Yup, but there are reasons for it. He had an open bite before, which
you do not want to reproduce. How does it look with the closed or
minimally opened lip?

Joel

Thanks for the advice. I tried counting from 40 to 77 and it seems OK. I was
having a hard time talking yesterday. And I still can't get my Zs right.
The dentist shortened them up a bit since my last post. Not quite to the
blue line in the picture but it seems to have helped some. I'm not sure if
they need to be shortened any more, maybe a little. Here is a new picture
link with them shortened up a bit: http://tinyurl.com/68k47
I will certainly ask him to use temporary cement for the finals. Thanks!

Rich

That's what the dentist said. I might be able to live with them until
Wednesday!

8 had rct and needed a crown, 9 needed rct, all front teeth except a couple
on bottom where almost totally composite material, three of them were build
ups, some crowns, LOTS of fillings, lots of pain, and so on. In my view I
just thought my front teeth were shot. Plus they were transparent on the
bottom, you could literally see through some places and most all of my teeth
were grayish colored. The "before" picture didn't have the flash on the
camera so it was hard to see all that. And I like these temps, I can eat
anything now and it doesn't hurt! Thanks.

Rich

I agree. When the patient is unhappy,
I give him Sr. Fibonacci's phone number!

Joel

**

Leonardo Pisano Fibonacci

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Born: 1170 in (probably) Pisa (now in Italy)
Died: 1250 in (possibly) Pisa (now in Italy)

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Leonardo Pisano is better known by his nickname Fibonacci. He was the
son of Guilielmo and a member of the Bonacci family. Fibonacci himself
sometimes used the name Bigollo, which may mean good-for-nothing or a
traveller. As stated in [1]:-

Did his countrymen wish to express by this epithet their disdain for a
man who concerned himself with questions of no practical value, or
does the word in the Tuscan dialect mean a much-travelled man, which
he was?

Fibonacci was born in Italy but was educated in North Africa where his
father, Guilielmo, held a diplomatic post. His father's job was to
represent the merchants of the Republic of Pisa who were trading in
Bugia, later called Bougie and now called Bejaia. Bejaia is a
Mediterranean port in northeastern Algeria. The town lies at the mouth
of the Wadi Soummam near Mount Gouraya and Cape Carbon. Fibonacci was
taught mathematics in Bugia and travelled widely with his father and
recognised the enormous advantages of the mathematical systems used in
the countries they visited. Fibonacci writes in his famous book Liber
abaci (1202):-

When my father, who had been appointed by his country as public notary
in the customs at Bugia acting for the Pisan merchants going there,
was in charge, he summoned me to him while I was still a child, and
having an eye to usefulness and future convenience, desired me to stay
there and receive instruction in the school of accounting. There, when
I had been introduced to the art of the Indians' nine symbols through
remarkable teaching, knowledge of the art very soon pleased me above
all else and I came to understand it, for whatever was studied by the
art in Egypt, Syria, Greece, Sicily and Provence, in all its various
forms.

Fibonacci ended his travels around the year 1200 and at that time he
returned to Pisa. There he wrote a number of important texts which
played an important role in reviving ancient mathematical skills and
he made significant contributions of his own. Fibonacci lived in the
days before printing, so his books were hand written and the only way
to have a copy of one of his books was to have another hand-written
copy made. Of his books we still have copies of Liber abaci (1202),
Practica geometriae (1220), Flos (1225), and Liber quadratorum. Given
that relatively few hand-made copies would ever have been produced, we
are fortunate to have access to his writing in these works. However,
we know that he wrote some other texts which, unfortunately, are lost.
His book on commercial arithmetic Di minor guisa is lost as is his
commentary on Book X of Euclid's Elements which contained a numerical
treatment of irrational numbers which Euclid had approached from a
geometric point of view.

One might have thought that at a time when Europe was little
interested in scholarship, Fibonacci would have been largely ignored.
This, however, is not so and widespread interest in his work
undoubtedly contributed strongly to his importance. Fibonacci was a
contemporary of Jordanus but he was a far more sophisticated
mathematician and his achievements were clearly recognised, although
it was the practical applications rather than the abstract theorems
that made him famous to his contemporaries.

The Holy Roman emperor was Frederick II. He had been crowned king of
Germany in 1212 and then crowned Holy Roman emperor by the Pope in St
Peter's Church in Rome in November 1220. Frederick II supported Pisa
in its conflicts with Genoa at sea and with Lucca and Florence on
land, and he spent the years up to 1227 consolidating his power in
Italy. State control was introduced on trade and manufacture, and
civil servants to oversee this monopoly were trained at the University
of Naples which Frederick founded for this purpose in 1224.

Frederick became aware of Fibonacci's work through the scholars at his
court who had corresponded with Fibonacci since his return to Pisa
around 1200. These scholars included Michael Scotus who was the court
astrologer, Theodorus Physicus the court philosopher and Dominicus
Hispanus who suggested to Frederick that he meet Fibonacci when
Frederick's court met in Pisa around 1225.

Johannes of Palermo, another member of Frederick II's court, presented
a number of problems as challenges to the great mathematician
Fibonacci. Three of these problems were solved by Fibonacci and he
gives solutions in Flos which he sent to Frederick II. We give some
details of one of these problems below.

After 1228 there is only one known document which refers to Fibonacci.
This is a decree made by the Republic of Pisa in 1240 in which a
salary is awarded to:-

... the serious and learned Master Leonardo Bigollo ....

This salary was given to Fibonacci in recognition for the services
that he had given to the city, advising on matters of accounting and
teaching the citizens.

Liber abaci, published in 1202 after Fibonacci's return to Italy, was
dedicated to Scotus. The book was based on the arithmetic and algebra
that Fibonacci had accumulated during his travels. The book, which
went on to be widely copied and imitated, introduced the Hindu-Arabic
place-valued decimal system and the use of Arabic numerals into
Europe. Indeed, although mainly a book about the use of Arab numerals,
which became known as algorism, simultaneous linear equations are also
studied in this work. Certainly many of the problems that Fibonacci
considers in Liber abaci were similar to those appearing in Arab
sources.

The second section of Liber abaci contains a large collection of
problems aimed at merchants. They relate to the price of goods, how to
calculate profit on transactions, how to convert between the various
currencies in use in Mediterranean countries, and problems which had
originated in China.

A problem in the third section of Liber abaci led to the introduction
of the Fibonacci numbers and the Fibonacci sequence for which
Fibonacci is best remembered today:-

A certain man put a pair of rabbits in a place surrounded on all sides
by a wall. How many pairs of rabbits can be produced from that pair in
a year if it is supposed that every month each pair begets a new pair
which from the second month on becomes productive?

The resulting sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
(Fibonacci omitted the first term in Liber abaci). This sequence, in
which each number is the sum of the two preceding numbers, has proved
extremely fruitful and appears in many different areas of mathematics
and science. The Fibonacci Quarterly is a modern journal devoted to
studying mathematics related to this sequence.

Many other problems are given in this third section, including these
types, and many many more:

A spider climbs so many feet up a wall each day and slips back a fixed
number each night, how many days does it take him to climb the wall.
A hound whose speed increases arithmetically chases a hare whose speed
also increases arithmetically, how far do they travel before the hound
catches the hare.
Calculate the amount of money two people have after a certain amount
changes hands and the proportional increase and decrease are given.

There are also problems involving perfect numbers, problems involving
the Chinese remainder theorem and problems involving summing
arithmetic and geometric series.

Fibonacci treats numbers such as 10 in the fourth section, both with
rational approximations and with geometric constructions.

A second edition of Liber abaci was produced by Fibonacci in 1228 with
a preface, typical of so many second editions of books, stating that:-

... new material has been added [to the book] from which superfluous
had been removed...

Another of Fibonacci's books is Practica geometriae written in 1220
which is dedicated to Dominicus Hispanus whom we mentioned above. It
contains a large collection of geometry problems arranged into eight
chapters with theorems based on Euclid's Elements and Euclid's On
Divisions. In addition to geometrical theorems with precise proofs,
the book includes practical information for surveyors, including a
chapter on how to calculate the height of tall objects using similar
triangles. The final chapter presents what Fibonacci called
geometrical subtleties [1]:-

Among those included is the calculation of the sides of the pentagon
and the decagon from the diameter of circumscribed and inscribed
circles; the inverse calculation is also given, as well as that of the
sides from the surfaces. ... to complete the section on equilateral
triangles, a rectangle and a square are inscribed in such a triangle
and their sides are algebraically calculated ...

In Flos Fibonacci gives an accurate approximation to a root of 10x +
2x2 + x3 = 20, one of the problems that he was challenged to solve by
Johannes of Palermo. This problem was not made up by Johannes of
Palermo, rather he took it from Omar Khayyam's algebra book where it
is solved by means of the intersection of a circle and a hyperbola.
Fibonacci proves that the root of the equation is neither an integer
nor a fraction, nor the square root of a fraction. He then continues:-

And because it was not possible to solve this equation in any other of
the above ways, I worked to reduce the solution to an approximation.

Without explaining his methods, Fibonacci then gives the approximate
solution in sexagesimal notation as 1.22.7.42.33.4.40 (this is written
to base 60, so it is 1 + 22/60 + 7/602 + 42/603 + ...). This converts
to the decimal 1.3688081075 which is correct to nine decimal places, a
remarkable achievement.

Liber quadratorum, written in 1225, is Fibonacci's most impressive
piece of work, although not the work for which he is most famous. The
book's name means the book of squares and it is a number theory book
which, among other things, examines methods to find Pythogorean
triples. Fibonacci first notes that square numbers can be constructed
as sums of odd numbers, essentially describing an inductive
construction using the formula n2 + (2n+1) = (n+1)2. Fibonacci
writes:-

I thought about the origin of all square numbers and discovered that
they arose from the regular ascent of odd numbers. For unity is a
square and from it is produced the first square, namely 1; adding 3 to
this makes the second square, namely 4, whose root is 2; if to this
sum is added a third odd number, namely 5, the third square will be
produced, namely 9, whose root is 3; and so the sequence and series of
square numbers always rise through the regular addition of odd
numbers.

To construct the Pythogorean triples, Fibonacci proceeds as follows:-

Thus when I wish to find two square numbers whose addition produces a
square number, I take any odd square number as one of the two square
numbers and I find the other square number by the addition of all the
odd numbers from unity up to but excluding the odd square number. For
example, I take 9 as one of the two squares mentioned; the remaining
square will be obtained by the addition of all the odd numbers below
9, namely 1, 3, 5, 7, whose sum is 16, a square number, which when
added to 9 gives 25, a square number.

Fibonacci also proves many interesting number theory results such as:

there is no x, y such that x2 + y2 and x2 - y2 are both squares.

and x4 - y4 cannot be a square.

He defined the concept of a congruum, a number of the form ab(a + b)(a
- b), if a + b is even, and 4 times this if a + b is odd. Fibonacci
proved that a congruum must be divisible by 24 and he also showed that
for x, c such that x2 + c and x2 - c are both squares, then c is a
congruum. He also proved that a square cannot be a congruum.

As stated in [2]:-

... the Liber quadratorum alone ranks Fibonacci as the major
contributor to number theory between Diophantus and the 17th-century
French mathematician Pierre de Fermat.

Fibonacci's influence was more limited than one might have hoped and
apart from his role in spreading the use of the Hindu-Arabic numerals
and his rabbit problem, Fibonacci's contribution to mathematics has
been largely overlooked. As explained in [1]:-

Direct influence was exerted only by those portions of the "Liber
abaci" and of the "Practica" that served to introduce Indian-Arabic
numerals and methods and contributed to the mastering of the problems
of daily life. Here Fibonacci became the teacher of the masters of
computation and of the surveyors, as one learns from the "Summa" of
Luca Pacioli ... Fibonacci was also the teacher of the "Cossists", who
took their name from the word 'causa' which was first used in the West
by Fibonacci in place of 'res' or 'radix'. His alphabetic designation
for the general number or coefficient was first improved by Viète ...

Fibonacci's work in number theory was almost wholly ignored and
virtually unknown during the Middle ages. Three hundred years later we
find the same results appearing in the work of Maurolico.

The portrait above is from a modern engraving and is believed to not
be based on authentic sources.

Article by: J J O'Connor and E F Robertson

Click on this link to see a list of the Glossary entries for this page

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List of References (21 books/articles)
A Quotation


A Poster of Fibonacci
Mathematicians born in the same country


Cross-references to History Topics
Mathematical games and recreations

A chronology of pi

An overview of the history of mathematics

The trigonometric functions

Prime numbers

A history of Zero

Arabic numerals

The Golden ratio