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=test=

$$a^b$$ $$a^b a^b$$ $$a^b = a^b$$ $$a^b \to a^b$$ $$a^b \rightarrow a^b$$ $$a^b c^d$$ $$a^b = c^d$$ $$a^b \to c^d$$ $$a^b \rightarrow c^d$$

=Mathematics good article nominations= My nominations:

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=van Eck's sequence= In recreational mathematics van Eck's sequence is an integer sequence that starts with a0=0. For n>=0, if there exists an m < n such that am = an, take the largest such m and set an+1 = n-m; otherwise an+1 = 0. Thus the first occurrence of an integer in the sequence is followed by a 0, and the second and subsequence occurrences are followed by the size of the gap between the most recent two occurrences.

The first few terms of the sequence are:


 * 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5 ...

The sequence was named by Neil Sloane after Jan Ritsema van Eck, who contributed it to the On-Line Encyclopedia of Integer Sequences in 2010.

Properties
It is know that the sequence contains infinitely many zeros and that it is unbounded.

It is conjectured, but not proved, that the sequence contains every positive integer, and that every pair of non-negative integers apart from (1,1) appears as consecutive terms in the sequence.

Penrose tiling: Construction by L-system
The rhombus (P3) Penrose tiling can be drawn using the following L-system:



Here 1 means "draw forward", + means "turn left by angle", and − means "turn right by angle" (see turtle graphics). The [ means save the present position and direction to restore them when corresponding ] is executed. The symbols 6, 7, 8 and 9 do not correspond to any action; they are there only to produce the correct curve evolution.

Georgy Fedoseevich Voronoy
Born: 28 April 1868 in Zhuravka, Poltava guberniya, Russia (now Ukraine) Died: 20 Nov 1908 in Warsaw, Poland

Georgy Voronoy studied at the gymnasium in Priluki graduating in 1885. He then entered the University of St Petersburg, joining the Faculty of Physics and Mathematics.

After graduating from St Petersburg in 1889, Voronoy decided to remain there and work for his teaching qualification. He was awarded a Master's Degree in 1894 for a dissertation on the algebraic integers associated with the roots of an irreducible cubic equation.

Voronoy lectured at Warsaw University, being appointed professor of pure mathematics there. He wrote his doctoral thesis on algorithms for continued fractions which he submitted to the University of St Petersburg. In fact both Voronoy's master's thesis and his doctoral thesis were of such high quality that they were awarded the Bunyakovsky prize by the St Petersburg Academy of Sciences.

Later Voronoy worked on the theory of numbers, in particular he worked on algebraic numbers and the geometry of numbers. He extended work by Zolotarev and his work was the starting point for Vinogradov's investigations. His methods were also used by Hardy and Littlewood.

In 1904 Voronoy attended the Third International Congress of Mathematicians at Heidelberg. There he met Minkowski and they discovered that they were each working on similar topics.

The concept of Voronoi diagrams first appeared in works of Descartes as early as 1644. Descartes used Voronoi-like diagrams to show the disposition of matter in the solar system and its environs.

The first man who studied the Voronoi diagram as a concept was a German mathematician G. L. Dirichlet. He studied the two- and three dimensional case and that is why this concept is also known as Dirichlet tessellation. However it is much better known as a Voronoi diagram because another German mathematician M. G. Voronoi in 1908 studied the concept and defined it for a more general n-dimensional case.

Very soon after it was defined by Voronoi it was developed independently in other areas like meteorology and crystalography. Thiessen developed it in meteorology in 1911 as an aid to computing more accurate estimates of regional rainfall averages. In the field of crystalography German researchers dominated and Niggli in 1927 introduced the term Wirkungsbereich (area of influence) as a reference to a Voronoi diagram.

http://www.comp.lancs.ac.uk/~kristof/research/notes/voronoi/voronoi.pdf