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Transforming the Fibonacci sequence into a matrix equation:
1. All my math hubs have something about God in their final paragraph. Well, in this hub it is the last four(4) paragraphs.
2. There has been some really good articles already written about Fibonacci numbers( FN) and the Fibonacci sequence( FS) right here at Hubpages; therefore, I'll link you to some of those now:
3. Flysky wrote, the Fascinating World of Fibonacci Numbers. He gives interesting history of Fibonacci numbers, and he has included a good video showing how Fibonacci numbers permeate all of God's creation. Someone in the comments section also left an informative link
4. Scramble wrote, Fibonacci Numbers & Golden Ratio in Nature. He addresses several ways the FNs show up in Nature, and he has a math limit that approaches the golden ratio.
5. Pemekwulu( link to his profile) is probably the FN guru of hubpages. He has written at least 7 very interesting hubs on them, and the hub, Table of Contents for a Course in FNs, indicates he has written a book about the FNs.
6. Dzymslizzy wrote a very lucid hub titled, The FS: Math for the Non-Math Brain. She promised to explain it in a way anyone would understand, and that promise was fulfilled.
WHAT IS THE FIBONACCI SEQUENCE?
7. The Fibonacci Sequence( FS) is a recurrence relation( sometimes called a difference equation). It is a sequence( an ordered list of numbers) in which each number( or term) is found by applying a formula to the previous two terms. This formula is simple; we simply add the previous two terms to determine the term of interest. If we want to know the value of the 9th term( defined at L6) of L1 we simply add 21 + 34 = 55( the 9th term). The FS is not an arithmetic sequence because an arithmetic sequence is defined as having a "common difference," which the FS does not have; therefore, the formula--- an = a + (n -- 1)d--- that is used to find the nth term of an arithmetic sequence cannot be used to find the nth term of the FS.
THE DIFFICULTY OF FINDING THE NTH TERM OF THE FIBONACCI SEQUENCE:
8. Because the FS is a recurrence relation defined by the recurrence equation of L2, it makes the job of finding the nth term very difficult because you have to determine all the previous terms in order to find it. So if you want to know the 300th term you must first find the previous 299 terms so that L2 can be used: X300 = X299 + X298. By converting L2 into a matrix equation we can find the 300th term, 1000th term, or whatever term.
LET 'S GET STARTED:
9. L1 is the Fibonacci sequence; L2 is the formula that generates it, and L3,4, and L5 gives examples of how L2 is used. L6 defines the terms beginning with the 0th term.
10. We must transform L2 into a matrix, but before we can get L2 converted to a matrix, we have to get it to look like a matrix. You cannot call something a duck until you can get it to look like a duck and quack like a duck, so also before L2 can be called a matrix, it must look like a matrix and quack( behave) like a matrix.
11. This is done at L7. We define Yn as given at the beginning of L7, and then rewrite L2 in the form of the last 2 terms of L7. L8 just gives the details of what we did and why we could do it. As hub#12.1( Solving 4 Simultaneous Equations . . . ) explains, those 3 dots that form a triangle at L9 means "therefore". At L10 we put together L9 and L7 to form 2 simultaneous equations, and we add the zero there so it will look like a matrix. This is similar to what we did to the first equation of L1 of Hub#12.1. There was no X4 there, so we gave it a coefficient of zero to make sure no X4, an oxygen atom, was going to be there, but it is still part of that matrix; therefore, element, a1,4 has a value of zero. Hub#12.1 goes into further detail on simultaneous equations and their matrix representation. Anyway back to this hub. These are all variables at L10, and it is to be understood that the variables on the right side of those 2 equations each have coefficients of 1, except element( explained at hub#21.1) a2,2 ,which has zero as a coefficient. We do to the 2 simultaneous equations of L10 what we did to the 4 simultaneous equations of Hub#12.1( more detail given there), we convert them into a matrix at L11. T.elia has a very thorough hub on multiplying matrices( link to her hub), and that is what is done at L11. It is the matrix way of describing simultaneous equations. Notice if you take the matrix brackets away, and also the two matrices in the middle that are being multiplied, then L11 would look identical to L10.
OUR MATRIX EQUATION:
12. L12 defines the matrices in L11 and L13. Matrix A at L12 was constructed from L10 and it represents the coefficients of those variables in L10. For example, 1 times Xn-1 is Xn-1 for element a1,1 and one times zero is zero for element a2,2 . Elements are explained in hub#12.1. Notice that matrix A is constructed so that as matrix multiplication is performed with matrix Fn-1 we get the same equations as we have at L10. Hub#12.1 explains matrix multiplication, and I have links there that explain it further. L13 is concise notation for L11 and both of them represents our matrix equation.
HOW THIS MATRIX EQUATION REPRESENTS THE FS:
13. From here to L24 is nothing but matrix multiplication. T.elia in her very thorough hub, Multiplying Matrices, does one of the best jobs I've seen in showing us how to do this. I know I'm being redundant but if you do not know how to do this, then check out this link and then come back to finish this hub. At L14,15,16 and 17 we use our matrix equation of L11 or L13 to find various terms( defined at L6) of the FS. We progressively keep multiplying to find the 0th, 1st, 2nd, 3rd, 4th terms from L14 to L17, but then you will see a remarkable pattern. Patterns in math are often used to solve complicated concepts and problems.
A REMARKABLE REVELATION!!!
14. What comes next is--in esoteric mathematical terms--really, really cool! We begin to repeat at L18,19, and 20 what we did at L14,15,16, and 17 but we are going to decompose L14.15 and 16 into their constituent parts at L18,19 and 20. You will notice that F1 equals AF0 at L14; therefore, we plug this value of F1 into L18 and we wind up with A2 times F0. L19 is the matrices written out so you can get a closer look at what is going on. L20 takes it one step further with F3 . I'm sure you see the pattern. L21 and 22 does the actual matrix multiplication as explained by T.elia. L23 raises matrix A to the 7th power, and 21 is the 7th term of the FS, 13 is the 6th term and 8 is the 5th term.
THE 4,OOOTH TERM OF THE FS:
15. I raised matrix A to the 4,000th power with my Ti-89 calculator and found element a1,1 , which is the 4,000th term, to be 6.4575 X 10835. This matrix also gave me the 3,999th term as 3.9909 X 10835 , and the 3,998th term is 2.4665 X 10835 . This remarkable calculator will give me the exact number but I did not want to type out 836 numbers.
BUT WHAT ABOUT THE 1,000,000TH TERM?
16. My Ti-89 calculator is about 6 or 7 years old. Maybe the newer ones will handle this request, but mine will get buried with such a request. What will we do?!?!?! WHAT WILL WE DO? The fate of the universe depends on us finding the millionth term. We just can't give up! We just can't! "Failure is not an option."
WE WILL NOT GIVE UP; WE MUST SAVE THE UNIVERSE!
17. We must find the millionth term, and the billionth term, and the trillionth term! We can not let the universe implode because we can't determine the answers to these questions! "FAILURE IS NOT AN OPTION."
THE EIGENVALUES OF THE FIBONACCI SEQUENCE:
18. It would probably take us 317,100 years to find the trillionth term with just adding the terms on a calculator. I'm sure high end calculators and computers can do it, but in part 2( The eigenvalues of the Fibonacci Sequence) of this hub we begin to determine a simple formula that will give us the nth term, and with it we can determine the trillionth term. I say "begin" because in part 2 we find the eigenvalues of the FS, and in subsequent parts we will go through the arduous arithmetic using the eigenstuff( eigenvalues, eigenfunction and eigenvectors) to determine that simple formula. This will be like boot camp--good training--for us because you may remember that in hub#12.2(Finding the Determinant and Inverse . . . ) we want to raise that 4 X 4 matrix to the 300th power, a rational power, a complex power, and find its 7th root. To do that we need to determine its eigenstuff( this word is not in the dictionary--it is my invention). Finding the eigenvalues of our simple 2 X 2 matrix in this hub will be good practice in preparation to determine the eigenvalues of the much more complicated DNA matrix of hub#12.2.
HOW DID GOD DESIGN THE FIBONACCI SEQUENCE INTO HIS CREATION?
19. Scramble( link above and here) has a picture near the end of his hub demonstrating how the length of the bones of a finger follow the FS. Although the average chemical bond lengths in our bones is about 240pm( picometers; a picometer is 10-12 meters), I'm going to assume it is 140pm because I think that is a closer average for all the chemical bond lengths in our bodies. I will be referring to this average bond length in subsequent hubs. Since bond lengths are "in general, the sum of the radii of the atoms and or ions constituting the bond" then we have a good idea of an atom's size involved in the bond. Two radii of identical atoms tells us the diameter of a single atom, and in this case it is 140pm on average. This is small ! To give you an idea of how small, if we make the atom the size of a 6 foot tall man, then the man in comparison( keeping the proportion the same) would be nearly 15 million miles tall( 1.4845 X 107 miles). From the atom's perspective, the construction project of making those 4 bones associated with the finger presented in Scramble's hub is enormous. If the atoms were our size then it would be like us taking on a project in which the bone at the tip of the finger would have a length of 275.1 LYs( light years or a distance of 1.616 X 1015 miles), the 2nd bone is 412.6 LYs, the 3rd is 687.7 LYs, and the 4th is 1,100.3 light years long. At least that would fix the unemployment problem( date of this hub is March of 2012). If we assume the bond length for each atom is 140pm( as I said it is actually more for bone than other structures of the body) then the length of each bone in atoms is 108.857million for the tip, then 163.286M for the 2nd bone, 272.143M for the 3rd, and 435.429M for the 4th bone associated with the finger. It would take an entire encyclopedia to expound on the complexity God, through Jesus Christ, designed into the electromagnetic force(emf) within each atom to construct the bone geometry and to follow the FS. For millions of bond lengths the emf constructs the bones in a somewhat( although very specific) linear fashion but then "decides" to end each bone according to the Golden Ratio( ratio of Fibonacci numbers) and prepare the geometry of each end so it can act as a joint to be used to do what God designed the fingers to do. The morphology and physiology of bones is unfathomably complex---as is everything else of our bodies from individual cells to our brains---and yet the emf must prepare these bones for ligaments( connects bones), tendons( connects muscles to bones), arteries( supplies oxygenated blood to bones), veins( carries deoxygenated blood from the bones), capillaries( forms a network between arterioles and venules), lymphatic vessels, etc, etc, etc. The emf can do this because God, through Jesus Christ, has designed and taught( Job 38;36) the emf to execute these constructions.
GOD, THE CREATOR BY JESUS CHRIST( JOHN 1:3; 1COR.8:6; EPH.3:9; COL.1:16,17) OF THE FS AND THE GOLDEN RATIO:
20. I didn't directly answer the question that headed the previous paragraph, did I? I don't know how He did it nor do I know why He did it. As to why, I suspect it provides the best efficiency and maximization with the mathematical boundaries God structured into His creation. Boundaries are a big thing in mathematics, so it shouldn't be surprising that they are important in God's creation, and mathematics is the language we use to gain greater understanding of God's creation. The universe and all of its constituents are finite, although that is a tough thing to argue when one encounters quasars, black holes and many other things; nevertheless, finite, by definition, has boundaries. The efficiency of the FS must operate within those boundaries. In the hub, DNA, a Witness for God, a good argument is made for the enormous informational density stored in DNA; however, I also mentioned that the actual informational density is much greater than what was described. So, the answer to that question is, in my opinion, God designed the Fibonacci sequence into His creation via the electromagnetic force( emf). It is the emf that constructs a DNA molecule, and it is the emf that structures the FS into God's creation. But how? How does that happen?
GOD STRETCHES HEAVEN BY HIS UNDERSTANDING:
21. Big deal, right? It doesn't take any understanding to stretch a rubber band; it just takes brute force. For all we know it may take enormous force to stretch space; nevertheless, it should not take any understanding . . . unless . . . well, unless God has a purpose to stretch space in a specific way. Then we have entered into the realm of understanding. But why? Why would God want the stretch space in a specific way. Well first of all it pops up all the time in our formulas--the dimensional part of our formulas, and several hubs could be written about that alone. So here is my take on that, but I'm just going to mention it, and expound upon it later in subsequent hubs. Jeremiah 51:15 says, " He has made the earth by His power, He has established the world by His wisdom, and has stretched out the heaven by His understanding." Then Job 38:36 says, " Who has put wisdom in the inward parts? or Who has given understanding to the heart?" Since the emf stores information within the geometric structure of DNA, and since it has to have access to that information in order to do that--and the millions of other things it does--I believe it is receiving information from the structure of space itself. I think God is storing information within the structure of space and He does this by stretching it in very specific ways.
THE MOST SIMPLEMINDED CONJECTURE MANKIND HAS EVER UPCHUCKED:
22. One final thought since we are on the subject of God's creation. Darwinian Evolutionary Theory is the most simpleminded thinking ever vomited upon an easily duped society. I think that is why God invented negative numbers. One would have to have zero thought, zero reasoning ability, and zero amount of logical and analytical ability to believe God had nothing to do with creating the universe, but to explain it away with the asinine conjectures spewed from evolutionists . . . well, now we are in the realm of negative numbers, and negative brain power. A moron who believes in Jesus Christ is smarter than an erudite who does not believe--much smarter! That moron will go to heaven for eternity, but the erudite, with clever, albeit erroneous, reasoning pitted against God will be in torment forever.