Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature
Our modern word "algorithm" does not just apply to the rules of arithmetic but means any precise set of instructions for performing a computation whether this be
] a method followed by humans, for example:
(.) a cooking recipe;
(.) a knitting pattern;
(.) travel instructions;
(.) a car manual page for example, on how to remove the gear-box;
(.) a medical procedure such as removing your appendix;
(.) a calculation by human computors : two examples are:
(*) William Shanks who computed the value of pi to 707 decimal places by hand last century over about 20 years up to 1873 - but he was wrong at the 526-th place when it was checked by desk calculators in 1944!
(*) Earlier Johann Dase had computed pi correctly to 205 decimal places in 1844 when aged 20 but this was done completely in his head just writing the number down after working on it for two months!!
] or mechanically by machines (such as placing chips and components at correct places on a circuit board to go inside your TV)
] or automatically by electronic computers which store the instructions as well as data to work on.
Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature
** The Fibonacci Numbers **
In Fibonacci's Liber Abaci book, chapter 12, he introduces the following problem (here in Sigler's translation - see below): How Many Pairs of Rabbits Are Created by One Pair in One Year
A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. He then goes on to solve and explain the solution: Because the above written pair in the first month bore, you will double it; there will be two pairs in one month.
One of these, namely the first, bears in the second montth, and thus there are in the second month 3 pairs;
of these in one month two are pregnant and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month;
...
there will be 144 pairs in this [the tenth] month;
to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.
To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the abovewritten pair in the mentioned place at the end of the one year. You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months.
beginning 1 first 2 second 3 third 5 fourth 8 fifth 13 sixth 21 seventh 34 eighth 55 ninth 89 tenth 144 eleventh 233 end 377
Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature
Did Fibonacci invent this Series?
Fibonacci says his book Liber Abaci (the first edition was dated 1202) that he had studied the "nine Indian figures" and their arithmetic as used in various countries around the Mediterranean and wrote about them to make their use more commonly understood in his native Italy. So he probably merely included the "rabbit problem" from one of his contacts and did not invent either the problem or the series of numbers which now bear his name.
D E Knuth adds the following in his monumental work The Art of Computer Programming: Volume 1: Fundamental Algorithms errata to second edition:
Before Fibonacci wrote his work, the sequence F(n) had already been discussed by Indian scholars, who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes. The number of such rhythms having n beats altogether is F(n+1); therefore both Gospala (before 1135) and Hemachandra (c. 1150) mentioned the numbers 1, 2, 3, 5, 8, 13, 21, ... explicitly.
Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature
Naming the Series
It was the French mathematician Edouard Lucas (1842-1891) who gave the name Fibonacci numbers to this series and found many other important applications as well as having the series of numbers that are closely related to the Fibonacci numbers - the Lucas Numbers: 2, 1, 3, 4, 7, 11, 18, 29, 47, ... named after him.
Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature
Fibonacci's Mathematical Books
Leonardo of Pisa wrote 5 mathematical works, 4 as books and one preserved as a letter:
Fibonacci's Liber Abaci translated by L E Sigler, Springer Verlag (2002), 672 pages
available for the first time in English in 2002 celebrating it's 800th anniversary, as a translation with notes of Fibonacci's Liber Abaci (The Book of Calculating) from 1202 but revised in 1228.
One of the problems in this book was the problem about the rabbits in a field which introduced the series 1, 2, 3, 5, 8, ... . It was much later (around 1870) that E Lucas named this series of numbers after Fibonacci.
The Book of Squares
his largest book: an annotated translation into English of Leonardo Fibonaci's 1225 AD version of Liber quadratorum by L E Sigler, 1987, Academic Press, 124 pages.
Starting with a brief biography of Fibonacci, this is an interesting and ingenious book on all sorts of questions about expressing a number as the sum of two, three of four square numbers (or squared fractions).
If we can express a square number also as the sum of two other square numbers then Pythagoras' Theorem tells us that we have three sides of a right-angled triangle and this is Fibonacci's first Proposition. It seems that he was familiar with Euclid's Elements which also contains (Book 10, Proposition 29), Lemma 1) the same method of constructing all sets of three numbers that are the sides of a right-angled triangle. even though Fibonacci does not use the algebraic notation we do today, it is marvellously clear in its desriptions of the processes and algorithms and Sigler's notes show the algebraic notation to explain Fibonacci's process as we would write them today.
just thought a historical and mathematical basis would help!
