Quasicrystals are aperiodic structures that still have some regularities. Both the spatial and frequency domain representations of quasicrystals are quite beautiful. See also this more recent post for more quasicrystal entertainment. Here is the classic diffraction pattern from a synthesized icosahedral quasicrystal

You can create two-dimensional quasicrystals by summing together more than 3 plane waves. For example, here are some spatial domain two-dimensional quasicrystals :

4-fold :
5-fold :
7-fold :

[overflow gallery]

The picture at the beginning of this post is an x-ray diffraction pattern, which is a little bit like the Fourier transform of a quasicrystal lattice. Here is a simulated 2D Fourier spectrum from a 7-fold 2D quasicrystal :

You can create your own images like this by rendering large two-dimension quasicrystals as described here or here. The frequency domain images can be rendered using ImageJ, or your own personal 2d FFT code. In ImageJ, open an image, and then create a frequency domain image by clicking Process→FFT→FFT. If you rendered a N-fold quasicrystal, you should see 2N points arranged in a circle around the origin, corresponding to your N plane waves. If you applied some nonlinear image operation, like contrast enhancement or thresholding, to the spatial domain image, you will have created some harmonics and overtones of your original N plane waves, which should appear as a constellation of other points that themselves are arranges on a quasi-crystal lattice.

Quasicrystals can be related to more abstract mathematical concepts. For instance, this article finds that some quasicrystals are related to sorting algoritms :
Current research in abstract tiling theory examines tilings of high rotational symmetry in collaboration with Remy Mosseri and co-workers. Possibly the limit of high rotational symmetry may prove easier to analyze than specific finite symmetries (10-fold, for example) of direct physical interest. Surprisingly, rhombus tilings are related to algorithms for sorting of lists. Counting the number of distinct tilings enumerates simultaneously the number of equivalance classes of sorting algorithms, a problem previously considered by computer scientist D.E. Knuth. Our random tiling theory implies an upper bound of log(2) for the tiling entropy per vertex, consistent with a conjecture by Knuth. Click here for a preprint on this research.


The Mechanization of Man

One of the most common critiques of human enhancement is that it would lead to viewing human beings as machines or creatures (in Francis Fukuyama's words) rather than as people; that the mechanization implied by enhancement is a threat to viewing others as individuals, and therefore human rights and freedoms. As in the case of cyborg theory, I believe that Fukyama's concern is at least fifty years too late. In many cases, we are already treated as components, and in the significant power relationships of our life, human enhancement cannot further mechanize us.

There are two basic ways to view the universe, the aesthetic and the scientific. A scientific viewpoint sees the universe as a system to be understood. There are elements and relationships between them. It is objective and universal, in that any two entities using a scientific methodology should arrive at the same conclusion. The aesthetic is subjective and personal, the universe is composed of experiences, each one different for every time and every person. This is not to imply any kind of value judgment, the aesthetic and the scientific have their place.

The relevance to the enhancement debate is that for every significant power relationship in your life, between yourself and your workplace, school, the government, you are treated scientifically. Organizations have no subjectivity, no locus of "I" with which to experience the universe. They are a collection of rules, applied with varying degrees of fairness. Organizations deal with you not as a person, but as test scores, resumes, dossiers. A human being is not an individual, it is a component to be processed, improved, reformed.

The illusion that we have rights or freedoms vis a vis these organization originates in the aesthetic interactions that we have with their representatives, bosses, teachers, police, and our personal relationships. As the former are illusory, a social artifact, we can dismisses their value, and therefore the claimed threat to institutional human rights. That leaves our personal relationships, and will those change with enhancement? Clearly not, we address each other, not our belongings, or our education, or any other kind of non-technological "enhancement" that we already possess.

The mechanistic view is inherent in the system. All that human enhancement could do is make us better machines. The human essence, if such a thing exists (I enjoy Chris Beck's formulation of the soul as something non-algorithmic), remains what it is.


Science in the Media

I just got back from a PCAST briefing on K-12 Science, Technology, Engineering, and Math education. As we all know STEM is totally broken, and aside from the usual platitudes, we don't have great ideas about how to fix it. There was one interesting thing: Kids these days spend 12 hours a day accessing media (guilty as charged). Only 10% of characters in media are scientists or engineers, and 70% of them are portrayed as the bad guy. We need some new positive role-models.

The Real Adventurers of Richard Feynman: Nobel prizing winning physicist and international playboy travels the world from Los Alamos to Rio de Janeiro, solving mysteries, seducing beautiful women, and stopping the evil machinations of VIPER, a sinister organization bent on destroying the world with nuclear weapons. Featuring cameos from historical figures, and the Feynman Action Science Team of Bobby Kennedy, Marilyn Monroe, and Time-Traveling Teddy Roosevelt.

The Lab: You've seen The Office, right? Like that, but with science. NBC already has two shows that are basically The Office, so one more can't hurt.

Chrome Streets: A cyberpunk action/RPG in the style of Mass Effect. The player is an underground scientist, battling megacorporations for the fate of mankind. Heavy transhumanist themes, as the game revolves around your continual human advancement. Skills are divided between Robotics/Biotech/Nanotech tracks, each one offering a different combination of powers. The game is supported by a robust crafting/research system that mimics real science. Written by Charles Stross, and with a better title.

EDIT: Apparently there's a Deus Ex 3 coming out. Dues Ex was one of the best games of all time, and pretty much exactly the above. Do it right, Eidos, or I will go to French-Canadia and go all JC Denton on you.


An Illogical Argument ( from a friend )

In light of Beck's recent post, I have dug up the following illogical proof. I think he and I spent a some 52 hours trying to parse this and convert it to more formal logic. We vowed never to touch this stuff again :

The statement that there is no absolute truth is in and of itself an absolute, and if the statement were true, it would be self contradictory, and hence the statement is itself false.

Therefore, there is no such truthful statement. the statement that there is an absolute truth is also an absolute, and if the statement is true, it is not self contradictory. hence this statement by itself is not necessarily true nor false. if we assume that this statement is neither true nor false then we arrive at a contradiction, because the negation of the statement that there is absolute truth is the statement that there is no absolute truth.

Therefore, if one is true, the other must be false since both are absolute statements about truth or falsity. the statement that an absolute truth exists which is both true and false is false since we have shown that the negation of there existing an absolute truth is false. so, it must be that there is at least one absolute truth.

"Universality of the human mind is much more important than immortality"

So I had an interesting discussion with a particular someone not too long ago. Don't remember how it started exactly, but it was basically like, why are you doing this research, what does it mean to you... etc.

One of the things that came out of this very quickly was that, I realized that I really do believe that limits on the power of algorithms represent limits on what we can know, what we can do and be as people.

So then there's the issue, but that's so depressing if we are just computers. And my response, to my surprise, was along the lines of, its way worse if we really do have souls that we can't peer into or explain or simulate. Because then there's no possibility for communication, and the machines can't simulate eachother, unless the souls are basically identical and then they can simulate eachother.

I ended up arguing that, as long as we are all generic machines, then we know that our ideas can always be explained to the other robots and can live on forever. If we have souls, then this is fucked, because if the souls are computationally distinct, then it may be possible for one person's thought not to have an analogue in another person's thought, so communication is impossible, and all of the ideas this person produced that made them distinct must die because they cannot be comprehended and passed on.

Which is totally the opposite from how people usually argue this issue. They say, to live on, there must be a soul, so I would rather believe in that. Really? You'd rather believe that your most treasured thoughts and ideas can never be passed on, and instead, you will die and take them with you from this world? And, I suppose, spend the afterlife still unable to communicate them to your peers, or to receive their treasured ideas?

Universality of the human mind is much more important than immortality.


Bits of Intellectual Property

At what point do information artifacts cease to become copyrighted works of art and become patentable inventions ? The distinction between aesthetics and utility would seem to clarify this, but in an information based economy the two become less and less distinguishable. Code is data, data is code, and certain innocent prime numbers become illegal.

Copyright is the set of exclusive rights granted to the author or creator of an original work, including the right to copy, distribute and adapt the work. Copyright lasts for a certain time period after which the work is said to enter the public domain. Copyright applies to a wide range of works that are substantive and fixed in a medium.
In the United States, a design patent is a patent granted on the ornamental design of a functional item. Design patents are a type of industrial design right. Ornamental designs of jewelry, furniture, beverage containers (see Fig. 1) and computer icons are examples of objects that are covered by design patents.
Note that copyright verses design patent has nothing to do with the medium. A mass produced copy of a marble sculpture can be copyrighted, and a mass produced copy of a marble coffee cup would be design-patented. The critical feature it would seem, is weather the art is attached to something useful, or is purely valued for aesthetic purposes. However, this distinction is undefined. Modern art created objects of minimalist, utilitarian design that were also clearly works of art (see here). Furthermore, information can itself be a technology. The distinction between code (patentable) and data (copyrighted) has been dissolved my modern programming paradigms. Music is arguably a technology for manipulating emotional and cognitive state. Whether a coffee cup is a useful object or an art object changes depending on how many you sell and what your clients decide to do with them. Modern music seems to be made for production, to the point where some of it is legally in the same class as disposable coffee cups.

For every partition between the set of things covered by copyright, and the set of things covered by design patents, I can construct an object that lies directly on this partition. This indicates that the two sets are connected and must share at least on object. Existing law acknowledges the intersection, (e.g. The Statue of Liberty ).

The distinction boils down to the number of bits representing the idea and the probability of collision between artifacts. There are still objects that clearly fall under copyright, and objects that clearly fall under design patent, but the digital economy has created ever more exceptions to this classification.

Design patents cover features with low information content and a high probability of collision. Alternatively, (since we may be working with physical objects with an infinite amount of state ) the similarity threshold for design patents is higher than that for copyrighted works. Generally speaking, design patents will cover the "gist" of the decorative appearance of objects, and will therefore contain far fewer bits than might be required to exactly reproduce the design.

Copyrighted tends to cover objects with high information content where collisions are improbable. Almost all written and printed works contain sufficient entropy that the probability of collision between two original works is near 0. The lyrics to the song "Happy Birthday" have exceptionally low information content as far as copyright goes and, arguably somewhere close to 50 bits, using 1.0 bits per letter and assuming that the repetitive structure reduces essentially to

"eval('happybirthdaytoyou'*4;s/toyou/dearname line3)"

This gives p(collision) as 2^-50. However, if each of the last 7 billion people to live uttered a mere 100,000 sentences of equivalent entropy over the course of their lifespan, you'd get approximately 50% chance of collision ((1-(1/2^50))^(7000000000*100000)=0.537). Although this estimate is off, due to the number of people on earth who actually speak English, most works are quite a bit longer than Happy Birthday and therefore will not collide in practice.

Incidentally this gives a heuristic for the expiration of copyright : The length of time a copyright is granted can be proportional to the raw number of novel bits contained within the work. Perhaps an article could remain copyrighted until you reach a 50% probability of another human producing the same work by random chance. This creates the unfortunate incentive to simply append a lot of unrelated works together. Perhaps copyrighted material could be sectioned into substrings which provide no mutual information about eachother, although this enters into a weirdness of information theory about which I am not equipped to reason.

The only sense I can make of the distinction is that design patents must be verified before production, and copyright violations are settled in expensive lawsuits after production. It makes economic sense to thoroughly check objects with a high probability of collision before mass production, but to be lazy with objects of a low probability of collision. Of course, for physical artifacts the number of effective bits is subjective and depends on how discerning the consumer is. Have you ever heard two people arguing for hours over the relative merits of two cars/shoes/operating-systems that appear functionally indistinguishable to you? In light of this, it almost makes more sense to let the author ( yes, 3D objects with utility are works of authorship ) decide on what risks to take : expensive, fixed cost patent search, or prohibitively expensive possible lawsuit ? I bet we can write a whole set of optimization equations to determine what is economically most favorable based on probability of collision, and the cost of resolving these collisions, but I need to go to sleep.

How does one tell if two files are "equal" up to the equivalence class of "copyright" ? Computer files can be encoded in a multitude of ways, producing file strings of seemingly different information content. The key is that a file string, in union with a decoding method, makes up the information artifact. The file string itself is useless without a consensus decoding scheme. Large prime numbers are not illegal, but a specific large prime together with the decoding scheme "this is a zipped c source for a DVD decryption algorithm" becomes a copyright violation on the DVD decoding algorithm. This is much the same way that possessing a sequence of numbers, eg 4-8-15-16-23-42 is not in itself illegall, unless you also have the information "This is the combination to the safe at the bank downtown". For a working definition, let us say that a tuple (file, decoder) are equivalent if they must, at some point, be translated into the same string of bits to become usable, or ultimately compute the same function. Note that the general problem of determining if two algorithms compute the same function is undecidable, but most specific instances can be evaluated in finite time.

This problem is complicated by the advent of lossy compression algorithms. If I compress an audio book into a low bit rate Ogg Vorbis format, I will loose most of the original file information, and can not reconstruct it. We might be able to get away with "equivalent information content as appropriate for the medium", which for books might reasonably be defined as the string of text representing the work. This case would also cover transcribing the audio book to text using speech recognition. However, this definition is still vague. For instance, the quality of the narration in an audio book is arguably part of the product and would be lost with heavy compression schemes.

We might be able to get away with another definition : a lossy compression of a copyrighted work is a violation if the tuple (file, decoder) could not, with high probability, have been arrived at without access to the original file in some form. This can be formalized by saying that the compressed file contains a very specific subset of the information content of the original, and effectively is a copyright violation on this subset of information, but not the whole file.

So, how many bits do you need for a copyright violation ? How many bits do you need for a patent violation on an algorithm ?

Another scenario, dealing with algorithm patents. An algorithm may be patented because it is exceptionally fast, or has some desirable properties. So, say Alice has patented a super good face recognition algorithm with excellent accuracy and time constant, and Bob then develops a wholly different algorithm with the same speed and accuracy. Alice's algorithm was phrased in terms of digital signal processing, and Bob's algorithm used a lot of group theory and graph data-structures. The patent office decides that these are clearly pretty different algorithms, and grants both patents. Later on, a mathematician comes along and says "Aha! by the following isomorphism Alice's and Bob's face algorithms are mathematically equivalent", and furthermore "All possible algorithms with this time constant are isomorphic to Alice's and Bob's algorithms", and even worse "No one can do better, theoretically, than Alice or Bob". So, not only are Alice's and Bob's inventions provably identical, but all future inventions attempting to solve the problem will also be provably identical. How do you resolve this ? Do you revoke the patent on whoever filed second ? This seems pretty clearly unfair. It could get even worse : Alice's and Bob's algorithms could be, theoretically, identical in speed and accuracy, but the equivalence between them could be uncomputable. Is this really, legally, a different situation from the case where the equivalence is computable(provable) ?