15 November 2014

Penrose tiling in Helsinki

Downtown Helsinki I stepped on a pedestrian street tiled with the standard nonperiodic Penrose (kite and dart) tiling.
This tiling consists of two basic shapes, the kite and the dart, both derived by taking a canonical pentagon inscribed in a circle, splitting it into 5 triangles with common vertex the center of the circle, and then making a variation on one of these isosceles triangles: take the side of the triangle which corresponds to a chord of the circle and make an inwards bump to obtain the dart (blue figure below) and an outwards one to obtain the kite (red figure).
Then follow some rules on how to join copies of these pieces so as to completely cover the plane. The result is a non-periodic pattern: no finite portion of it can describe the whole tiling. In particular, the tiling has no translational symmetry and is self-similar. Here is another picture and below it my attempt to show you its basic shapes. Kites are red, darts are blue.

The interesting thing with this tiling is that it appears as if it will repeat itself after a while, but it won't (this is a theorem). Nevertheless it is not random because it is created from a set of specific rules.

The tiling was discovered first by Roger Penrose 40 years ago. It was known that one could produce non-periodic tilings with a finite number of shapes but Penrose managed to do this with only 2. In nature, there are materials (quasicrystals) exhibiting such behaviors. Since the Penrose tiling is based on the pentagon, the so-called golden ratio plays a fundamental role. Indeed, if we call  A, B, C, D, E the vertices of a canonical pentagon (in the ordered traversed when going around in one direction) and let X be the point of the intersection of the chords AC and BE then, using similar triangles, we see that AX/AB = AB/AC. (The triangles ABX and ACB are similar, i.e., one is a scaled version of the other.) If we let AB=a and AX=b, then we see that AB=a and XC=a, so AC=AX+XC = b+a. The equality of the ratios above then becomes b/a = a/(a+b), so if we let  φ be the ratio b/a, we have φ = 1/(1+φ) which means that φ2 + φ = 1. But (φ+(1/2))2 = φ2 + φ + (1/4) = 1 + (1/4) = 5/4, and so φ = (√5 -1)/2, a number known and used since times immemorial.

If you have java installed and enabled on your browser, you can play with trying to create variations of non-periodic tilings using the Penrose tiling applet. (Or see the PhD thesis of Craig Caplan.)

But the interesting thing is what a then young PhD postdoctoral physicist, Peter Lu, found out some 10 years ago in (the Islamic) Darb-i Imam shrine in Isfahan, Iran, dating from 1453. He observed that the patterns forming the wall decorations form a non-periodic tiling, just as the Penrose tiling. In fact, you can see the kites and darts in the picture below.
He then wrote a paper with (P Steinhardt) analyzing this.I think that, since then, non-periodic patterns have been discovered in other places in the Islamic world. And businesses have grown out of it.

The fascinating thing about this discovery is two-fold. First, its mathematical interest and the fact that non-periodic tilings had been discovered more than 500 years ago. Second, the fact that they had been discovered empirically. Which makes us wonder why on earth would those Muslim decorators be interested in creating something so complex. My reasoning is as follows. It is known that, in Islam, people are very restricted with what kind of things they are allowed to decorate their temples/mosques/shrines. Gods and the like are not allowed. Human forms are not allowed. Animals or plants are not allowed (exception: in Iran, but that is, I am being told, a remnant of the pre-Islamic religion). Any concrete objects are not allowed. This is why Muslims have very few things they can play with: abstract patterns, tilings, geometric figures. But, even within this restricted framework, humans' minds can be quite creative. A human has an innate need to be free, to explore, to wonder, to create. When authority and religion imposes restrictions and rules, humans will try as much as they can to break them. It seems that this is a prime example of the innate need for freedom of expression.

11 November 2014

Visiting Aalto University

Here are a few pictures from the main building of Aalto university in Helsinki,  designed by the great architect (a genius,  according to Frank Lloyd Wright, or, as others have put it, the Finnish Frank Lloyd Wright).
An eloquent humanist, as well as one of the great architects and designers of the 20th century, Alvar Aalto breathed life and warmth into modernism, placing emphasis on "organic" geometry; supple, natural materials; and respect for human feeling.
Finnish architecture and design are some elements of this country I noticed quite some time ago.

30 October 2014

Unfortunate business names

Tonight, I noticed the following sign of a cycle shop in Uppsala:
From where I was sitting, I read it as SKITOTAL. Depending on where you separate the word it may be something stinky.

One can understand why IKEA in Thailand didn't notice that Swedish product names meant something offensive or sensitive in Thai (see Wall Street Journal article here) but one wonders how come that a Swedish company didn't notice the stinkiness of its name in Swedish.

28 October 2014

Universities and tabloids

Have you noticed that, more and more, university web pages resemble tabloids?

The first page below is from a major university: "Best sex positions for women with bad backs" is on the front page.
The second page is from a major tabloid: "Just when we had sex, I noticed..." is on the front page.
Both are catch phrases of similar type. Their goal is to attract the customer's [sic] attention so that they click and read further, and, possibly, contribute some money. By subscribing, in the case of the tabloid, or by contributing towards the 250 thousand dollar goal, in the case of the university (top right corner of first image).

Some time ago we used to think that universities were serious institutions of higher learning and research. With some exceptions, of course, this is not the case any more. A large number of academic institutions are usurping the terms "research" and "teaching" and use them for services that have nothing to do with the original meaning of the words.

19 October 2014

The eerie silence

I recently read a book, "The Eerie Silence", by physicist Paul Davies. Paul Davies is the head of project SETI. The book is about the search for extraterrestrial life/intelligence. Of course, to-date, there has been no hint of any life whatsoever outside our own planet. Nevertheless,  the 50-year old project SETI, apparently now privately funded, is alive. There are many excellent reviews of the book on the Internet, for example, on the Guardian (see here and here), the New York Times, Goodreads, Science News, and others. The book is, indeed, interesting. It debunks UFO stories, discusses the issue of whether life is a commonplace in this galaxy (or in the universe)--with no conclusions, of course, the issues of habitable zone and multiple biospheres on Earth, the probability of intelligent life elsewhere (and Drake's "equation"), the need for less anthropocentric search methods, the possible ways that aliens might communicate with us (which may be far from what we currently think of or use), the inability we might have in even recognizing advanced extraterrestrial technology, various philosophical issues, what would happen if we ever recognized that life existed, and an optimistic conclusion.

Now, all that is great, we need to be optimistic, we need to keep searching and wondering and, as is well known, the answer to the question "is there life elsewhere" would be profound regardless of whether it is positive or negative. But the book is rather long and tends to get a bit boring at times. Drake's "equation" for instance is hardly anything remarkable. It's just a back-of-the-envelope calculation that anyone with high school knowledge can think of (except that data may be missing, and they still are). At times, there are diversions towards religion, history or philosophy. What I found remarkably shallow is the author's claim that it was monotheistic religions (and, by this, he means the Abrahamic religions) are conducive to science. Namely, Davies claims, in the book and elsewhere, that, as opposed to Hinduism, the Abrahamic religions hold that the universe had a beginning. He also claims
The Greek philosophers taught that humans could come to understand the world by the existence of reason, which achieved its most disciplined form in the rules of logic and mathematical theorems that followed therefrom. They asserted that the world wasn't arbitrary or absurd, but rational and intelligible, even if confusing and complicated. However, Greek philosophy never spawned what today we would understand by the scientific method, in which nature is `interrogated' via experiment and observation, because the Greek philosophers' touching belief that the answers could all be deduced by pure reason alone.

Meanwhile, monotheism increasingly shaped the Western world view during the formative stages of science. Judaism represented a decisive break with almost all contemporary cultures by positing an unfolding cosmic narrative based on linear time.

The concept of linear time, and a universe created by a rational being and ordered according to a set of immutable laws, was adopted by both Christianity and Islam, and was the dominant influence in Europe at the time of Galileo. The early scientists, who were deeply religious, regarded their work as uncovering God's plan for the universe, as revealed through hidden mathematical relationships. What we now call the laws of physics they saw as thoughts in the mind of God. Without belief in a single omnipotent rational lawgiver, it is unlikely that anyone would have assumed that nature is intelligible in a systematic, quantitative way, mirrored by eternal mathematical forms.
Davies' claims suffer from a number of historical and logical inaccuracies.
  1. It is true that ancient scholars had not fully developed the scientific method, but it is not true that they only relied on things they could do in their heads. Indeed, the name of Archimedes is never mentioned in the book. Neither is any mention of the Antikythera mechanism. It is true that these things belong, perhaps, to the domain of engineering, but it is clear that nobody could have built them by thinking only, without any kind of experimentation. The claim that "nature [was not] `interrogated' via experiment and observation does not seem to be correct.
  2. When Davies speaks of monotheistic religions, he means the Abrahamic ones. There have been other monotheistic religions which are not included in his `reasoning', for instance, Zoroastrianism.
  3. Davies speaks of what--he thinks--monotheistic religious scientist achieved several centuries after these religions were invented, but he never mentions what happened, for example, when Christianity became the official religion of the Roman Empire. Who got rid of all work in mathematics and reason, up to that point, if not the new monothestic religion (or the particular version which the emperors adopted)? 
  4. The claim that "linear time" and "universe which has a single beginning" are both mentioned by monotheistic religions does not imply that monotheism implies these concepts. It is, merely, an accident that these concepts were adopted by the Jews (and hence by Christians and Muslims). The implied implication is not valid. 
  5. "Early scientists were deeply religious." We've heard this argument many times. But why is not Davies considering the obvious fact that those scientists had to be religious in order to be allowed to do what they were doing. Yes, some of them believed in god (there was no alternative anyway), some did not, but everyone had to pretend and behave as if they actually believed. So we will never know the truth. Imagine, for instance, the future historian who will claim that "in 20th. c. America, every president was highly religious and always appeared to pray in public". We, of course, know that without appearing to be religious they stand no chance of getting elected.
This is a weak point of the book. Other than that, I liked it, but, as I said, I could have read the same things in half the space.

Davies seems to be a smart person. So let us examine why he often digresses to praise monotheism. Well, there are several reasons, among which I can identify at least two:
  1. First, he's director of SETI which is privately funded, so he needs to please donors. Many of them (by virtue that they come from a religious country) are probably religious.
  2. Second, he probably likes awards. For example, he's a recipient of the Templeton prize. This is a very peculiar prize because it is given to all kinds of people, including ones who have caused harm. The prize has been criticized by Richard Dawkins ("[the Templeton prize is] usually [given] to a scientist who is prepared to say something nice about religion"),  Sean  Carroll (people cannot take Templeton research grants when they do not support Templeton's beliefs) and Martinus Veltman ("the Templeton prize bridges the gap between sense and nonsense")
Davies' claims that monotheism is conducive to science gives religionists ground to support their irrational beliefs. Some Muslims believe that the Quran contains science and they sometimes quote Davies as scientific support of this ridiculous claim.

On the Christian front, Davies seems to be a friend of John Lennox who likes to use "mathematics" and "logic" and "science" to support his religious claims. (The worst of all, in this conversation, is that Davies and Lennox, a physicist and a mathematician, discuss "specified complexity", a bogus concept invented by William Dembski for the sole purpose of promoting creationism.)

Last but not least, I can't fail but notice that the website (mentioned in the Eerie Silence) IETI (invitation to extraterrestrial intelligence, created because if--they claim--aliens get in touch with us, they might do so over the Internet) contains 100 individuals (inviting aliens) among which a certain Sohail Inayatullah who "brings an Islamic and Indian tantric perspective to understanding the Other, space travel, and alternative futures."

The upshot of all this is that, by trying to please everyone, including those who have nothing to do with science, one ends up having their work used for the purposes of the those who do nonsense.


What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant