Digiting Solutions

quasi - state

Quasicrystals are structural forms that are ordered but not periodic. They form patterns that fill all the space though they lack translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only 2, 3, 4, and 6-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders, for instance 5-fold.

Aperiodic tilings were discovered by mathematicians in the early 1960s, but some twenty years later they were found to apply to the study of quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography. Quasicrystals had been investigated and observed earlier but until the 80s they were disregarded in favor of the prevailing views about the atomic structure of matter.

Roughly, an ordering is non-periodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original. The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled; i.e. the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two dimensions. The ability to diffract comes from the existence of an indefinitely large number of elements with a regular spacing, a property loosely described as long-range order. Experimentally the aperiodicity is revealed in the unusual symmetry of the diffraction pattern, that is, symmetry of orders other than 2, 3, 4, or 6. The first officially reported case of what came to be known as quasicrystals was made by Dan Shechtman and coworkers in 1984. (read more) (the basics)

Is God A Mathematician?

Albert Einstein once wondered: "How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?" Indeed, Newton formulated a mathematical law of gravity which he himself could verify (given the observational results of his day) to an accuracy of no better than four percent. Yet, the law proved to be precise to better than one part in a million! How is that possible? Or take the example of knot theory – the mathematical theory of knots. It evolved as an obscure branch of pure mathematics. Amazingly, this abstract endeavor suddenly found extensive modern applications in topics ranging from the structure of the DNA to "string theory" – the candidate for an ultimate theory of the subatomic world.

And this is not all. The famous logician Bertrand Russell argued that logic and mathematics are really the same thing. "They differ as boy and man", he said, "logic is the youth of mathematics and mathematics is the manhood of logic." So how can we explain these incredible powers of mathematics? How come that stock option pricing and the agitated motion of pollen suspended in water can be described by the same mathematical equation?

At an even more fundamental level, are we merely discovering mathematics, just as astronomers discover previously unknown galaxies? Or, is mathematics simply a human invention? These (and many more) are the questions that Mario Livio is attempting to answer in "Is God A Mathematician?" The book reviews the ideas of great thinkers from Plato and Archimedes to Galileo and Descartes, and on to Russell and Gödel. It offers a lively and original discussion of topics ranging from cosmology to the cognitive sciences, and from mathematics to religion. The focus on the scientific and practical applications of the fascinating insights of great minds will appeal to a wide audience. (read more)

10,000 Monkeys

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.

In this context, "almost surely" is a mathematical term with a precise meaning, and the "monkey" is not an actual monkey, but a metaphor for an abstract device that produces a random sequence of letters ad infinitum. The theorem illustrates the perils of reasoning about infinity by imagining a vast but finite number, and vice versa. The probability of a monkey exactly typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time of the order of the age of the universe is minuscule, but not zero. (read more)

Grigori Perelman

You can read the book: Perfect Rigor

I am reading the book right now. Since I am given the opportunity by Oberon to write in this site; here you have it.

My point is that Mathematical Science appears anywhere for no particular reason. At some point in Greece, a group of free thinkers started the Western Tradition of Mathematics and Philosophy. Why Greece? Put your favorite answer here. Mine is that a brotherhood developed between a group of like minded individuals there; maybe encouraged by an open mind about homosexual love. Now jump to middle XX^th Russia. In the most unlikely place, two men fall in love: Pavel Alexandrov, and Andrey Kolmogorov .

That is as unlikely as a butterfly wing flapping: producing one of the most successful Mathematics traditions to appear on this Earth since the classic Greek period.

I believe that the ways of the World are mysterious.