All the mathematical results in this book have been discovered or invented by the four authors. The reason for developing the book in this way, rather than producing a traditional text on number theory, is to preserve the styles of the original authors in the various parts rather than to homogenize the writing.
T. ATANASSOV
G. SHANNON
C. TURNER
Coupled Recurrence Relations
Number Trees
Fibonacci Vector Geometry
COUPLED RECURRENCE RELATIONS
Sequences
- T h e four 2-F-sequences
- The Scheme (2.2)
- Remark on a new direction for a generalization of the Fibonacci sequence
- Fibonacci sequence via arithmetic progression
Here we will discuss an approach to an interpretation of the Fibonacci sequence as an arithmetic progression. 1995: Note on a new direction for a generalization of the Fibonacci sequence., The Fibonacci Quarterly 3 3 , No.
NUMBER TREES
Introduction
It was then shown how the generalized Fibonacci numbers can be used to color convolution trees so that the tree shades create a generalization of Zeckendorf's theorem and its dual [21, 1985b]. In the sequence of Fibonacci convolution trees {Tn} given in [20, 1985a], the sum of the weights assigned to the nodes of Tn is equal to the nth term of the convolution of {Fn} and {Cn}.
Let (Na,Nb) represent the number of a's and the number of b's at a given level of the tree.
Generalized tableaux
As explained elsewhere [23], the formation rule arises directly from the construction of the trees. The proof follows from the initial conditions and the ordinary repetition relation (2.2) for {Usn} to get Xk+i,m, and then from the partial repetition.
Gray code of the cube, and recurrences
Third-order coloured trees
Matrix representations of coloured trees
It can be seen that the sum of the weights of the nodes in the Tijtk tree is given by Wijtk, where. There the sum of the weights of the trees colored at the nodes by the Fibonacci numbers, Fn, is given by the convolution.
FIBONACCI VECTOR GEOMETRY
The Fibonacci Vector—some Elementary Results
Solutions of 'squares' equations
It is easy to show [32] that we can replace F by L in the above equation (2), and thus obtain a solution comparable to the four-square equation, in terms of the Lucas numbers.). The presentation has a poetic style; so I was tempted to call it Square Dance in Fibonacci Numbers, an ode with an infinite number of verses.
Geometric properties of Fibonacci triangles Some elementary geometric results concerning the geometry of the trian-
Each of them is the hypotenuse of a right triangle, made with two of the reference axes. v)(a). If we take the origin Q(0,0,0) as a fourth point connected to the vertices of the F-triangle, we have defined a tetrahedron T„ (actually a sequence of tetrahedra if n is allowed to vary).
Integer-vector Recurrence Equations
And then, provided that the initial vectors are integer vectors (i.e. vectors with integer coordinates), and the coefficients in the equation are suitably chosen integers, the iteration will produce a sequence of vectors of integers. Given initial vectors of integers xi and x2, iteration (2.1) will generate a complete vector sequence.
Fibonacci Vector Sequences
Geometric Properties of G
Thus there is also a left limit ray (call it L') and a right limit ray (call it L") for the vector sequence G. Proof: The plane AQAB is given by \QA.QB$mLAQB, which is half the size of the vector product a x b.
T h e Limit Rays of Vector Sequence G
For convenience, we will define this last as the shortest vector of the case (ii) sequence. The direction ratios are independent of the values of aj and o2; therefore the same limit rays apply to all Fibonacci vector sequences in the plane determined by these a and b.
We first consider how points in ir0 define Fibonacci vectors and Fibonacci vector sequences. We will denote the union of these three sets of vectors by G = {G„} and call it the general vector Fibonacci sequence.
The Honeycomb of Points in 7To
The set of vertices of all the triangles forms the set of integer vertices in -K0. The Fibonacci Honeycomb—the plane TTQ and axes The centers of the hexagons are the 5-points;.
Some Properties of B-points
We can call the points in B the B points (bees!) of ir0; and the points in the H .ff points (Honey points). Similarly, we can connect the points of the Lucas vector sequence to obtain the Lucas polygon.
Convergence Properties of the Polygons
We find that the result is independent of the choice of a and 6 (except that at least one must be non-zero). This means that all (except F(0,0)) sequences of Fibonacci vectors tend to the same ray Q-Poo, which has direction cosines u.
Some Theorems about Triangles, Lines, and Quadrilat- erals
We have already given some results on triangles related to Fibonacci vector polygons in the previous chapter. Finally, we will define and study transformations of triangles by means of equilateral triangle constructions, and apply them to Fibonacci vector polygons.
T h e Impossibility of a (90,45,45) Integer Triangle
In this chapter we present some theorems about the triangles that appear in the Fibonacci plane 7r0. We will only deal with triangles at grid points; we will call these whole triangles.
The Ubiquity of (60,60,60) Integer Triangles
In the proof of Theorem 5.3 we showed that all expressions between parentheses, in the above coordinates, are congruent with 0 (mod 3). We have seen how certain triangles with integer vertices can be drawn in the honeycomb plane, while others cannot.
Some triangle constructions
T h e ET-transform set £ of a triangle
Constructing triangles ABC, for which P',Q',R' are coininear It is possible that the three inner points in Z are coininear, as the following right diagram shows. Then a point C can be found on the perpendicular bisector of AB, such that the three inner transformation points of AABC are collinear.
A n associated Diophantine equation
For any given solution there are 35 others which are equivalent up to permutations of x, y, z terms and u, v, w terms. iii). It is natural, since we are working in the honeycomb plane, that we ask whether there are classes of solutions of (*) which are either Fibonacci vectors (6-tuples) or which can be characterized in terms of the Fibonacci numbers.
I?T-transforms of Fibonacci vector polygons
In Chapters 2 and 3, we defined the recurrence equations of whole vectors and studied some geometric properties of the vector sequences they generate. Finally, we focus on the class of Fibonacci vector sequences in the honeycomb plane and study their various aspects and transformations.
Eight Example Vector Sequences
We then state and prove several general theorems about type I and type II generated vector sequences. Note that vector sequences (1), (2) and (3) are basic Fibonacci, Lucas and Pell sequences.
Vector Sequence Planes
Inherent Transformations of Planes
However, we can choose (see below) the pseudo-inverse of H~ which will perform the back-transformation we want. The inherent matrix H(l,l) for the honeycomb plane has the following properties in its powers Hn and H~n.
Vector Recurrence Relations
Type I: The matrix/vector equation, order 1
Alternatively, we can replace H with any 3 x 3 integer matrix (say T) and choose the elements of T such that different types of vector sequences emerge from the inverse relation. This sequence uses Fibonacci numbers twice for its coordinates, with all vectors in the Pell plane.
Some nice comparisons can be made of T with: for example, they have the same characteristic equation. Proof: The first three terms of the sequence generated by type I iteration are x, xH, a n d x i ?2.
Some Geometric Observations (continuation from Chap- ter 2)
Demonstration 1 (recapitulation)
Fibonacci number sequence, the ratio Fn+i/Fn tends to a, regardless of the choice of starting numbers. are parallelograms with equal area. are straight line segments, with X3,X4,X5 etc. d) All terms of the vector series lie in the plane defined by the three points Q,a, b. It is curious how the inductive process of arriving at proof of these identities is all taken care of by the geometry of the plane and the Type II vector sequence generated within it.
Demonstration 2
On the branches we show points from other Fibonacci vector sequences; and we claim that in all branches of the entire set of stacks all integer vectors lie in the 7To plane (above the UV-boundary line). We have drawn the first three central hexagons with dots to create a spider web image of the system.).
Demonstration 3
This is our simplified, pictorial representation of the set of all Fibonacci vector sequences, to be stored in our mind spaces. We have drawn in the first three central hexagons, dotted, to create a spider web image of the system.). A final note is that we can think of the two branches of a chimney Xm as the envelope of all the m Fibonacci vector polygons that zig-zag upwards inside the chimney.
Summary
It is easy to show* that / „ —> a, the golden ratio, as n —> oo (which we could easily derive by geometrical observations on the ascending Lucas vector polygon, but perhaps not rigorously).
Introduction
For now, c and d will be left out of consideration* (by setting them both equal to 1), in order to define the simplest possible iteration of the Fibonacci trace. The sequences of objects it generates will be called traces in S, or simply traces.
A n example in Vector Geometry: rectilinear spirals
An analysis of the above formulas for the expressions, however, shows that, in general, the joined points of the line form a right-angled spiral, which proceeds systematically from one to the next of three mutually perpendicular arms or directions. Thus, the Fibonacci vector-multiplicative sequence is an interesting [3D]-vector figure, a spiral on three mutually perpendicular arms, which can be placed anywhere in Z3 with a suitable choice of initial vectors a and b.
On Tracks in Groups
Perhaps we might expect them to appear as exponents, since the vector products were performed in repetitions, not additions; however, this was by no means guaranteed at the start of the investigation. It is obvious that the infinite Fibonacci vector sequence (which zigzags inside the chimney (see ch. 6) is a Fibonacci vector sequence in (Z3, + ) and that the set of all possible Fibonacci vector sequences (the set of sequences) exactly covers the integers points of the Fibonacci honeycomb plane (note that these points form a group when adding vectors).
Tracksets and Spectra of the Groups of Order 4
Orbital sets of the fourteen groups orders 1 to 8 (i) The eight cyclic groups:. 1) The two groups of order 4 have different trace sets; so are the two of order 6; so are the five of order 8. Therefore, most, perhaps all, of the properties given here about trace sets of cyclic groups may already be known.
Equivalence of Tracksets and Groups
An interesting question to ask is: How does a permutation of the elements of Z„ work, when applied to all the elements of a group and its tracks. For example, holding 0, permuting the elements of Zn\ { 0 } in any possible way leaves the trace sets of Ci, C2, C3 and the Four-group V unchanged.
The track set remains the same, but with different symbols for its elements; therefore it tells the same story about the group, whose symbols have changed in the same way. Some track sets are even invariant under all permutations applied to the set of their elements (except their identity).
Some Operations with Tracksets
Checking if a binary relation is a group relation
It should be clear that these postulates correspond directly to the postulates usually given as axioms for a group, namely: Gl (closure), G2 (existence of an identity element), G3 (existence of unique inverses) and G4 (associative law applies). Then we checked to see if 3.5 was the first pair of elements that wasn't in the first two numbers, so number r3 started with that.
Determining subgroups
As an aside, we note that in an early paper on groups, around 1855, Arthur Cayley announced that he had found three groups of order 6.
Properties of trackset-spectra of cyclic groups
Disregarding the 1-period identical track that appears in each spectrum, there are one or two periods in the P-spectrum Cp if p is prime.
Conjectures on tracksets of cyclic groups
- Equivalence with a linear Fibonacci recurrence
- A Knot from a Trackset
Its diagram is shown below. a) The C4 node (b) The V4 node 7.8 On aesthetics and applicability of track sets. To review examples of trace set use, and to discuss associated aesthetic values, let us first consider the trace sets of the cyclic groups (see the table in Section 3).
GOLDPOINT GEOMETRY
On Goldpoints and Golden-Mean Constructions
