FIBONACCI VECTOR SEQUENCES AND REGULAR POLYGONS Stuart D. Anderson 32141 Woodvine Drive Sorrento FL 32776 USA website: www.sdaphd.com email:
[email protected] and Dani Novak Department of Mathematics, Ithaca College 953 Danby Road, Ithaca NY 14850 USA email:
[email protected] Draft of 2009 March 22
ABSTRACT
It is well known that the Fibonacci sequence is related to the geometry of the regular pentagon by means of the Golden Ratio φ. In this paper, following the lead of George Raney [?] and Bjarne Junge [?], we generalize these notions. We define sequences of vectors of n ≥ 2 dimensions, which we call n-dimensional Fibonacci vector sequences, Fn . The original Fibonacci sequence is simply related to F2 , the 2-dimensional Fibonacci vector sequence. We study how the ratios of the components of the Fibonacci vectors are related. We also study the regular polygons with an odd number 2n + 1 of sides, in particular, the ratios of the lengths of their diagonals. We find that the n-dimensional Fibonacci vector sequence is related to the regular polygon with 2n + 1 sides by a set of n − 1 real numbers, which we call Golden Numbers and represent by φn (i), where 2 ≤ i ≤ n, as they are generalizations of the Golden Ratio φ = φ2 (2). The Golden Numbers are understood numerically as the limiting values of the ratios of components of the vectors of a Fibonacci vector sequence or geometrically as the ratios of the lengths of the diagonals to the length of a side of an odd-sided regular polygon.
1
1. INTRODUCTION Our intention in writing this paper is to present to a wider audience an amazing connection between Fibonacci vector sequences and regular polygons. Many of the results that we present here are not new, but were discovered and published more than forty years ago by George Raney [?]. However, these results are not widely known, probably because his paper is highly technical, employs sophisticated notation and provides rigorous proofs. Some of these results were also reported by Junge and Hoggatt [?]. We wish to bring these beautiful results to the attention of the larger mathematical community. We hope that it will encourage further research and that it will be found of special interest to teachers and students of mathematics at the college level and perhaps even the high school level. It has been known for centuries that the Fibonacci sequence is related to the Golden Ratio φ and thereby to the geometry of the regular pentagon. Basic background information on these topics is provided in Section 2. The purpose of this paper is to generalize these relationships. In Section 3, we define a generalization of the Fibonacci sequence, the Fibonacci vector sequences in n ≥ 2 dimensions. The original Fibonacci sequence is simply related to the two-dimensional Fibonacci vector sequence. We also show that each of these sequences generates a set of n − 1 real numbers. We call these Golden Numbers and represent them by φn (i), where 2 ≤ i ≤ n, as they are generalizations of the Golden Ratio φ, which is the smallest one of them, φ2 (2). We also provide a system of n − 1 equations that relates the n − 1 Golden Numbers in n dimensions. In the next section, Section 4, we begin our study of the geometry of the odd-sided regular polygons, that is, those having 2n + 1 equal sides and angles, where n ≥ 2. Such a polygon possesses n − 1 diagonals having different lengths, dn (i), where 2 ≤ i ≤ n, in addition to its side of length dn (1). We show that exactly the same system of relationships exists among the diagonal lengths of the polygon of 2n + 1 equal sides as among the Golden Numbers generated by the Fibonacci vector sequence in n dimensions. This relationship between the Golden Numbers generated by the Fibonacci vector sequences and the geometry of the diagonal lengths of the odd-sided regular polygons is quite remarkable. At this point in our investigation, we have discovered interesting relationships among the Golden Numbers and among the diagonal lengths, but we cannot yet calculate either. Therefore, in Section 5, we derive simple trigonometric formulas to calculate the diagonal lengths and the Golden Numbers, the first several of which we show in a Table. Finally, applying this formula for the diagonal lengths, we show that the relationship among the diagonal lengths, derived geometrically in Section 4, is, in fact, a trigonometric identity. We do not consider in this paper the case of the one-dimensional vector sequence and its geometric representation, because they are trivial. The n = 1 sequence is an unending string of 1’s, identical vectors of one component, the value of which is 1. Its geometric representation is the regular or equilateral triangle 2
of unit side, which has no diagonals. Also in this paper, we do not consider the Golden Numbers of index i = 1 to be genuine Golden Numbers, because they are trivial. For any n, the number φn (1) equals 1, for it represents the ratio of two identical quantities, the quantity being the first component of a Fibonacci vector or the length of the side of a regular polygon. Thus, for given dimension n, we generally say here that there are n − 1 Golden Numbers, φn (i) for 2 ≤ i ≤ n, meaning only the nontrivial ones.
2. THE FIBONACCI SEQUENCE AND THE REGULAR PENTAGON We assume that the reader is familiar with the Fibonacci sequence and with the geometry of the regular pentagon with which this sequence is connected. We summarize briefly here the well known results. The Fibonacci sequence is the sequence of integers F . F = (0, 1, 1, 2, 3, 5, 8, 13, · · · )
(1)
The sequence is generated from the seed sequence {F (0) = 0, F (1) = 1} by the recursive relationship F (i + 1) = F (i) + F (i − 1)
(2)
where F (i) is the member of the sequence of index i. The ratio of consecutive members of the sequence is defined by r(i). r(i) =
F (i) F (i − 1)
(3)
The recursive relationship between the ratios is expressed by r(i + 1) = 1 +
1 r(i)
(4)
In the limit that i → ∞, the ratio can be proved to exist and has the value r r = lim r(i) i→∞
(5)
which may be found as the solution to the recursion relation r =1+
1 r
(6)
or r2 − r − 1 = 0 The only positive solution to this equation is the value √ 1+ 5 r=φ= ≈ 1.618033989 2 which is of such significance that it has been given the name the Golden Ratio and the symbol φ.
3
(7)
(8)
Geometrically, the Golden Ratio is connected to the regular pentagon. A regular pentagon possesses five sides of equal length and, between adjacent sides, five equal interior angles of 108◦ =
3 5
· 180◦ . The
five equal diagonals of this figure form a regular pentagram, the familiar five-pointed star. The ratio of the diagonal length to the side length of a regular pentagon is φ, the Golden Ratio. Again, the purpose of this paper is to explore how these relationships can be generalized. First, we show how the original Fibonacci sequence is generalized by the Fibonacci vector sequences. Next, we show that the n-dimensional Fibonacci vector sequence generates n − 1 ratios, which we call Golden Numbers, because they are generalization of the Golden Ratio φ. Finally, we show how the Fibonacci vector sequences are related to the geometry of the regular odd-sided polygons, that is, polygons of 2n + 1 equal sides, where n is an integer n ≥ 2, which is the generalization of the regular pentagon. 3. FIBONACCI VECTOR SEQUENCES We now introduce the Fibonacci vector sequences, which generalize the original Fibonacci sequence. A Fibonacci vector sequence Fn is a sequence of vectors in Zn+ , that is, n-tuples of components that are nonnegative integers. We refer to the number n as the dimension of the vectors of the sequence. The original Fibonacci sequence F is simply related to the two-dimensional Fibonacci vector sequence, F2 , as you will soon see. The vectors of the sequence Fn are defined recursively. Described verbally, we say that the ith component of a vector is the sum of the last i components of the preceding vector. Therefore, the first component of a vector is equal to the last component of the preceding vector. Also, any component after the first component equals the sum of the preceding component of the same vector and the (n + 1 − i)th component of the preceding vector. The first vector in the sequence is given; it is the seed vector, a sequence of n − 1 zeroes followed by a one. Now, let us express this recursive definition of a Fibonacci vector sequence algebraically. Let Fn (k) represent the kth vector in the sequence Fn . The seed vector is given: Fn (0) = (0, 0, . . . , 0, 1). Let Fn (k, i) represent the ith component of the kth vector of the sequence Fn . Then, the sequence is defined recursively by the following system of n equations for 1 ≤ i ≤ n. Fn (k, i) =
j=n X
Fn (k − 1, j)
(9)
j=n+1−i
Thus, the equation for i = 1 states that the first component of a vector is identical to the last component of the preceding vector. Fn (k, 1) = Fn (k − 1, n)
(10)
The equations for 2 ≤ i ≤ n can then be stated more simply. Fn (k, i) = Fn (k, i − 1) + Fn (k − 1, n + 1 − i) 4
(11)
At this point, the reader is invited to work out from this definition the first few vectors of the Fibonacci vector sequences in the lowest few dimensions. Construct the Fibonacci sequence in two dimensions F2 to understand how it is related to the original Fibonacci sequence F . You will find that the sequence of the first components of the F2 vectors, F2 (k, 1), is the original Fibonacci sequence F . The sequence of the second components, F2 (k, 2), is also the original Fibonacci sequence, but shifted by one position. Table 1 provides an example of a Fibonacci vector sequence of slightly higher dimensionality, the one of four dimensions, F4 .
0
1
2
3
4
5
···
1
0
1
1
4
10
30
···
2
0
1
2
7
19
56
···
3
0
1
3
9
26
75
···
4
1
1
4
10
30
85
···
Table 1: The Four-Dimensional Fibonacci Vector Sequence Columns represent vectors F4 (k) for 0 ≤ k ≤ 5. Rows represent the components of these vectors, F4 (k, i) for 1 ≤ i ≤ 4. Now, define rn (k, i) as the ratio of the ith component to the first component of the kth vector. rn (k, i) =
Fn (k, i) Fn (k, 1)
(12)
The first of these ratios rn (k, 1) is obviously unity. rn (k, 1) = 1
(13)
The remaining n − 1 ratios are defined by the following recursive equations for 1 ≤ i ≤ n − 1. rn (k, i + 1) = rn (k, i) +
rn (k − 1, n − i) rn (k − 1, n)
(14)
From Raney [?], it follows that, for fixed n and i, the limit as k → ∞ of rn (k, i) exists. We have not been able to devise a proof shorter than his and invite the reader to try to do so. Accepting that the limit does exist, we denote it by rn (i). rn (i) = lim rn (k, i) k→∞
(15)
The n values of rn (i) for 1 ≤ i ≤ n may be viewed as the components of a vector in Rn , that is, a vector of n components, which are real numbers. The first of these n ratios is trivial. rn (1) = φn (1) = 1 5
(16)
The remaining n − 1 ratios are determined by the following system of n − 1 equations for 1 ≤ i ≤ n − 1, which we call the golden system of equations in n dimensions rn (i + 1) = rn (i) +
rn (n − i) rn (n)
(17)
which may also be written as rn (n) rn (i + 1) − rn (i) − rn (n − i) = 0
(18)
Of all possible solutions of this system of equations for a given dimension n ≥ 2, we are interested here in only one solution, the one that consists of a set of positive numbers only, for they represent the limiting values of ratios of the components of the vectors of a Fibonacci vector sequence, which are positive numbers only. We call this unique solution of the golden system of equations the Golden Vector, φn , and its n all positive components, the Golden Numbers, φn (i), 1 ≤ i ≤ n, for they are the generalization of the Golden Ratio φ = φ2 (2). Note that the first of these n Golden Numbers is the trivial one, φn (1) = 1, and that the other n − 1 are the positive solutions of the golden system of equations. For the case of dimension n = 2, the golden system of equations, (??) or (??), reduces to a single equation in the variable r2 (2), the only positive root of which is the Golden Number φ2 (2). This equation in the variable r2 (2) is identical with the equation (??) or (??) in the variable r, the only positive root of which is the Golden Ratio φ. The Golden Vector in this case is φ2 = (φ2 (1), φ2 (2)) = (1, φ).
4. RELATIONSHIPS OF REGULAR POLYGON DIAGONAL LENGTHS In this section, we will derive a geometric interpretation of the golden system of equations (??) or (??). We will show that the all positive solutions of the golden system of equations correspond to the lengths of the diagonals of the odd-sided regular polygons, those with an odd number 2n + 1 of equal sides, where n ≥ 2 and the sides are of unit length. Consider a regular polygon with an odd number 2n + 1 of vertices. Connecting its adjacent vertices are 2n + 1 sides of equal length, which we represent as dn (1). Connecting its nonadjacent vertices are its diagonals, of which there are n − 1 types, the lengths of which we represent as dn (i), for 2 ≤ i ≤ n, from the shortest, of length dn (2), to the longest, of length dn (n). In this notation, the subscript to d is the number n of types of edges, where an edge is either a side or a diagonal. The argument of dn , contained in parentheses (), is the diagonal or edge index, the (smaller) number of sides between the two vertices that the edge connects. Of each type of edge, there are 2n + 1 instances. Altogether, there are n · (2n + 1) edges in the complete diagram of a regular polygon, or the complete graph, of 2n + 1 vertices. Such a perfectly symmetric diagram is beautiful to behold, but, because we assume that you are already familiar with it and because it is too complicated for our purposes, we have not provided such a diagram here. 6
Instead, we refer you to Figure 1, which shows only one of an odd-sided regular polygon’s 2n + 1 sides, only five of its vertices and only a few of its diagonals. The five selected vertices, labeled A, B, C, D, and E, are located on the polygon’s circumscribed circle. Five arcs of the circle are labeled with their angular sizes, shown as integral multiples, 1, i or n − i, of 2αn , where the significance and value of the angle αn will be given shortly. The one side shown, CD, is labeled with its length dn (1). The diagonals shown are labeled with their lengths dn (i), dn (i + 1), dn (n − i), dn (n − i + 1) and dn (n). Angles between diagonals are labeled with their angular sizes, 1, i, i + 1, n − i, n − i + 1, n, n + i and 2n − i, in units of αn . In a regular polygon, any angle between two sides or diagonals is an integral multiple of a minimum angle αn αn =
π 2n + 1
(19)
where π ≈ 3.141592654 is the equivalent in radian measure of 180◦ . Similarly, any arc that connects two vertices is an integral multiple of the minimum arc, 2αn , the side arc. To save space in Figure 1, the value of each angle or arc is indicated by an integral multiple of the minimum angle or arc, rather than by the value of the angle or arc itself. In Figure 1, the point of intersection of the diagonals BD and CE is labeled Z. This point Z divides the diagonal BD, of length dn (i + 1), into two parts, BZ of length dn (i), and ZD, the length of which we denote as e. dn (i + 1) = dn (i) + e
(20)
Now, observe that there are at least two similar isosceles triangles in the figure, CAD and ZED. Each has two equal angles of nαn and a single angle of αn . Of course, the sum of the three interior angles of these triangles is (2n + 1) αn = π. The two equal sides of triangle CAD are of length dn (n) and its base is of length dn (1). Similarly, in the triangle ZED, the side length is dn (n − i) and its base length is e. The similarity of the two triangles may be expressed by equating the ratios of their side lengths to their base lengths. dn (n − i) dn (n) = dn (1) e
(21)
dn (1)dn (n − i) dn (n)
(22)
Solve this for e. e=
Substitute this value into the equation (??) that expresses the division of BD into two parts and we have dn (i + 1) = dn (i) +
dn (1)dn (n − i) dn (n)
(23)
which may also be written dn (n) dn (i + 1) − dn (i) − dn (1)dn (n − i) = 0
7
(24)
This system of equations expresses the relationships among the lengths of the diagonals of an odd-sided polygon. They should look familiar. If we set the polygon side length to one, dn (1) = 1, they are exactly the same as the equations (??) and (??) that express the relationships among the Fibonacci vector component ratio limits, with the d’s for diagonal lengths replacing the r’s for ratios. We conclude that the diagonals of the regular polygon of 2n + 1 vertices and the component ratio limits of the n-dimensional Fibonacci vector sequence satisfy the same non-linear golden system of equations. Based on Raney’s paper (again, we could not find a simple proof but invite the reader to try to devise one), we conclude that the all positive solutions to both systems of equations are identical and unique.
5. LENGTHS OF REGULAR POLYGON DIAGONALS The equations we have just derived are pleasing for the way in which they interrelate the lengths of the diagonals of an odd-sided polygon. But they are difficult to use when it comes to actually calculating those lengths. So in this section, we will derive a simple trigonometric formula to calculate the length of any one of the diagonals of such a polygon. Furthermore, we will provide in this section a proof of the equations (??) and (??) that express the relationship of the lengths of the diagonals, a proof which is the trigonometric equivalent of the geometric proof of the preceding section. First, in a regular polygon of 2n + 1 vertices, let us find the length of a diagonal of index i, dn (i). In Figure 1, note that Y is the point of intersection of the diagonals AD and CE. Denote the length of the segment CY as p and that of Y A as q. These are two of the sides of the triangle CAY . In this triangle, the side of length p is opposite the angle of αn , while the side of length q is opposite the angle of iαn . Apply the law of sines to this triangle. q p = sin αn sin iαn
(25)
Rewrite this equation to express the ratio of q to p. q sin iαn = p sin αn
(26)
Now, observe that the triangle CAY is similar to the triangle DBC. The similarity of these two triangles can also be expressed to yield the q to p ratio. q dn (i) = p dn (1)
(27)
Thus, setting these two expressions for the ratio of q to p equal to one another, we have our result, a simple trigonometric formula to calculate the length of the diagonal of index i, dn (i), as a multiple of the side length dn (1). dn (i) = dn (1)
iπ sin( 2n+1 ) sin iαn = dn (1) π sin αn sin( 2n+1 )
8
(28)
Also, to return briefly to the world of Fibonacci vector sequences, we have by analogy a simple formula to calculate the Fibonacci vector component ratio limits, φn (i), the Golden Numbers of which we spoke earlier. φn (i) =
iπ ) sin( 2n+1 sin iαn = π sin αn sin( 2n+1 )
(29)
Table 2 displays the first few Golden Numbers calculated from this formula.
2
3
4
5
2
1.618034
3
1.801938
2.246980
4
1.879385
2.532089
2.879385
5
1.918986
2.682507
3.228707
3.513337
···
···
···
···
···
···
···
Table 2: The Golden Numbers φn (i) Rows represent the dimension n and columns represent the index i of the numbers. For given dimension n, the indices i range over 2 ≤ i ≤ n. We do not include here the trivial Golden Numbers φn (1) = 1. The Golden Numbers φn (i) of only the lowest four dimensions 2 ≤ n ≤ 5 are shown here. The top number, φ2 (2), is the Golden Ratio φ. But we view these all as Golden Numbers, because they are calculated from the same formula, (??), and have similar interpretations.
We will now use equation (??), the trigonometric formula for the length of a diagonal, to prove the result of the previous section, the relationship of the diagonal lengths, equation (??) or (??). We will first use trigonometric identities to transform the result and then show this transformed result is, in fact, a trigonometric identity. Begin by substituting equation (??), with the appropriate values of the diagonal index, into equation (??) and removing common factors of dn (1) and sin αn . sin αn · sin(n − i)αn = sin nαn · sin(i + 1)αn − sin iαn
(30)
Now, employ these trigonometric identities sin β · sin γ =
1 2
cos(β − γ) − cos(β + γ)
9
(31)
cos(−β) = cos β
(32)
cos(n + i)αn + cos(n − i + 1)αn = cos(n − i)αn + cos(n + i + 1)αn
(33)
to find the transformed result.
Now, let us examine the trigonometric identity that the cosine function is antisymmetric about 90◦ or π 2
radians. cos( π2 − β) + cos( π2 + β) = 0
Into this identity, substitute for
π 2
(34)
the equivalent expression π 2
= (n + 21 )αn
(35)
β = (i − 21 )αn
(36)
cos(n + i)αn + cos(n − i + 1)αn = 0
(37)
and for β the expression
This trigonometric identity now reads
An equivalent form of this identity is found by changing i to −i. cos(n − i)αn + cos(n + i + 1)αn = 0
(38)
Set these two forms of the identity equal to one another and what do we have? It is exactly equation (??), the transformed result. The result is simply a trigonometric identity. This completes the trigonometric proof of the relationship of the diagonal lengths, equation (??) or (??).
6. CONCLUSION Following the lead of George Raney, we have come to the following conclusions. Fibonacci vector sequences Fn may be defined as sequences of vectors of an arbitrary number of dimensions n ≥ 2. The n-dimensional Fibonacci vector sequence, Fn , generates n − 1 Golden Numbers, φn (i), for 2 ≤ i ≤ n, which can be calculated by a simple trigonometric formula.The Golden Numbers are represented geometrically by the ratios of the lengths of the diagonals to the length of a side of a regular polygon of 2n + 1 vertices. The original Fibonacci sequence is simply related to the vector sequence in two dimensions, F2 ; it is the sequence of the first (or of the second) components of the vectors. The original sequence generates a single positive root, the Golden Ratio φ, which equals φ2 (2), the only Golden Number generated by the vector sequence in two dimensions F2 . The regular pentagon with its five equal sides and its inscribed pentagram of five equal diagonals is recognized as the case n = 2 of the regular polygon of 2n + 1 = 5 vertices, with its 2n + 1 = 5 sides and 2n + 1 = 5 diagonals of n − 1 = 1 type. 10
Fibonacci vector sequences generalize the original Fibonacci sequence. But other generalizations of the original Fibonacci sequence have been claimed, for example, the so-called higher order Fibonacci sequences, specifically, the tribonacci, tetrabonacci, etc. sequences. Such sequences are simpler to define and describe than the Fibonacci vector sequences that we have discussed here. However, they are not known to have a general geometrical representation. On the contrary, the Fibonacci vector sequences are represented geometrically by regular polygons of 2n + 1 vertices, which generalize the original sequence’s geometric representation by the regular pentagon. Therefore, we believe that the Fibonacci vector sequences are unique, the most natural and powerful generalization of the original Fibonacci sequence. We are amazed that the relationships between the original Fibonacci sequence, its one Golden Number, and the regular pentagon can be so readily generalized to relationships between the n-dimensional Fibonacci vector sequence, its n − 1 Golden Numbers and the regular polygon of 2n + 1 vertices. We are astounded that a relationship between such simple sequences of numbers and the lengths of polygon diagonals should even exist.
REFERENCES
References [1] Bjarne Junge and V. E. Hoggatt, Jr., “Polynomials Arising from Reflections across Multiple Plates,” Fibonacci Quarterly, 11:3, 285-291 (1973). [2] George N. Raney, “Generalization of the Fibonacci Sequence to n Dimensions,” Can. J. Math., 18, 332-349 (1966).
AMS Classification Number: 11B39
11
FIGURE
Figure 1: One side and several selected diagonals of a regular (2n + 1)-sided polygon.
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