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Nauka Znanost
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Nauka Znanost

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Coxeter–Conway friezes

The grid of numbers in the picture is an example of a Coxeter–Conway frieze. The frieze takes the form of an infinitely wide checkerboard of positive integers, satisfying the following two conditions. 

1. There are finitely many rows, and all of the entries in the top and the bottom row must be equal to 1.

2. If a, b, c and d are four numbers surrounding a particular empty square to the square's left, top, bottom and right respectively, then the numbers must satisfy the unimodular law, meaning that ad–bc = 1.

Coxeter–Conway friezes are named after H.S.M. Coxeter (1907–2003) and John H. Conway (1937–), who studied them in the 1970s. Conway and Coxeter found a remarkable connection between their friezes and triangulations of convex polygons, the details of which are as follows.

A triangulation of a convex n-gon is a way to divide the n-gon into non-overlapping triangles in such a way that the corners of each triangle are three of the corners of the original n-gon. For example, the picture shows a triangulation of a heptagon (or 7-gon) which, inevitably, consists of 7–2 = 5 triangles. 

The width of a Coxeter–Conway frieze is the number of rows in the frieze excluding the top and bottom rows of 1s; for example, the frieze in the picture has width 4. Conway and Coxeter proved that, for a fixed n, there is a one to one correspondence between Coxeter–Conway friezes of width n–3 on the one hand, and triangulations of a convex n-gon on the other hand. 

To see how this works, let us look at the seven corners of the heptagon, working clockwise from the lowest corner shown. The number of triangles meeting at each of the seven corners is 2, 2, 1, 4, 2, 1, 3. These are exactly the entries of the second to top row of the frieze! The same idea can be used to match every frieze of width 4 with a triangulation of a heptagon, and vice versa. It follows from this result that the entries in any given row of a frieze of width n–3 must form a periodic cycle of length 7, and indeed this is the case for the frieze in the picture.

The friezes and their corresponding triangulations are closely related to triangulations of n-gons in the Farey graph; this is closely related to the Farey–Ford tessellation, which I posted about a few weeks ago (https://plus.google.com/101584889282878921052/posts/GrZP3ajnRGU). Friezes also have connections to other important areas of modern mathematical research, including the cluster algebras of Fomin and Zelevinsky.

Relevant links
Wikipedia entry on John H. Conway: http://en.wikipedia.org/wiki/John_Horton_Conway

Wikipedia entry on H.S.M. Coxeter: http://en.wikipedia.org/wiki/Coxeter

An arXiv paper by Sophie Morier-Genoud, Valentin Ovsienko and Serge Tabachnikov, which gives more details on some of the topics mentioned above: http://arxiv.org/abs/1402.5536

(Note: Usually, names of mathematicians are listed in alphabetical order, but for some reason, Coxeter is usually listed before Conway in the context of these friezes.)

#mathematics #sciencesunday #spnetwork arXiv:1402.5536
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Nauka Znanost

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Mersenne primes and perfect numbers

A Mersenne prime is a prime number of the form 2^n – 1, and a perfect number is a number N whose factors (other than N itself) add up to N. For example, 2^3 – 1 = 7 is a Mersenne prime, and 28 is a perfect number, because its factors add up to 28: 1 + 2 + 4 + 7 + 14 = 28.

In order for a number of the form 2^n – 1 to be prime, the number n itself must be prime. The reason for this is that if k is a factor of n, it can be shown that 2^k – 1 is a factor of 2^n – 1. For example, since 15 = 3x5, it follows that 2^{15} – 1 = 32767 cannot be prime, because it has 2^3 – 1 = 7 as a factor (as well as 2^5 – 1 = 31).

On the other hand, if p is a prime number, it may or may not be true that 2^p – 1 is prime. For example, if p = 7, then 2^7 – 1 = 127 is a Mersenne prime, but if p = 11, then 2^{11} – 1 = 2047 = 23x89 is not a prime number.

It turns out that Mersenne primes and perfect numbers are closely related to each other. For example, if p = 3, the Mersenne prime M = 7 = 2^3 – 1 corresponds to a perfect number N, in this case 28. The perfect number N can be obtained from the original prime number p using the formula N = 2^{p–1}x(2^p–1), or from the Mersenne prime M using the formula N = M(M+1)/2. 

The method above of producing perfect numbers from Mersenne primes was known to Euclid, who discussed it in volume 7 of his 13 volume treatise, the Elements, around 300 BC. However, it was not until the 18th century that Leonhard Euler proved that every even perfect number can be obtained from a Mersenne prime in this way.

The first four perfect numbers, 6, 28, 496 and 8128, which correspond to the Mersenne primes 3, 7, 31 and 127, were known to early Greek mathematics. The fourth perfect number, 33550336, which corresponds to the Mersenne prime 8191 = 2^{13} – 1, was discovered in the 15th century. 

Currently, 48 Mersenne primes are known, the largest of which is 2^{57885161} – 1. This number, which is also the largest known prime number, is 17425170 digits long. As a consequence of this, there are 48 known even perfect numbers, the largest of which is 34850340 digits long.

It is not known whether there are infinitely many Mersenne primes, or whether there are infinitely many perfect numbers. More fundamentally, it is not known whether any odd perfect numbers exist, although it is known that if an odd perfect number does exist, it must have at least 1500 digits. The mathematician J.J. Sylvester wrote in 1888 that
... a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number] — its escape, so to say, from the complex web of conditions which hem it in on all sides — would be little short of a miracle.

Relevant links
Mersenne primes on the On-Line Encyclopedia of Integer Sequences (https://oeis.org/A000668) and Wikipedia (http://en.wikipedia.org/wiki/Mersenne_prime)

Perfect numbers on the On-Line Encyclopedia of Integer Sequences (https://oeis.org/A000396) and Wikipedia (http://en.wikipedia.org/wiki/Perfect_number)

The Great Internet Mersenne Prime Search (http://www.mersenne.org/)

Wikipedia entry on J.J. Sylvester (http://en.wikipedia.org/wiki/James_Joseph_Sylvester)

A post by me about J.J. Sylvester (https://plus.google.com/101584889282878921052/posts/PJG9awS4MD5)

+James Grime of Numberphile on perfect numbers: 8128 and Perfect Numbers - Numberphile

Last, but not least, some stand-up comedy from UK comedian Dave Gorman about perfect numbers: Maths - Dave Gorman Stand Up

#mathematics #sciencesunday
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Astronauti se iz svemira vraćaju zaobljenijeg srca - Kad se astronauti nalaze u bestežinskom stanju gdje nema gravitacije, njihovi mišići ne moraju raditi toliko snažno kao i na Zemlji. Tijekom http://ow.ly/2FvVSS
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Kad se astronauti nalaze u bestežinskom stanju gdje nema gravitacije, njihovi mišići ne moraju raditi toliko snažno kao i na Zemlji. Tijekom dugih misija,...
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Map presenting the Revolutions of 1848 in Europe, another time in history when capitalism trumped law, equality, the environment, and almost any sense of human safety and welfare; the same year a stunning political manuscript was published by two guys -- Karl Marx and Friedrich Engels.
http://goo.gl/AFGuwT
http://www.marxists.org/archive/marx/works/1848/communist-manifesto/ch01.htm
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Believe it or not, Santa doesn't actually own the North Pole! And several countries want that sweet, sweet, icy land: http://dne.ws/18xGfSA
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A Lindenmeyer System is a grammar with simple recursive rules, which produce strings of symbols that can be interpreted as graphics commands in programs such as Python or Turtle to draw a line, turn by a certain angle, etc. The result of various grammars and ways to interpret them can be quite beautiful fractal images. +Andreas Wilhelm, a German Applied Computer Science BSc. who blogs about Computer Science and Math, has been making art using L-Systems (en.wikipedia.org/wiki/L-system). His blog, avedo.net/554/lindenmeyer-systems-a-python-adventure/ describes his work with plenty of interesting examples. Thanks to +Appropouture for finding the animation I posted here, which comes from verbalairways.tumblr.com.
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Rare sarcophagus and Egyptian scarab found in Israel

Israeli archaeologists have unearthed a rare sarcophagus featuring a slender face and a scarab ring inscribed with the name of an Egyptian pharaoh, Israel's Antiquities Authority said Wednesday.

More: http://www.usatoday.com/story/news/world/2014/04/09/sarcophagus-egyptian-scarab-found-in-israel/7508587/

Photo: The sarcophagus found at Tel Shadud, an archaeological mound in the Jezreel Valley. (Photo: Israel Antiquities Authority via AP)
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"A volcanic island in the Pacific Ocean has merged with its neighbour to form one landmass, the US space agency says.
The merged island lies some 1,000km (621mi) south of Tokyo, the result of eruptions on the seafloor that have spewed enough material to rise above the water line.
In November 2013, a new island sprouted near to Nishino-shima, another volcanic landmass that last expanded in the 70s.
Four months later, the new and old islands are one island.
The newer portion of the island - which was referred to as Niijima - is now larger than the original Nishino-shima landmass.
The merged island is slightly more than 1km (3,280ft) across".
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