Monday, February 23, 2009

Nikolai Ivanovich Lobachevsky reporting on 1826, February 23...

1826, February 23 (that's February 11 in "old-style") - Nikolai Ivanovich Lobachevsky reported on a geometry in which the fifth postulate was not true to the session of the department of physics and mathematics at the University of Kazan.

From wikipedia, under Nikolai Lobachevsky's work:
Lobachevsky's main achievement is the development (independently from János Bolyai) of a non-Euclidean geometry, also referred to as Lobachevskian geometry. Before him, mathematicians were trying to deduce Euclid's fifth postulate from other axioms. Euclid's fifth is a rule in Euclidean geometry which states (in John Playfair's reformulation) that for any given line and point not on the line, there is one parallel line through the point not intersecting the line. Lobachevsky would instead develop a geometry in which the fifth postulate was not true. This idea was first reported on February 23 (Feb. 11, O.S.), 1826 to the session of the department of physics and mathematics, and this research was printed in the UMA (Вестник Казанского университета) in 1829–1830. Lobachevsky wrote a paper about it called A concise outline of the foundations of geometry that was published by the Kazan Messenger but was rejected when the St. Petersburg Academy of Sciences submitted it for publication.

The non-Euclidean geometry that Lobachevsky developed is referred to as hyperbolic geometry. Lobachevsky replaced Euclid's parallel postulate with the one stating that there is more than one line that can be extended through any given point parallel to another line of which that point is not part; a famous consequence is that the sum of angles in a triangle must be less than 180 degrees. Non-Euclidean geometry is now in common use in many areas of Mathematics and Physics, such as general relativity; and hyperbolic geometry is now often referred to as "Lobachevskian geometry" or "Bolyai-Lobachevskian geometry".

Some mathematicians and historians, have wrongfully claimed that Lobachevsky stole his concept of non-Euclidean geometry from Gauss, which is untrue - Gauss himself appreciated Lobachevsky's published works very highly, but they never had personal correspondence between them prior to the publication. In fact out of the three people that can be credited with discovery of hyperbolic geometry - Gauss, Lobachevsky and Bolyai, Lobachevsky rightfully deserves having his name attached to it, since Gauss never published his ideas and out of the latter two Lobachevsky was the first who duly presented his views to the world mathematical community.

Lobachevsky's magnum opus Geometriya was completed in 1823, but was not published in its exact original form until 1909, long after he had died. Lobachevsky was also the author of New Foundations of Geometry (1835-1838). He also wrote Geometrical Investigations on the Theory of Parallels (1840) and Pangeometry (1855).

Another of Lobachevsky's achievements was developing a method for the approximation of the roots of algebraic equations. This method is now known as Dandelin-Gräffe method, named after two other mathematicians who discovered it independently. In Russia, it is called the Lobachevsky method. Lobachevsky gave the definition of a function as a correspondence between two sets of real numbers (Dirichlet gave the same definition independently soon after Lobachevsky).

Carl Friedrich Gauss (30 April 1777 – 23 February 1855)

Johann Carl Friedrich Gauss - "Princeps mathematicorum", died in Göttingen on February 23, 1855.

Friday, February 20, 2009

Fibonacci modulo m

The general problem of the periods of the Fibonacci sequence modulo m is definitely non-trivial (with the case m = p - prime - playing a very important role). A quick reference can be found here (The Fibonacci Sequence Modulo M). Also see the article (in pdf) on "The Fibonacci sequence modulo p^2...".

An example that teachers use relatively often as a middle-school problem: "Find the period of the sequence of the last digits of the Fibonacci numbers"! That will correspond to the modulus m=10, the answer is 60, and the elements of the period are
0,1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,1,7,8,5,3,8,1,9,0,
9,9,8,7,5,2,7,9,6,5,1,6,7,3,0,3,3,6,9,5,4,9,3,2 ,5,7,2,9,1
In the previous blog entry, the modulus m is 2011 (that is the 305-th prime) and the period of the Fibonacci sequence modulo m is 2010.

Wednesday, February 18, 2009

Distribution of the quadratic residues and nonresidues of the Fibonacci numbers modulo a prime p


I was just curious to visualize a plot of the cumulative sum of the sequence of quadratic symbols of the Fibonacci numbers modulo p (p - prime) within a period of that sequence. Here it is, for p=2011. These graphs stimulate the students' interest in deeper mathematical topics. They may be seen as useful educational tools.

Wednesday, February 4, 2009

Octacube

OCTACUBE: a sculpture designed by Dr. Adrian Ocneanu, Professor of Mathematics at Penn State and built by the machinists in the Penn State Engineering Shop. Jill Grashof Anderson (PSU '65, Mathematics) sponsored the sculpture, dedicated to the memory of her husband, Kermit Anderson (PSU '65 Mathematics), killed in the World Trade Center terrorist attack on 11 September 2001. The octacube encodes a rich variety of structures arising in advanced areas of Mathematics and Physics (Quantum Field Theory). For details, see Adrian Ocneanu's commentary and the Penn State octacube page.

Tuesday, February 3, 2009

Professor Solzhenitsyn!

In 1941, Alexandr Solzhenitsyn (1918-2008) graduated from Rostov University (currently Southern Federal University) - where he studied Mathematics and Physics. In this picture he is shown teaching mathematics to his children. For more details, see the "Solzhenitsyn mathematician" note from the "Math in the Media" magazine.