Math (Несколько текстов для зачёта), страница 4

(F5a) {w[sub 1] + w[sub 2] + ... + w[sub 2n - 1] = w[sub 2n]/k {w[sub 2] + w[sub 4] + ... + w[sub 2n] = w[sub 2n + 1] - 1/k

These extensions of Fibonacci properties convince us that these sequences are special and are worthy of further investigation. We can prove them by modifying the standard proofs of statements (F1) to (F5) for the Fibonacci sequence. See, for example, Hoggatt (1969). As an example, we prove property (F3) for the generalized sequences. That is, we prove

(F3a) w[sub n, sup - 2] - w[sub n - 1] x w[sub n + 1] = (-1)[sup n + 1],

for all integral n is greater than or equal to 1, where W[sub n + 1] = k x w[sub n] + w[sub n - 1], using mathematical induction.

We first note that when n = 1, the identity holds, since w[sub 1, sup 2] - w[sub 0] x w[sub 2] = 1 - 0 = (-1)[sup 2]. We assume that statement (F3a) is true for a particular value of n. Adding the quantity k x w[sub n] x w[sub n + 1] to each side of statement (F3a) and simplifying give

[Multiple line equation(s) cannot be represented in ASCII text]

Thus, statement (F3a) holds for n + 1 as well. The proof is complete.

NEARLY GOLDEN RATIOS

The golden ratio is the number

phi = 1 + Square root of 5/2

= 1.6180339887....

It arises in a variety of geometric contexts as a length or a ratio of lengths. See, for example, section 4 of Hoggatt (1969). We focus on a surprising property of the number phi, namely,

(F6) 1/phi = phi - 1,

and two connections between phi and the Fibonacci sequence,

(F7) [Multiple line equation(s) cannot be represented in ASCII text]

and

(F8) phi[sup n] = F[sub n] x phi + F[sub n - 1].

Property (F6) states that phi and its reciprocal differ only by 1, an integer, even though each is an irrational number with nonrepeating, nonterminating decimal expansion. An impressive way to demonstrate this property to students is to ask them to enter phi as

1 + Square root of/2

in their calculators, then use the reciprocal key. The decimal part does not change. Properties (F7) and (F8) hint at a complex intertwining of the golden ratio and the Fibonacci sequence. In particular, property (F7) tells us that the ratios of successive Fibonacci numbers,

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21,...,

approach the golden ratio phi. Property (F8) relates that every positive integral power of phi is a multiple of phi plus a constant, and these constants come from the Fibonacci sequence. For example,

phi[sup 2] = 1 x phi + 1, phi[sup 3] = 2 x phi + 1, phi[sup 4] = 3 x phi + 2,....

Let us see how properties (F6) to (F8) relate to the generalized Fibonacci sequences. We begin by considering how to verify property (F7). Since the terms of the Fibonacci sequence are defined by F[sub n + 1] = F[sub n] + F[sub n-1], the ratios of successive terms in the Fibonacci sequence are given by the equation

(1) F[sub n + 1]/F[sub n] = F[sub n] + F[sub n - 1]/F[sub n] = 1 + F[sub n - 1]/F[sub n],

for integral n is greater than or equal to 1. As n gets larger and larger, the ratio on the left-hand side of equation (1) approaches a limit, Which we call r. A calculus class can derive a proof. That is,

(2) F[sub n + 1]/F[sub n] arrow right r

as n arrow right Infinity. Then the ratio on the far right-hand side of equation (1) must approach 1/r, since it is a ratio of Fibonacci numbers in the reverse order of those on the left-hand side of equation (1). In other words,

(3) F[sub n - 1]/F[sub n] arrow right 1/r

as n arrow right Infinity. Putting equations (2) and (3) together with equation (1), we get

(4) [Multiple line equation(s) cannot be represented in ASCII text]

as n arrow right Infinity. It follows that r[sup 2] = r + 1 , or r[sup 2] - r - 1 = 0. By using the quadratic formula to solve this last equation, we arrive at the positive value of

r = 1 + Square root of 5/2

= phi,

the golden ratio. Equation (4) actually proves property (F6), since now we know that phi = r.

What happens if we repeat this process with any of the generalized Fibonacci sequences? We first consider some examples. Using the entries in table 1 when k = 2, we see that the successive ratios of this generalized Fibonacci sequence are

When k = 3, the ratios are

2/1, 5/2, 12/5, 29/12, 70/29, 169/70, 408/169,...,

with decimal approximations

2, 2.5, 2.4, 2.416, 2.41379..., 2.41428..., 2.41420..., ....

When k = 3, the ratios are

3/1, 10/3, 33/10, 109/33, 360/109, 1189/360, 3927/1189,...,

with decimal approximations

3, 33, 3.3, 3.30, 3.30275..., 3.3027, 3.30277...,....

Each sequence of ratios seems to be approaching a definite number. What numbers are they?

We can proceed in a manner analogous to the way that we derived equation (4). In the case in which k = 2, we have

(5) w[sub n + 1]/w[sub n] = 2 x w[sub n] + w[sub n - 1]/w[sub n] = 2 + w[sub n - 1]/w[sub n].

If the ratios on the left-hand side are converging to r, then the ratios on the far right-hand side of this equation are approaching 1/r. As n arrow right Infinity, equation (5) becomes

r = 2 + q/r.

Solving for r, we find that the positive value for r is

2 + Square root of 2[sup 2] + 4/2 = 1 + Square root of = 2.4142135624....

When k = 3, equation (5) becomes

r = 3 + 1/r,

with positive solution

r = 3 + Square root of 3[sup 2] + 4/2

= 3 + Square root of 13/2

= 3. 3027756377....

In general, for the sequence w[sub n] defined by w[sub n + 1] = k x w[sub n] + w[sub n - 1], we find that the ratios of successive terms in a generalized Fibonacci sequence approach r[sub k], which is the solution of

(F7a) r[sub k] = k + 1/r[sub k].

These numbers, which we call nearly golden ratios, are given by the formula

r[sub k] = k + Square root of k[sup 2] + 4/2.

Just like the golden ratio, phi, each of the numbers r[sub k] differs from its reciprocal by an integer because (F7a) can be rewritten as

1/r[sub k] = r[sub k] - k.

Letting k run through all nonnegative integers, we have a complete list of positive real numbers whose reciprocals have the same decimal part as the numbers themselves, resulting in an interesting exercise for students. The proof begins with equation (F7a). Table 2 gives some examples.

What about property (F8)? Its more general form is

(F8a) r[sup n, sub k] = w[sub n] x r[sub k] + W[sub n -1].

I leave it to the reader to verify the proof using induction.

CONCLUSION

This brief introduction is not nearly the whole story. We do not know the whole story about the Fibonacci sequence itself, so how could we give a complete account of these Fibonacci-like sequences. Properties (F1) through (F8) are merely a sampler of some of the well-known Fibonacci properties, identities, and connections that the newer Fibonacci-like sequences satisfy. I hope that you and your students are motivated to set off on the trail of more.

TABLE 1 Generalized Fibonacci Sequences

Legend for Chart:

A - k

B - The First Few Terms of w[sub n]

A B

2 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741,...

3 0, 1, 3, 10, 33, 109, 360, 1189, 3927, 12 970,...

4 0, 1, 4, 17, 72, 305, 1292, 5473, 23 184, 98 209,..

5 0, 1, 5, 26, 135, 701, 3640, 18 901, 98 145,...

TABLE 2 The First Four Nearly Golden Ratios and Their Reciprocals

Legend for Chart:

A - k

B - r[sub k]

C - 1/r[sub k]

A B C

1 1.6180339887 ... 0.6180339887 ...

2 2.4142135623 ... 0.4142135623 ...

3 3.3027756377 ... 0.3027756377 ...

4 4.2360679775 ... 0.2360679775 ...

5 5.1925824035 ... 0.1925824035 ...

GROUP SYMMETRIES CONNECT ART AND HISTORY WITH MATHEMATICS

Source: Mathematics Teacher, May2000, Vol. 93 Issue 5, p364, 7p, 6 diagrams, 29c

Author(s): Natsoulas, Anthula

THROUGHOUT HISTORY, different cultures have produced designs to be used as ornamentation, as part of ceremonies, and as religious symbols. Many of these designs are mathematical in nature, and their bases are often the transformations of reflection and rotation in the plane. The images form groupings that appear to have an underlying unity. Thus, history and art merge to create a medium through which students can study the concrete operations of reflection and rotation in the plane, as well as the more abstract concept of symmetry groups. The resulting patterns give students a sense of the potential for creativity inherent in mathematics. Exploring group symmetries within the context of such designs furnishes enriching experiences, connects art and history to mathematics, enhances the understanding of transformations in the plane, and shows the common underlying structure of algebra and geometry. Students should have the opportunity to see connections within mathematics and between mathematics and the various arenas of human activity and should develop an understanding of the types of reasoning that form the basis of mathematical thought.

All cultures participate in the six mathematical activities of counting, locating, measuring, designing, playing, and explaining (Bishop 1988); but designing results in some of the richest and most diverse outcomes. The beautiful designs created by different cultures mirror the uniqueness of their histories. Peoples of the Eastern and Western worlds have used mathematical ideas to create patterns in woven fabrics; ornamentation for religious objects and places of worship; and adornment of the walls, floors, and ceilings of the homes of nobles. A significant amount of mathematics, including the principles of symmetrical relationships, is implicit in such designs.

This article focuses on two types of symmetries--rotation and reflection, their underlying structure as a mathematical group, and their presence in the designs of diverse cultures. Patterns created by applying these symmetry operations offer students a visual image of closure, identity, inverse, and associativity, which form the axiomatic basis of algebra. Through patterns, this article intuitively develops the concept of symmetry groups and gives formal definitions of rotation and reflection symmetry and symmetry groups.

The design examples in this article focus primarily on those of Cyprus and Ethiopia, two nations whose mathematical art is not well known. The mosaics of Cyprus, typical of those found throughout the Roman world, date to between the fourth and eighth centuries C.E. and contain many intricate geometric patterns. It is believed that at one time designs for mosaics were collected in pattern books.

The form of Christianity introduced in Ethiopia in the first half of the fourth century and the art forms that developed from it became an integral part of the lives of its people. The Ethiopians developed elaborately designed crosses that they used both as jewelry and in religious processions. In the town of Lalibela, an important center of medieval Ethiopia, several rock-hewn churches built during the thirteenth century include geometric patterns.

REFLECTION AND ROTATION SYMMETRY

A symmetry is defined to be a motion of an object such that the appearance of the object is unchanged. A reflection symmetry is determined by a line, called the line of reflection, through which the original object is reflected. For each point of the original object, its distance to the line is the same as the distance of its corresponding image point. A rotation symmetry is determined by a rotation of the object around a fixed point called the rotocenter. The amount of rotation can be expressed as a fraction of a full turn or by the degrees of rotation in a counterclockwise direction.

Figure 1 includes a range of designs from Ethiopia and Cyprus that display different kinds of symmetry. The teacher can ask students to group those items that appear to have the same kinds of symmetry. A set of objects that have the same kinds of symmetry belongs to the same symmetry group. Thus, in figure 1, items (a) and (d) both belong to the same symmetry group, since rotations of 180 degrees or reflections around a vertical or horizontal line through the center return the design to its original appearance. Similarly, the interior part of the cross in item (e) and the circular portion of item (f) belong to the same symmetry group, since both exhibit 90 degree rotation symmetry. In like manner, the two designs enclosed within the circles in item (c) belong together, since both exhibit 60 degree rotation symmetry. The reader should explore the various reflection and rotation symmetries of item (b).

The Ethiopian cross, excluding its base, in figure 2 has both reflection and rotation symmetry. It contains four lines of reflection. If the figure is rotated through a one-quarter turn, a one-half turn, or a three-quarter turn--or equivalently, 90 degrees, 180 degrees, and 270 degrees, respectively--the appearance of the object remains unchanged. These symmetries are shown in figure 3 with a second Ethiopian cross, again excluding the base shown at the top of the figure. Since the figures are hand carved, the curved lines may not all line up precisely, but the artist clearly had such symmetries in mind when creating the figure.

The mosaic design shown in figure 4 (p. 366) is from Kourion in Cyprus; it contains the same symmetries as the Ethiopian cross. Students can test the rotation symmetries with a piece of tracing paper on which a coordinate axis is drawn or with two overhead transparencies. They can place the origin at the rotocenter on top of a copy of the design, trace an outline of one of the arms, and rotate the paper or transparency to show the symmetry.

ROTATION AND REFLECTION SYMMETRIES IN THE SQUARE