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Nauka Znanost
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Satellite View of the U.S. Wrapped in a Frozen Blanket: http://1.usa.gov/1qo6Lwg 
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Continued fractions

This animation by Lucas Vieira Barbosa illustrates the continued fraction representation of the golden ratio, φ (or phi), which is approximately equal to 1.618. The number φ is the larger root of the equation x^2–x–1=0, which means that φ = 1+(1/φ). The animation expresses this latter equation very well.

A continued fraction is an expression obtained by representing a number as the sum of its integer part and the reciprocal of another number, and then (recursively) writing this other number as a continued fraction. 

The continued fraction expansion of a number may be written in brief as a sequence of integers, which in the case of the golden ratio would be simply be [1;1,1,1,1,...]. More generally, the continued fraction [a0;a1,a2,a3,...] corresponding to the integer sequence (a0, a1, a2, a3,...) would correspond to the number a0+1/(a1+(1/(a2+1/(a3+...)))).

In a continued fraction, the first term a0 may be negative, but all the other terms will be nonnegative integers. The continued fraction of an irrational number will give an infinite sequence, whereas the continued fraction of a rational number will terminate after a finite number of steps. This representation of a number as an integer sequence will be unique for an irrational number, whereas a rational number will correspond to two such sequences. For example, the rational number 7/3 can be written either as [2;3] = 2 + 1/3 or as [2;2,1] = 2 + 1/(2 + 1/1).

The number e, which is approximately equal to 2.71828, is the base of the natural logarithm. It is an irrational number whose decimal expansion looks more or less random. Despite this, the continued fraction of e has a pattern to it: e = [2;1,2,1,1,4,1,1,6,1,1,8,...] (http://oeis.org/A003417). 

In contrast to this, neither the decimal expansion nor the continued fraction representation of the number π (approximately 3.14159) shows any obvious pattern: π = [3;7,15,1,292,1,1,1,2,1,3,1,...] (http://oeis.org/A001203). Truncating this sequence after two steps, we obtain [3;7] = 3 + 1/7, which is a familiar approximation to π. Truncating the sequence after the large entry of 292 will give a very accurate rational approximation to π, namely 103993/33102; this has the first nine decimal places correct.

I was reminded of continued fractions by my recent post about Srinivasa Ramanujan (1887-1920), who was extremely good at using them. There is a famous anecdote about Ramanujan being asked the answer to a brain-teaser type problem from a magazine. As soon as the problem was read out to him, Ramanujan said "Please take down the solution" and dictated a continued fraction. When asked how he solved the problem, Ramanujan replied:

Immediately I heard the problem, it was clear that the solution was obviously a continued fraction; I then thought, “Which continued fraction?” and the answer came to my mind. It was just as simple as this.

And in case you are wondering, this method of solving problems sounds baffling and intimidating to other professional mathematicians.

Relevant links
Wikipedia on continued fractions: http://en.wikipedia.org/wiki/Continued_fraction

Details of the anecdote on Ramanujan and continued fractions: http://anecdotesandallthat.blogspot.com/2012/01/mahalanobis-on-ramanujan.html

A post by me about Ramanujan and taxi cab number 1729: https://plus.google.com/101584889282878921052/posts/QtuFkDQnDQ6

A recent post by me about Ramanujan and pi: https://plus.google.com/101584889282878921052/posts/74oomcTuJoV

Lucas Vieira Barbosa's tumblr page, where this animation came from: http://1ucasvb.tumblr.com/about

Lucas contributes many mathematical diagrams and animations to Wikipedia. If you like his work, there is a way to make a donation to it on the above page, so that he can keep producing these wonderful graphics.

#mathematics #scienceeveryday
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A spiral in a furnace | Hubble Space Telescope
This new Hubble image is a snapshot of NGC 986—a barred spiral galaxy discovered in 1828 by James Dunlop. This close-up view of the galaxy was captured by Hubble’s Wide Field and Planetary Camera 2 (WFPC2).

NGC 986 is found in the constellation of Fornax (The Furnace), located in the southern sky. NGC 986 is a bright, 11th-magnitude galaxy sitting around 56 million light-years away, and its golden center and barred swirling arms are clearly visible in this image.

Barred spiral galaxies are spiral galaxies with a central bar-shaped structure composed of stars. NGC 986 has the characteristic S-shaped structure of this type of galactic morphology. Young blue stars can be seen dotted amongst the galaxy’s arms and the core of the galaxy is also aglow with star formation.

To the top right of this image the stars appear a little fuzzy. This is because a gap in the Hubble data was filled in with data from ground-based telescopes. Although the view we see in this filled in patch is accurate, the resolution of the stars is no match for Hubble’s clear depiction of the spiral galaxy.

Credit: ESA/Hubble & NASA

+Hubble Space Telescope 
+NASA Goddard 
+European Space Agency, ESA 
+Space Telescope Science Institute 

#NASA   #Space   #Astronomy   #Hubble  #Galaxy #Barred #Spiral #NGC986 #Fornax #Telescope #Cosmos #Universe #ESA
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“Impossible” by Regolo Bizzi

This picture, from the “Impossible” series by Regolo Bizzi, demonstrates that isometric graph paper is more useful than you might have thought for the purposes of creating apparently impossible geometric structures.

The picture is based on the Penrose triangle. This impossible shape, which is familiar from the art of M.C. Escher, is named after Roger Penrose and his father Lionel Penrose, who popularised it in the 1950s. However, it was first created by the Swedish artist Oscar Reutersvärd in 1934.

Regolo Bizzi lives in Tumba in Sweden, and there is much more of his art online. Some of it is similar in spirit to this picture, but some is very different.

Relevant links

Regolo Bizzi on deviantART, Flickr and Instagram:
http://odonodo.deviantart.com
http://www.flickr.com/photos/regolo
http://instagram.com/regolo54

Wikipedia on the Penrose triangle: http://en.wikipedia.org/wiki/Penrose_triangle

Wikipedia on Oscar Reutersvärd: http://en.wikipedia.org/wiki/Oscar_Reutersvärd

(Found via Ed Southall on Twitter.)

#art #mathematics  
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In 2014, bright planet Jupiter near radiant point of Leonid meteor shower.

Peak tonight: http://bit.ly/1xxyxrW
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