The following passage is the problem which the solution is the famous Fibonacci sequence. This is the English translated passage, and it is somewhat difficult to read because the way we view mathematics in modern times is through numerical relations and equations, but when this was originally written the nice convenient mathematical notation that we commonly use today was not widely accepted. It is kind of funny because in this work, Liber Abaci, Fibonacci was trying to present and persuade Europeans to use the Hindu Arabic number system that we commonly use today. Note: The following passage is difficult to understand completely. I have produced an easier to read summary of this problem. So please do not spend too much time on this passage. I want to show this passage to see how different modern mathematics is to medieval mathematics. If you do not care about the difference from medieval mathematics and modern mathematics, then I suggest that you skip this passage. “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. 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 month, 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; in this month 3 pairs are pregnant, and thus there are 5 pairs in the month; in this month 3 pairs are pregnant, and in the fourth month there are 8 pairs, of which 5 pairs bear another 5 pairs, these are added to the 8 pars making 13 pairs in the fifth month, but another 8 pairs are pregnant, and thus there are in the sixth month21 pairs; to these are added the 13 pairs that are born in the seventh month; there will be 34 pairs in this month; to this are added the 21 pairs that are born in the eight month; there will be 55 pairs in this month; to these are added the 34 pairs that are born in the ninth month; there will be 89 pairs in this month; to these are added again the 55pairs that are born in the tenth month; there will be 144 pairs in this 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 above written 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 add 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 above written sum of rabbits, namely 377, and thus you can in order find it for an unending number o months.” (1, p. 404-5) Now I will present my interpretation of this problem. If you begin with one pair of rabbits, and after one month each pair of rabbits produces another pair of rabbits, and a new born pair of rabbits is mature after one month. How many pairs of rabbits will you have after one year? Month Adult Pairs Newborn Pairs Total For the Month 1 1 1 2 2 2 1 3 3 3 2 5 4 5 3 8 5 8 5 13 6 13 8 21 7 21 13 34 8 34 21 55 9 55 34 89 10 89 55 144 11 144 89 233 12 233 144 377
This table is the solution to the problem in a notation that is far easier to read. Keep in mind at this time that in Europe numbers were written in roman numerals, so a huge portion of mathematics was presented in a similar fashion as this problem. The Fibonacci sequence is (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …). So what does this silly solution involving rabbits have to do with anything else? This question is difficult to answer because this “silly” solution pops up in almost every branch of science. Before I attempt to answer anything further I need to introduce a number that has many names, but I will refer to this number as the “Golden Ratio.” The golden ratio is a number that has very strange and unusual properties. The golden ratio is commonly known as the Greek letter φ, pronounced “phi.”   This is both the algebraic and numerical representations of the golden ratio. The 3 dots represent the fact that the number continues infinitely. The Fibonacci sequence: The Series, the Application, the Web Page φ, the golden ratio is a never ending, never repeating number. This number has even more bizarre properties of multiplication of itself. What I mean is that when you multiply φ by itself, you obtain the result, φ ×φ = φ2 = (1.6180339887…) × (1.6180339887…) = 2.6180339887… = φ + 1. So φn (φ is multiplied by itself n-times)= n.6180339887… = φ + n-1, (think of n being any counting number that is greater than 0). (If you do not believe me, then try it, but DO NOT use the 1.6180339887…, because after the second time the values mess up. Instead use the 1 + square root of 5, divided by 2 term.) In numbers this means: 1/ φ = φ-1 = 0.6180339887…= 1.6180339887... - 1 wikipedia.org  F(n) is the n number of the Fibonacci Sequence. wikipedia.org A quick way to summarize this method is as follows. Lets say that you have 1.1 by the floor function 1.1 = 1, or if you have 5.8 by the floor function 5.8 = 5. You do this with the F(n) equation. Now the question is “How does any of this relate to geometry?” There are many answers to this question and I do not have time to show very many answers, but I will show a couple. The first case according to Mario Livio, is called “Squaring” Rectangles. The “Squaring” Rectangles case states, “If you sum up an odd number of products of successive Fibonacci numbers …, then the sum is equal to the square of the last Fibonacci number used in the products” (2, p. 103). Sum up five products, (1 x 1) + (1 x 2) + (2 x 3) + (3 x 5) + (5 x 8) =1 + 2 + 6 + 15 + 40 = 64 =82 8 is the fifth Fibonacci number so this case holds. Now it is time to interpret this in a geometric way. Area, A, for rectangles and squares is just length multiplied by width. A = (Length) x (Width) This figure is based off of figure 30 of (2, P. 104). A very interesting geometric object that has a crucial relationship with the golden ratio and the Fibonacci sequence is the logarithmic spiral. The exact workings of this spiral are not important, but what is important is to see its relationship between the golden ration and the Fibonacci sequence. The easiest way of thinking of a logarithmic spiral is by looking at an example of the chambered nautilus. "For example, as the mollusk inside the shell of the chambered nautilus grows in size, it constructs larger and larger chambers, sealing off the smaller unused ones. Each increment in the length of the shell is accompanied by a proportional increase in its radius, so that the shape remains unchanged" (2, P. 116-7).  Answers.com  This is the logarithmic spiral. Notice how similar this is with the chambered nautilus. Answers.com Now it is time to link this spiral with the Fibonacci sequence. Looking back at the “Squaring” Rectangles, If you take all of the rectangles and make the ratio of the bigger side to the smaller side to be the golden ratio, then you get this relationship.  KeplersDiscovery.com As some of you may have guessed, this logarithmic spiral shows up all of the time in nature. In fact it is very difficult to find an area of science that the spiral does not appear in, or to find a branch of science were the Fibonacci sequence is absent. I will post a couple of pictures that have these spirals within.  A low pressure area over Icelandshows an approximately logarithmic spiral pattern Answers.com  Answers.com Interpretation The relation between the Fibonacci sequence and the golden ratio is yields some of the most interesting results. One of the things that interested me about this subject was that a simple problem involving the mating of rabbits was able to describe the shape of some galaxies. This relation can describe various problems involving rabbits, sunflowers, daisies, galaxies, and storms. I do not know very much about this subject. What I have just shown was just a very limited amount of relations that the Fibonacci sequence produces. One of the other points I wanted to get across is that every time you hear about something involved with the golden ratio, that the Fibonacci sequence is related to that something also. Sources
1.) Pisano, Leonardo, Liber Abaci, A translation Into Modern English of Leonardo Pisano's Book of Calculation. Translated by L.E. Sigler. New York : Springer, c2002. Notes: The Liber Abaci is important because it presented the problem that produced the famous Fibonacci Sequence. If it were not for this source then this project would not be relevant. 2.) Livio, Mario, The Golden Ratio: The Story of Phi The World’s Most Astonishing Number. New York : Broadway, c2003. Notes: Mario Livio is the head of the Science Division at the Space Telescope Science Institute. He obviously has knowledge in the area of math, physics and astronomy. In the book he describes the golden ratio in context to all three areas. |
