Showing posts with label Fibonacci sequence. Show all posts
Showing posts with label Fibonacci sequence. Show all posts

Wednesday, 22 June 2011

On This Day in Math - June 22


The mathematical education of the young physicist [Albert Einstein ] was not very solid, which I am in a good position to evaluate since he obtained it from me in Zurich some time ago.
~Hermann Minkowski

EVENTS
1633 Galileo, under threat of torture from the inquisition, was forced to "abjure, curse, and detest" his Copernican heliocentric views.
The recantation of GALILEO took place in the Great Hall of the former monastery of Santa Maria sopra Minerva, then the headquarters of the Dominican order. This is where he supposedly said "E pur si muove" (Nevertheless, it does move). For a long time, these words were believed to be a much later invention, but they probably date back to c1643 [Fahie, pp. 72 75]. Galileo was never officially imprisoned except for the few hours between his trial and the sentencing. In 1992, the Vatican officially declared that Galileo had been the victim of an error.

In 1675, the Royal Greenwich Observatory was created by Royal Warrant in England by Charles II. Building designed by Sir Christopher Wren (who was also a Professor of Astronomy) was commenced 10 Aug 1675 and finished the following year by John Flamsteed was appointed as the first Astronomer Royal. Its primary uses were in practical astronomy - navigation, timekeeping, determination of star positions. In 1767 the observatory began publishing The Nautical Almanac, which established the longitude of Greenwich as a baseline for time calculations. The almanac's popularity among navigators led in part to the adoption (1884) of the Greenwich meridian as the Earth's prime meridian (0° longitude) and the international time zones.*TIS

1799 France adopted the metric system of weights and measures. *VFR

1902 In response to a letter from Bertrand Russell dated 16 June 1902, Gottlob Frege responded with characteristic scientific honesty that “your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic.” [van Heijenoort, From Frege to G¨odel, 125–128] *VFR
Russell had found a class of contradictions to Frege's 1879 Begriffsschrift. This contradiction can be stated as "the class of all classes that do not contain themselves as elements".

1978 evidence of the first moon of Pluto was discovered by astronomer James W. Christy of the Naval Observatory in Flagstaff, Ariz. when he obtained a photograph of Pluto that showed the orb to be distinctly elongated.. Furthermore, the elongations appeared to change position with respect to the stars over time. After eliminating the possibility that the elongations were produced by plate defects and background stars, the only plausible explanation was that they were caused by a previously unknown moon orbiting Pluto at a distance of about 19,600 kilometers (12,100 miles) with a period of 6.4 days. The moon was named Charon, after the boatman in Greek mythology who took the souls of the dead across the River Styx to Pluto's underworld. *TIS

2004 Humans are officially slow learners... In 2004, a study led by Richard Doll was published in the British Medical Journal, the first research that quantified the damage over the lifetime of a generation, based on a 50-year study of a group of almost 35,000 British doctors who smoked. The study found that almost half of persistent cigarette smokers were killed by their habit, and a quarter died before age 70. Further, those who quit by age 30 had the same life expectancy as a nonsmoker. Even quitting at age 50 saved six more years of life over those who continued smoking. At age 80, 65% of non-smokers were still alive, but only 32% of smokers. Fifty years before, Doll published in the same journal the first report of a study that linked cigarette smoking to lung cancer*TIS

2011 One of the 15th century copies of a manuscript of Fibonacci's Liber Abacci that was owned by Boncompagni and was until recently in Brown University Maths library is for sale, by auction, on June 22, 2011, in New York and is estimated to fetch in excess of $120,000.

BIRTHS
1837 Paul Gustav Heinrich Bachmann born. He wrote (1892–1923) a five volume survey of the state of number theory including an evaluation of the various methods of proof. He also devoted time to composing, playing the piano, and serving as a music critic for various newspapers. *VFR
1860 Mario Pieri (22 June 1860 in Lucca, Italy - 1 March 1913 in S Andrea di Compito (near Lucca), Italy) Pieri's main area was projective geometry and he is an important member of the Italian School of Geometers. However, after he moved to Turin, Pieri became influenced by Peano at the University and Burali-Forti who was a colleague at the Military Academy. This influence led Pieri to study the foundations of geometry.
In 1895 he set up an axiomatic system for projective geometry with three undefined terms, namely points, lines and segments. He improved on results of Pasch and Peano and then, in 1905, Pieri gave the first axiomatic definition of complex projective geometry which does not build on real projective geometry.
In 1898 Pieri published the memoir The principles of the geometry of position through the Academy of Sciences of Turin. Russell was impressed by this memoir and wrote, in his Principia, "This is, in my opinion, the best work on the present subject." *SAU
1864 Herman Minkowski born. The motto on his Akademie-Schrift was “Rien n’est beau que le vrai, le vrai seul est aimable” (Nothing is beautiful but the truth, only the truth is lovable). *VFR He developed the geometrical theory of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity. By 1907, Minkowski realised that the work of Lorentz and Einstein could be best understood in a non-euclidean space. He considered space and time, which were formerly thought to be independent, to be coupled together in a four-dimensional "space-time continuum". Minkowski worked out a four-dimensional treatment of electrodynamics. His idea of a four-dimensional space (since known as "Minkowski space"), combining the three dimensions of physical space with that of time, laid the mathematical foundation of Albert Einstein's general theory of relativity.*TIS My favorite Minkowski story from Constance Reid's Hilbert, Once in a topology lecture he brought up the Four-color theorem. "This theorem has not been proved, but that is because only mathematicians of the third rank have occupied themselves with it" he announced with unusual arrogance. "I belive I can prove it." He began on the spot to work out the problem and continued over several classes to develop the work. After several weeks he entered one rainy day and a crash of thunder accompanied his entrance. Turning to his students he announced, "Heaven is angered by my arrogance, My proof is defective."
1866 Kazimierz Żorawski (June 22, 1866 – January 23, 1953) was a Polish mathematician. His work earned him an honored place in mathematics alongside such Polish mathematicians as Wojciech Brudzewski, Jan Brożek (Broscius), Nicolas Copernicus, Samuel Dickstein, Stefan Banach, Stefan Bergman, Marian Rejewski, Wacław Sierpiński, Stanisław Zaremba and Witold Hurewicz.[citation needed]
Żorawski's main interests were invariants of differential forms, integral invariants of Lie groups, differential geometry and fluid mechanics. His work in these disciplines was to prove important in other fields of mathematics and science, such as differential equations, geometry and physics (especially astrophysics and cosmology).*Wik

1880 Alfred Rosenblatt born. He worked in analysis and probability theory. *VFR

1906 Ott-Heinrich Keller was a German mathematician who worked on algebraic geometry and topology*SAU

1910 Konrad Zuse, inventor of the first fully functional programmable digital computer. *VFR

1920 James H. Pomerene American computer pioneer. In Apr 1946 he joined John von Neumann and Herman Goldstine in their newly organized Electronic Computer Project at the Institute for Advanced Study in Princeton, New Jersey. This project was to build a parallel stored-program computer. He designed the adder portion of the arithmetic unit and then was entirely responsible for the development and construction of the electrostatic (Williams tube) memory and became the chief engineer of the project 1951-56. Then he joined IBM to assist development of the HARVEST computer, a special system built for the National Security Agency. It had two levels of program control and also had a tape and tape library system that was fully automatic and of great capacity.*TIS

DEATHS

1389 Giovanni Dondi died. In 1381 he built one of the earliest geared equatoria driven by clockwork. There is a model of it in the Smithsonian. It has a heptagonal frame with a planet on each face. Dials show the time of sunrise, sunset, movable feasts, and the nodes of the moon’s orbit. *VFR

1450  Jamshid al-Kashi was an Islamic mathematician who published some important teaching works and anticipated Stevin's work on decimals.*SAU

1925 Felix Klein died. Curiously, this was the birthday of his dear friend Minkowski. *VFR German mathematician whose synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programm, profoundly influenced mathematical development. He created the Klein bottle, a one-sided closed surface. A Klein bottle cannot be constructed in Euclidean space. It is best pictured as a cylinder looped back through itself to join with its other end. However this is not a continuous surface in 3-space as the surface cannot go through itself without a discontinuity. It is possible to construct a Klein bottle in non-Euclidean space.*TIS

1936 Moritz Schlick, philosopher of science and leader of the Vienna Circle, was murdered by a deranged former student, on the steps of an academic building. *VFR

1977 Harold Calvin Marston Morse developed variational theory in the large with applications to equilibrium problems in mathematical physics, a theory which is now called Morse theory and forms a vital role in global analysis*SAU

1990 Ilya Mikhaylovich Frank Russian physicist who, with Tamm, theoretically explained the mechanism of Cherenkov radiation. In 1934, Cherenkov discovered that a peculiar blue light is emitted by charged particles traveling at very high speeds through water. Frank and Tamm provided the theoretical explanation of this effect, which occurs when the particles travel through an optically transparent medium at speeds greater than the speed of light in that medium. This discovery resulted in the development of new methods for detecting and measuring the velocity of high-speed particles and became of great importance for research in nuclear physics. For this, Frank received the Nobel Prize for Physics in 1958 (jointly with Pavel A. Cherenkov and Igor Y. Tamm).*TIS

Credits:
*VFR = V Frederick Rickey, USMA
*TIS= Today in Science History
*Wik = Wikipedia
*SAU=St Andrews Univ. Math History

Saturday, 15 January 2011

Math and Poetry

My beautiful wife Jeannie is a poet, so I like the idea that the connections between math and poetry goes back a long way. It might surprise you to know that :
1) The first use of binary numbers
2) The first recorded illustration of what we now call Pascal's Arithmetic Triangle
3) The Fibonacci sequence
All appeared first in a book on Sanskrit poetry around 2000 BC, Pingala's "Chandahsutra".

Poetry in Sanskrit, and it seems in some modern languages, poems are described by the pattern of long and short syllables. Very little is known about the author's life, and in fact it seems that anything that is stated in one ancient text is contradicted by another.
What we know about the work comes from an upgrade of the text created by a 10th century mathematician named Halayudha.

Describing short and long syllables seems a natural stimulus for binary notation. Instead of zero's and ones, Pingala used (or we suspect that he used) symbols for syllables which were Guru (heavy-given two beats) or Laghu (light-given one beat).

To illustrate that there could only be eight line patterns with three syllables he lists them: LLL, LLH, LHL, HLL, LHH, HLH, HHL, HHH.

I think there is still some doubt about whether Pingala actually made a diagram similar to the arithemtic triangle (which they called meruprastāra) or if that was added by Halayudha. In any event, it seems quite easy to see that for n= 2,3,4 etc syllables, and count the number of short syllables we get

1 1 short or long
1 2 1 short short, short long, long short, long long
1 3 3 1

and the triangle is born.

But if you decided to count the number of lines by length (remember heavy syllables last twice as long as light ones)..

There can only be one line of length one, it has a single light accent.
There is also only one line of length two, a single heavy accent.
But for length three you can get two different types, HL, LH, or LLL
And by now you expect that somehow there will be five patterns for length four. Sure enough LLLL, LLH, LHL, HLL, HH.... and we have the Fibonacci sequence.

It is easy to see that the number of patterns of length N, would be all the patterns of length n-2 followed by a heavy accent, plus all the patterns of length n-1 followed by a light accent. This is just the recursive definition of a Fibonacci sequence.

And in honor of the Woman I love, and the 4000 year association between the things we love, here is a poem she wrote years ago in Japan. It was inspired by a friend, Idell Tong, who was in charge of something planned around the school. When Jeannie asked her how it was going, she replied, "Well, you know my philosophy, If you have a spare minute, worry." That became this:


TIME WELL SPENT
The Walls of your world are crumbling.
Your wrinkled brow is beaded with sweat,
All the plans you made are disintegrating,
Got a minute? FRET!

When the boss's deadline is looming,
You know your rat is losing the race,
And your best is just not good enough.
Now's the time to PACE!

You're stuck in rush hour traffic,
Urging the taxi driver to hurry,
He gives you the glance of annoyance.
Just sit back, scowl, and WORRY!

The BIG event is scheduled,
Here you are with nothing to do,
All the presentations are perfect.
Don't rest on your laurels....STEW.

Friday, 15 May 2009

Who Was Fibonacci Before He Was Fibonacci


Vlorbik picked up on my "factoid interlude" in my last blog that no one called Fibonacci by that name in his lifetime. When he asked, "Who did?", I realized that thousands of students (and their teachers) probably do not know that name was added well after his death. So here is the story, as well as I know it, from the notes on my Math Words Etymology page.
--------------------------------------
According to Paul J. Nahin, author of AN IMAGINARY TALE, the name Fibonacci was not common until centuries after Leonardo's death, and during his lifetime he was called Bigollo, a slang term for a loafer, and my wife's favorite term for me, drawn from the word bighellone. Julio Gonzalez Cabillon has written, "The name 'Fibonacci' most probably originated with the historian of mathematics Guillaume Libri (1803-1869)."

In September of 2001, Heinz Lueneburg posted a note that seemed to suggest that there may have been earlier uses than Julio suggested. He writes (with some editing by me):

I was in Rome and checked the Boncompagni paper
I quoted in my posting of August 28. The paper starts with the diskussion of what is known of persons of the Bonacci family other than Leonardo: One Matteo Bonacci is known because he is mentioned as a witness of the treaty Pisa and Genova signed on February 13, 1188.
Then he lists the names of authors who use the name "Leonardo Pisano".
Then he lists the names of authors who use the name "Fibonacci".
John Leslie 1820
Cossali 1797-99
Giovanni Gabriello Grimaldi 1790-1792
Libri 1838-1841
Chasles 1837
Nicollet 1811-1818
S. Ersch & I. G. Gruber 1818 and subsequent years
August de Morgan 1847. He also uses Bonacci.

Then he lists the names of authors who explain "Fibinacci = filio Bonacci"
Flaminio dal Borgo 1765
Tiraboschi 1822-1828
Ranieri Tempesti 1787
Giovanni Andres 1808-1817
Grimaldi 1790-1792
Libri 1838-1841

Then his arguments for Fibonacci = de filiis Bonacci follow.
Then he discusses the sobriquet Bigollone, Bigollo, Bigoloso.
Finally, in the major part of the paper, he discusses the various manuscripts of the various works of Fibonacci still in existence.
The question "who gets the credit?" is still open.
Regards, Heinz Lueneburg

My personal pick is Flaminio dal Borgo 1765

By the way, a personal footnote, On the last day of my first visit to Pisa I missed a turn and came upon the Via Fibonacci quite by accident.

And just a followup, we DO know who first invented the term "Fibonacci Sequence". It was created by Edouard Anatole Lucas, who is also known for inventing the "Towers of Hanoi" puzzle.

Wednesday, 1 October 2008

Fibonacci's Thumb

One of those things that goes around the internet math groups every now and then, is going around again.... things to do with the Fibonacci sequence. I call them Fibonacci Rules of thumb... Ok, you remember Fibonacci, he introduced the Arabic numbers to the west with his famous book, The Liber Abaci.. and had a problem about rabbits where the number each month was given by 1, 1, 2, 3, 5, 8, 13, 21, etc....... YEAH, the one in the DaVinci Code!

The most common Fibonacci Rule of Thumb is converting between miles and kilometers. Suppose you are zipping along in Canada and the speed limit is in Km/hr, and you wonder..."Gosh, just how fast AM I driving?". So take your speedometer reading in good old miles per hour, and think of the number in the fibonacci sequence nearest this one... suppose you are driving a modest 55 mph.. which just happens to be a number in the sequence. Find the next number in the sequence, in this case 89, and that is a really good estimate of your speed in Km/hr (OK, I think it works out to be more like 88.5, but you get the idea).

So why does that work? The typical conversion is miles = 1.609344 kilometers (I know, I just googled it); and if you take any Fibonacci number, and divide it by the one in front, you get about that number. As the numbers get larger and larger, the ratios get closer and closer to 1.618033989.... so the ratios are pretty close. Of course you could use it for lots of other things, and with a little torture, you can make it work for less usual measures... If you want to know how many seconds (appx) in N years, where N is a Fibonacci number, then you can just take the next Fibonacci number, double it, and multiply by 107... So for example, If you wanted to find the number of seconds in 5 years... the next Fibonacci number is 8... double that to get 16... and now add the seven zeros to get 160,000,000 seconds in five years. You can check that one on your own.

And if you are STILL not convinced, would you be impressed if I told you that for N electrons (where N is a Fibonacci number), you can find the charge in Couloumbs by taking the next Fibonacci number, and multiply by 10-19... NOW THAT IS USEFUL STUFF