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New Visual Perspectives on Fibonacci Numbers

Krassimir Atanassova, Vassia Atanassova, Anthony Shannon, and John Turner
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Charles Ashbacher
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No more solid evidence exists concerning the incredible complexity of the integers and the remarkable ingenuity of mathematicians than the continued finding of new results in the series of Fibonacci numbers. From the simple definition of F(0) = 0, F(1) = 1 and F(n) = F(n-1) + F(n-2) for n > 1, books have been written on the formulas that they satisfy and the many places in nature where they appear. The authors of New Visual Perspectives on Fibonacci Numbers define several modifications of the basic sequence and then pursue the consequences. As the title implies, they often rely on diagrams to demonstrate their results.

In the first sections, integer sequences called 2-Fibonacci sequences are defined. Several different forms are used, but the basic definition is to use two sequences and interlace the terms. For example, starting with the initial terms a(0) = c, b(0) = d, a(1) = e and b(1) = f, one definition of the sequences is

a(n+2) = b(n+1) + b(n) n ³0
b(n+2) = a(n+1) + a(n) n ³ 0

There are many different consequences of these definitions, and they prove many theorems, although obviously there are an enormous number of possibilities. While the sequences are interesting, they are not all that new, as I have seen similar sequences over the past several years in material originating from Florentin Smarandache.

In section 2, weighted binary trees where some form of Fibonacci process is used to weight and place the nodes are described. Connections are made between the trees and Gray codes as well as other types of recurrence trees. Once again interesting, but not extensively so.

The real excitement in the book starts on the first page of part B and continues to the end of the book. Three sets of three-dimensional vectors are defined at the beginning of part B:

The nth Fibonacci vector FV(n) = (F(n-1), F(n), F(n+1))

The nth Lucas vector LV(n) = (L(n-1), L(n), L(n+1))

The nth generalized Fibonacci vector GV(n) = (G(n-1), G(n), G(n+1))

Geometric consequences of these definitions lead to the Fibonacci Honeyomb Plane, where a plane is partitioned into hexagons defined by integer nearest neighbors, trigonometry in the Fibonacci plane and many other vector sequences that can be generated. This was mathematics that was new to me, and the authors presented it very well. Their results were easy to follow, and like all good mathematical expositions, future directions were clear, even though they were not necessarily explicitly stated.

A goldpoint is defined as one that forms a golden section in a segment. The sets of goldpoints defined by a set of line segments were then used to construct goldpoint geometries. Forming new figures from the goldpoints of a geometric figure opens up a whole new area in the construction of new figures, including fractals. It is also possible to tile the plane using figures with goldpoints, and the authors demonstrate many such tessalations. As I read this section, I thought of many additional figures that could be drawn by performing slight modifications of their basic definitions. Although it is not likely, I hope that someday I can find the time to program the generation of the figures.

In this book, the authors explored many applications of the Fibonacci-type sequences that are new and point the way to many additional lines of study. Their ideas were original and well-described and I recommend this book to anyone interested in iterative processes on integers.

Charlie Ashbacher ( is the principal of Charles Ashbacher Technologies, a company that offers state of the art computer training. He is also an adjunct instructor at Mount Mercy College in Cedar Rapids, Iowa, and at the end of this academic year, he will be three courses short of having taught every class in the math and computer science majors. A co-editor of the Journal of Recreational Mathematics, he is the author of four books in mathematics and one in computer programming.

  • Number Theoretic Perspectives — Coupled Recurrence Relations:
  • Introductory Remarks by the First Author
  • The 2–Fibonacci Sequences
  • Extensions of the Concepts of 2–Fibonacci Sequences
  • Other Ideas for Modification of the Fibonacci Sequences
  • Number Theoretic Perspectives — Number Trees:
  • Introduction — Turner's Number Trees
  • Generalizations Using Tableaux
  • On Gray Codes and Coupled Recurrence Trees
  • Studies of Node Sums on Number Trees
  • Connections with Pascal–T Triangles
  • Geometric Perspectives — Finonacci Vector Geometry:
  • Introduction and Elementary Results
  • Vector Sequences from Linear Recurrences
  • The Fibonacci Honeycomb Plane
  • Fibonacci and Lucas Vector Polygons
  • Trigonometry in the Honeycomb Plane
  • Vector Sequences Generated in Planes
  • Fibonacci Tracks, Groups, and Plus–Minus Sequences
  • Geometric Perspectives — Goldpoint Geometry:
  • On Goldpoints and Golden-Mean Constructions
  • The Goldpoint Rings of a Line-Segment
  • Some Fractals in Goldpoint Geometry
  • Triangles and Squares Marked with Goldpoints
  • Plane Tessellations with Goldpoint Triangles
  • Tessellations with Goldpoint Squares
  • Games with Goldpoint Tiles