Vox: Emmy Noether revolutionized mathematics - and still faced sexism all her life

http://www.vox.com/2015/3/23/8274777/emmy-noether

RAMANUJAN: Letters from an Indian Clerk

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Mysterious WWII encrypted message found in UK

UK spies unable to crack coded message from WWII carrier pigeon (CNN)

Eadem mutata resurgo

Commemorating Jacob Bernoulli...

Eadem mutata resurgo
"Though changed I shall rise the same"

Inscribed on Jacob Bernoulli's tombstone (he died on August 16, 1705 in Basel), this motto refers to the logarithmic (equiangular) spiral (N.B. the spiral that was actually imprinted on the tombstone is not equiangular). Through this self-similar object Jacob Bernoulli symbolically points to the ‘fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self ’ - as quoted in Mario Livio's book "The Golden Ratio..." (via here).

Jacob Bernoulli at the Mathematics Genealogy Project.

The Whirlpool Galaxy...

Johann Bernoulli (1667 - 1748) anniversary

Johann Bernoulli (1667 - 1748) was born on July 27, 7^3 years ago, in Basel, Switzerland. He was Euler's "mathematical parent".

And here is the... "sophomore's dream" - neat identities due to Johann Bernoulli (1697):

Lothar Collatz anniversary

Lothar Collatz, who proposed (in 1937) the celebrated "3n+1 problem", was born 100 years ago in Arnsberg, Germany.

Mac Tutor Biography

Lothar Collatz at MGP

Élie Cartan (1869-1951)

Élie Cartan had significant contributions in areas such as Lie theory, differential geometry, exterior differential forms, the theory of spinors (introduced by him in 1913), etc. Cartan died on May 6, 1951.

Dieudonné places Cartan right after Poincaré and Hilbert when it comes to the lasting influence in shaping modern mathematics. He was a speaker at the 1924, 1932 and 1936 International Mathematical Congresses. He lectured in Romania in 1931. The letters that he exchanged with Albert Einstein, Gheorghe Ţiţeica, Alexandru Pantazi and Gheorghe Vrânceanu, have been published (as mentioned in M. A. Akivis and B Rosenfeld - Élie Cartan (1869-1951), Providence R.I., 1993).

Paul Montel (1876-1975)

Paul Antoine Aristide Montel, Honorary Member of the Romanian Academy, advisor of Henri Cartan, Jean Dieudonné, Miron Nicolescu, Tiberiu Popoviciu and others (see Montel's entry at MGP), was born on April 29, 1876...

Max Planck on consciousness

"I regard consciousness as fundamental. I regard matter as derivative from consciousness. We cannot get behind consciousness. Everything that we talk about, everything that we regard as existing, postulates consciousness." (Max Planck - born on April 23, 1858)

link to quotation source
link to top picture source/credits (grave of Max Planck in Göttingen)
link to  bottom picture source/credits (NGC7090)
Mac Tutor Biography
Mathematics Genealogy Project - Max Planck
Planck units

Ph.D. mathematician and NFL champion

In some sense, the stunning 2010 Super Bowl XLIV victory of New Orleans Saints led by Drew Brees against the Peyton Manning's Indianapolis Colts (my favorite team) may be analogue to a similar event that happened in 1964. Then the NFL quarterback Frank Ryan led the Cleveland Browns to the 1964 NFL Championship title in a 27-0 victory against Johnny Unitas' Baltimore Colts. To this one might add the impressive 1966 season in which the Cleveland Browns' legend Ryan threw for 2974 yards and scored 29 touchdowns.

What is especially relevant for this particular blog is that Frank Ryan is also the recipient of a Ph.D. in Mathematics awarded by Rice University in 1965, with a most interesting thesis, "A Characterization of the Set of Asymptotic Values of a Function Holomorphic in the Unit Disc", and that among the references cited in the thesis are Luzin's "Leçons sur les ensembles analytiques et leurs applications", Sierpinski's "General Topology" (University of Toronto Press, 1952), and Stoilow's "Les propriétés topologiques des fonctions analytiques d'une variable", Ann. Inst. H. Poincaré, 2 (1932), 233–266. In 1966 Frank Ryan also published two fundamental papers on the set of asymptotic values of a function holomorphic in the unit disc in Duke Mathematical Journal (he also published in Pacific Journal of Mathematics, Mathematische Zeitschrift, Michigan Mathematical Journal, etc).

I will conclude with mentioning a recent mathematical event - the amazing, super-entertaining after-dinner talk "Resolved, that a Football is a Mathematical Object" delivered by Frank Ryan at the 2007 Ohio MAA Meeting held at Wittenberg (a talk which I will never forget).

Fibonacci numbers and the extreme and mean ratio - some history

Ruth Tatlow's article The Use and Abuse of Fibonacci Numbers and the Golden Section in Musicology Today (Understanding Bach, 1, 69-85, 2006) besides being very interesting in itself as a documented criticism of the "Golden numberism [that] has thoroughly infected musicology", incidentally provides useful references for those interested in the history of the representation of the EMR ("extreme and mean ratio" or "golden section") as the limit of the sequence of quotients F_n+1/F_n of consecutive Fibonacci numbers [1]. One may wonder who noticed this first? Leonardo Pisano (c. 1170 – c. 1250, also known as Fibonacci)? Not even close!

Evidence that this fact was noticed as early as the beginning of the 16th century was discovered by Leonard Curchin and Roger Herz-Fischler (handwritten annotation in the 1509 Luca Pacioli's edition of Elements [2]). When it comes to published work which associates the EMR with the limit of the sequence of quotients of consecutive Fibonacci numbers, Johannes Kepler wrote [3] the following, in 1611, about "this proportion that the geometers of today call divine":

It is impossible to provide a perfect example in round numbers. However, the further we advance from the number one, the more perfect the example becomes. Let the smallest numbers be 1 and 1... Add them, and the sum will be 2; add to this the greater of the 1s, result 3; add 2 to this, and get 5; add 3, get 8; 5 to 8, 13; 8 to 13, 21. As 5 is to 8, so 8 is to 13 approximately, and as 8 to 13, so 13 is to 21, approximately.

However, as communicated in 1995 by Peter Schreiber [4], in a rare book by the German reckoning master Simon Jacob (d. 1564) one can find a remark that the sequence of quotients of consecutive Fibonacci numbers approaches the EMR.

NOTES

[1] The term "extreme and mean ratio" goes back to Euclid's "Elements". Luca Pacioli introduced the "divine proportion" term in 1509, while the term "golden section" was introduced in 1835 by Martin Ohm. The exact value for the EMR is (1+sqrt(5))/2. The phrase "Fibonacci sequence" was coined by Edouard Lucas (1842-1891).
[2] Leonard Curchin and Roger Herz-Fischler, "De quand date le premier rapprochement entre la suite de Fibonacci et la division en extrême et moyenne raison?" (French) ["When were the first parallels drawn between the Fibonacci sequence and the golden section?"], Centaurus 28 (1985), no. 2, 129-138.
[3] Kepler, Johannes. Vom sechseckigen Schnee. (German) [On hexagonal snowflakes]. Translated from the Latin and with an introduction and notes by Dorothea Goetz. Ostwald's Classics of the Exact Sciences, 273. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987.
[4] Peter Schreiber, "A Supplement to J. Shallit’s Paper "Origins of the analysis of the Euclidean algorithm"", Historia Mathematica, Vol. 22, Issue 4, 422-424 (1995). http://poncelet.math.nthu.edu.tw/disk5/js/history/1033a.pdf

Bernhard Riemann

Georg Friedrich Bernhard Riemann (September 17, 1826 – July 20, 1866) - one of the greatest mathematicians of all time.

• Full Mac Tutor Biography
• Wikipedia entry
• The Mathematical Papers
• Mathematics Genealogy
• "On the Number of Primes Less Than a Given Magnitude" (pdf)

Dedekind has a deeply touching account of Riemann's final moments, his passing away being marked by the words of Lord's prayer. Riemann's tombstone (see here and here) in Biganzolo (Italy) refers to Romans 8:28 ("And we know that all things work together for good to them that love God, to them who are the called according to his purpose"):

Here rests in God
Georg Friedrich Bernhard Riemann
Professor in Göttingen
born in Breselenz, September 17th, 1826
died in Selasca, Juli 20th, 1866
---
Those, who love God, all things
must serve to its best manner.