New Visual Perspectives on Fibonacci Numbers

by Atanassov, Krassimir T.; Atanassova, V.; Shannon, A. G.; Turner, J. C.; Atanassov, Krassimir T.

• ISBN13: 9789812381347
• ISBN10: 9812381341
• Format: Paperback
• Copyright: 2002-11-01
• Publisher: World Scientific Pub Co Inc

Summary

This book covers new ground on Fibonacci sequences and the well-known Fibonacci numbers. It will appeal to research mathematicians wishing to advance the new ideas themselves, and to recreational mathematicians, who will enjoy the various visual approaches and the problems inherent in them.There is a continuing emphasis on diagrams, both geometric and combinatorial, which helps to tie disparate topics together, weaving around the unifying themes of the golden mean and various generalizations of the Fibonacci recurrence relation.Very little prior mathematical knowledge is assumed, other than the rudiments of algebra and geometry, so the book may be used as a source of enrichment material and project work for college students. A chapter on games using goldpoint tiles is included at the end, and it can provide much material for stimulating mathematical activities involving geometric puzzles of a combinatoric nature.

Table of Contents

Introduction
Foreword
Preface
Number Theoretic Perspectives
Coupled Recurrence Relations
Introductory remarks by the first author	p. 3
The 2-Fibonacci sequences	p. 7
Extensions of the concepts of 2-Fibonacci sequences	p. 29
Other ideas for modification of the Fibonacci sequence	p. 39
Number Trees
Introduction - Turner's Number Trees	p. 53
Generalizations using tableaux	p. 57
On Gray codes and coupled recurrence trees	p. 63
Studies of node sums on number trees	p. 71
Connections with Pascal-T triangles	p. 75
Geometric Perspectives
Fibonacci Vector Geometry
Introduction and elementary results	p. 85
Vector sequences from linear recurrences	p. 95
The Fibonacci honeycomb plane	p. 107
Fibonacci and Lucas vector polygons	p. 119
Trigonometry in the honeycomb plane	p. 123
Vector sequences generated in planes	p. 137
Fibonacci tracks, groups, and plus-minus sequences	p. 153
Goldpoint Geometry
On goldpoints and golden-mean constructions	p. 183
The goldpoint rings of a line-segment	p. 205
Some fractals in goldpoint geometry	p. 215
Triangles and squares marked with goldpoints	p. 229
Plane tessellations with goldpoint triangles	p. 245
Tessellations with goldpoint squares	p. 269
Games with goldpoint tiles	p. 293
Index	p. 309
Table of Contents provided by Blackwell. All Rights Reserved.