating these events of a global nature. We provide the feedback loop for these events, and generate more events which are simultaneous, distributed and everywhere. This is the true coming of the smart mob - the smart planet.

There is a new problem that we have that the ancients did not have. Their problem was too little information from remote lands, our problem is too much information from remote and near lands. We are forced to ignore certain events just so that we can remain sane, to the extent that, we start believing that those events DID NOT OCCUR AT ALL. There were 64 organized bomb explosions in 6 states of India over the last 6 months, but Indian media are far more interested in the fall of Wall Street, and the U.S. Presidential election. The Indian media wants to make sure that no one forgets the fall of Wall Street. This is important and necessary for the human psyche, to maintain a continuity in it's own narrative (the serialism of the narrative?), and pretend as if certain things did not happen at all. We even choose hidden meanings from real events, and reconstruct those events in our head. We use reality (remixed) to manufacture dreams, and inject them back into the socium. A global event assumes a more stronger reality than something that's happening down the street.

Some call it a meltdown, a crisis or crash....but those who float on a raft in the dreaming sea, they know....they've seen it coming before. It's called a tsunami.