The Quaterniontonic and Octoniontonic Fibonacci Cassini’s Identity: An Historical Investigation with the Maple’s Help

This paper discusses a proposal for exploration and verification of numerical and algebraic behavior correspondingly to Generalized Fibonacci model. Thus, it develops a special attention to the class of Fibonacci quaternions and Fibonacci octonions and with this assumption, the work indicates an investigative and epistemological route, with assistance of software CAS Maple. The advantage of its use can be seen from the algebraic calculation of some Fibonacci’s identities that showed unworkable without the technological resource. Moreover, through an appreciation of some mathematical definitions and recent theorems, we can understand the current evolutionary content of mathematical formulations discussed over this writing. On the other hand, the work does not ignore some historical elements which contributed to the discovery of quaternions by the mathematician William Rowan Hamilton (1805 – 1865). Finally, with the exploration of some simple software’s commands allows the verification and, above all, the comparison of the numerical datas with the theorems formally addressed in some academic articles.


INTRODUCTION
Undoubtedly, the role of the Fibonacci's sequence is usually discussed by most of mathematics history books. Despite its presentation in a form of mathematical problem, concerning the birth of rabbits' pairs, still occurs a powerful mathematical model that became the object of research, especially with the French mathematician François Édouard Anatole Lucas . From his work, a profusion of mathematical properties became known in the pure mathematical research, specially, from the sixties and the seventies.
With the emergence of the periodical The Fibonacci Quarterly, we register the force of the Fibonacci's model, with respect to their various ways of generalization and specialization. Thus, we can indicate the works of Brother (1965), Brousseau (1971), Horadam (1963;1967). From these works, besides the well-known the second order recurrence formula +2 = +1 + , ≥ 0, we also derive the following identity = (−1) +1 = (−1) −1 , for any integer 'n'.
Moreover, other studies found other ways for the process of generalization of the Fibonacci's model. Some of them employ methods of Linear Algebra (King, 1968;Waddill & Sacks, 1967), while others explore the theory of polynomial functions (Bicknell-Johnson & Spears, 1996;Tauber, 1968). In some articles, experts are interested in the extent of the fibonacci function in other numerical fields, like real numbers and complex numbers, relatively to its set of subscripts (Reiter, 1993;Scott, 1968).
From some trends work around the model, we express our interest in involving the process of Fibonacci's complexification model and the corresponding introduction of imaginary units. From the historical point of view, we can recall the Italian mathematician's work Corrado Segre (1863 -1924), involving the mathematical definition of the bicomplex numbers, indicated by { 1 + 2 j|z 1 = a + bi, 2 = + , j 2 = −1} . But, every conceptual element in the previous set may be still expressed as 1 + 2 ⋅ j = a + b ⋅ i + c ⋅ j + d ⋅ i ⋅ j, with the real numbers , , , ∈ . This abstract entity may also be represented by a + b ⋅ i + c ⋅ j + d ⋅ , with the set of operational rules 2 = 2 = −1, = = .
Given these elements and others that we will seek to discuss in the next sections, mainly some elements with respect to an evolutionary epistemological trajectory and, especially, an historical perspective. In this way, it may raise an understanding about the continued progress in Mathematics and some elements, which can contribute to an investigation about the quaternions and octonions of Fibonacci's sequence which is customarily discussed in the academic environment, however in relation to their formal mathematical value.
Moreover, in view of the use of software Maple, we will explore particular situations enabling a heuristic thought and not completely accurate and precise with respect to certain mathematical results. Such situations involve checking of algebraic properties extracted from current numerical and combinatorial formulations of the Fibonacci's model.
Thus, in the next section, we consider some elements and properties of quaternions or hyper-complex numbers (Kantor & Solodovnikov, 1989).
Halici (2015, p. 1) comments that the quaternions are a number system which extends to the complex numbers, first introduced by Sir William Rowan Hamilton (1805 -1865), in 1843. From the fact 2 3 = 4 = − 3 2 we say that the algebra is not commutative but associative, as, ( 2 3 )e 4 = 2 ( 3 4 ). On the other hand, the historical process, concerning the systematization process of quaternions demanded considerable time and effort, above all, the capacity of Hamilton 's imagination. We can see this from the comments due to Hanson (2006, p. 5).
Quaternions arose historically from Sir William Rowan Hamilton's attempts in the midnineteenth century to generalize complex numbers in some way that would be applicable to three-dimensional (3D) space. Because complex numbers (which we will discuss in detail later) have two parts, one part that is an ordinary real number and one part that is "imaginary," Hamilton first conjectured that he needed one additional "imaginary" component. He struggled for years attempting to make sense of an unsuccessful algebraic system containing one real and two "imaginary" parts. In 1843, at the age of 38, Hamilton had a brilliant stroke of imagination, and invented in a single instant the idea of a threepart "imaginary" system that became the quaternion algebra. According to Hamilton, he was walking with his wife in Dublin on his way to a meeting of the Royal Irish Academy when the thought struck him An element or factor that cannot be disregarded in the previous section concerning the role of the mathematical genius in the sense of obtaining an idea or an insight (Hadamard, 1945) in view of the consistent formulation of the set of quaternions. Moreover, an essential aspect pointed by Hanson (2006) relates precisely to the preparation and formulation of a formal mathematical definition process. We can, for example, see in  Brousseau (1965) discusses the extension's process to the Fibonacci's model Figure 2, Hanson comments some explication present in a bronze plaque that mentions explanatory words about the glorious and unexpected moment, that the professional mathematician, with a view to establishing a new mathematical set which is still studied nowadays (Hanson, 2006).

SOME HISTORICAL ASPECTS ABOUT THE OCTONIONS
In view of the formal properties of regular quaternions, mainly its dimensional properties, of course, after a certain time, a natural thought refers to increasing the dimensional set. Thus, from this dimensional elevation, occurred soon after the emergence of all octonions.
In this way, let Θ be the octonion algebra over the real number field . Keçiolioglu & Akkus (2014, p. 2) record that, from the Cayley-Dickson process, we can take any element ∈ Θ, therefore, it can be written as = ′ + ′′ , where ′, ′′ ∈ the real quaternion division algebra. On the other hand, the excerpt below shows the important role of Hamilton's student.
Less well known is the discovery of the octonions by Hamilton's friend from college, John T. Graves. It was Graves' interest in algebra that got Hamilton thinking about complex numbers and triplets in the first place. The very day after his fateful walk, Hamilton sent an 8-page letter describing the quaternions to Graves. Graves replied on October 26th, complimenting Hamilton on the boldness of the idea, but adding, "There is still some thing in the system which gravels me. I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties." And he asked: "If with your alchemy you can make three pounds of gold, why should you stop there?" (Conway & Smith, 2004, p. 9) Graves contributed to a first impulse to obtain some preliminary ideas about octônios, however, the fundamental elements for its definitive establishment, as explained Halici (2015), was awarded by Cayley, who defined his algebra, subject to certain formal mathematical rules. In this way, Halici (2015, p. 5) records the system of multiplication: Halici (2015, p. 6) comments that is well known the octonions algebra Θ is the real quaternion division algebra. Moreover, among all the real division algebras octonion algebra forms the largest normed division algebra.
From this point, with the appreciation of some definitions related Fibonacci model, we note a natural style of composition properties of the two different mathematical models. One factor that cannot be disregarded with respect to the historical time corresponding to the mathematical evolutionary process, especially to the sixties, and that contributed to the current research.
Before concluding, we recall the first articles that explored some fundamental properties, in order to formulate and define new conceptual entities. Thus, we find the following definitions.  Hanson (2006, p. 6) comments the iluminatory moment that Hamilton stablishes the rules for the quaternions set Definition 1: The nth Fibonacci quaternions is defined by = 0 + +1 1 + +2 2 + +3 4 , where the coefficients are de Fibonacci numbers. (Horadam, 1963).
Sometime later, we find a definition that involves the complexity of the process of the Fibonacci numbers and, respectively, a concern with their representation in the complex plan. We observe such a characterization in the next definition.
The collection of these mathematical definitions should convey to the reader an understanding about the evolutionary process of the Fibonacci model, shown adhered to the other models discussed, even like the quaternions and octonions. Soon after, we discuss in detail some invariants mathematical properties, when we examine closely each of the previous mathematical definitions.
In the theorem 3, we have indicated the octiotonic Cassini's version.
On the other hand, we know a strong tradition in the works in order to use the matrix approach and with the goal to derive some generalized results. Indeed, from the Halici (2012, p Surely, there are other forms of representation of Fibonacci quaternions and octonios. However, we see that its representation through a matrix, of second order, will be very useful, especially at the time of implementation of CAS Maple. By the software help, we will investigate the numerical behavior of some particular cases (Cassini's identity) and thereby conjecture a closed Cassini's formula, for some particular sets.

HISTORICAL INVESTIGATIONS WITH THE MAPLE'S HELP
In this section, we will indicate some basic commands and a command package that let you explore a set of numerical operations and algebraic with quaternions and octonions Fibonacci. The software will allow an especially verification and numerical exploration for particular sets, with a view to determining and formalizing certain properties. The aim of this worksheet is to define some procedures in order to make computations in a Fibonacci quaternion and octonion algebras over the field of rational number. Below the figure, we see that the Maple's command package that allows us to explore a series of operations with quaternions and octonios.
The Quaternions package allows the user to construct and work with quaternions in Maple as naturally as you can work with complex numbers. The list of procedures may should be inserted In the Figure 4, we can visualize some preliminar procedures, for the purpose of inserting some particular cases.
Following, we can declare the vectors that we want to work, taking into account a particular set. Below, we indicate the following vectors.
Regarding the Maple's use, we highlight the elements: (i) The software enables verifications of particular cases and properties related to the Fibonacci quaternions and Fibonacci octonions; (ii) The software allows the verification properties provided by classical theorems related to the Fibonacci quaternions and Fibonacci octonions, especially the most recently discussed in the literature; (iii) The software allows the description of a lot of special particular cases conditioned by newly formulated mathematical definitions; (iv) The software enables verification of properties related to a larger set of integer subscripts indicated in the scientific articles; (v) the software enables verification of a large number of individual cases in order to test mathematical conjectures; (vi) the software allows the correction of mathematical formulas in order to provide a correct description.
Finally, through the invariance numerical and algebraic of behavior, we can conjecture the intermediate steps of the inductive process. Thus, as in all verified cases, determine a simplified Formula for the Cassini's identity Casino and thus avoid certain mathematical errors that might be observed in other studies. Moreover, from the historical evolution of the quaternions and octonions' model, we can understand the current evolution of the research process around of the inherited Leonardo Pisano's model.