Symmetric Bends: How to Join Two Lengths of Cord (K & E Series on Knots and Everything, Vol. 8)

A bend is a knot securely joining together two lengths of cord (or string or rope), thereby yielding a single longer length. There are many possible different bends, and a natural question that has probably occurred to many is: is there a "best" bend and, if so, what is it?" Most of the well-known bends happen to be symmetric - that is, the two constituent cords within the bend have the same geometric shape and size, as well as an interrelationship. Such "symmetric bends" have great beauty, especially when the two cords bear different colours. Moreover, they have the practical advantage of being easier to tie (with less chance of error), and of probably being stronger, since neither end is the weaker. This book presents a mathematical theory of symmetric bends, together with a simple explanation of how such bends may be invented. Also discussed are the additionally symmetric "triply symmetric" bends. Full details, including colour pictures, are given of the "best 60" known symmetric bends, many of which were created by those methods of invention. This work will appeal to many - mathematicians as well as non-mathematicians interested in beautiful and useful knots. Table of Contents......Page 12 Preface......Page 8 1.1 Introduction......Page 16 1.2 Summary......Page 21 2.1 The Elementary and Other Well-known Symmetric Bends......Page 24 2.2 Three Classes of Symmetric Diagrams......Page 29 3.1 `Necessity': Geometric Theory of Symmetric Bends and their Aspects......Page 32 3.2 Planar Representations of Symmetric Bends......Page 38 3.3 α and β Square Lattice Diagrams for Symmetric Bends......Page 41 4.1 Mirror Image, Colour Interchange and Reverse......Page 44 4.2 Topological Considerations......Page 45 4.3 The CHAMELEON......Page 50 4.4 The Marginal Knots of a Symmetric Bend......Page 53 4.5 Reverse and Mixed Bends, and a Theorem Relating their Symmetries......Page 57 5.1 γ Diagrams for ⊕ Symmetric Bends......Page 64 5.2 Additional Symmetry: (γ , α ), (γ , β) and (γ ,γ) Diagrams......Page 67 5.3 Practical Reverse Invariance......Page 75 5.4 Triple Symmetry : the Geometry of |Rl+| ⊕ Symmetric Bends......Page 76 5.5 Single Colour Symmetric Bends......Page 79 6.1 Introduction......Page 82 6.2 Diagrams of the〈60〉......Page 91 6.3 Colour Plates of the〈60〉......Page 114 6.4 Notes on the〈60〉......Page 132 6.5 Two Conjectures......Page 148 7.1 Loops, Knots and Links......Page 150 7.2 A Remarkable `Almost Symmetric' Bend, and Some Related Hitches......Page 152 8.1 Diagram Invention......Page 158 8.2 An α or β , a y and a (γ , β) Outline......Page 163 8.3 Alternative ad hoc Method......Page 167 The International Guild of Knot Tyers......Page 172 Bibliography......Page 174 Index......Page 176 A bend is a knot securely joining together two lengths of cord (or string or rope), thereby yielding a single longer length. There are many possible different bends, and a natural question that has probably occurred to many "Is there a 'best' bend and, if so, what is it?"Most of the well-known bends happen to be symmetric -- that is, the two constituent cords within the bend have the same geometric shape and size, and interrelationship with the other. Such 'symmetric bends' have great beauty, especially when the two cords bear different colours. Moreover, they have the practical advantage of being easier to tie (with less chance of error), and of probably being stronger, since neither end is the weaker.This book presents a mathematical theory of symmetric bends, together with a simple explanation of how such bends may be invented. Also discussed are the additionally symmetric 'triply symmetric' bends. Full details, including beautiful colour pictures, are given of the 'best 60' known symmetric bends, many of which were created by these methods of invention.This work will appeal to many -- mathematicians as well as non-mathematicians interested in beautiful and useful knots. A bend is a knot securely joining together two lengths of cord (or string or rope), thereby yielding a single longer length. There are many possible different bends, and a natural question that has probably occurred to many is: “Is there a ‘best'bend and, if so, what is it?”Most of the well-known bends happen to be symmetric — that is, the two constituent cords within the bend have the same geometric shape and size, and interrelationship with the other. Such ‘symmetric bends'have great beauty, especially when the two cords bear different colours. Moreover, they have the practical advantage of being easier to tie (with less chance of error), and of probably being stronger, since neither end is the weaker.This book presents a mathematical theory of symmetric bends, together with a simple explanation of how such bends may be invented. Also discussed are the additionally symmetric ‘triply symmetric'bends. Full details, including beautiful colour pictures, are given of the ‘best 60'known symmetric bends, many of which were created by these methods of invention.This work will appeal to many — mathematicians as well as non-mathematicians interested in beautiful and useful knots. A bend is a knot securely joining together two lengths of cord (or string or rope), thereby yielding a single longer length. There are many possible different bends, and a natural question that has probably occurred to many is: "Is there a 'best' bend and, if so, what is it?"Most of the well-known bends happen to be symmetric - that is, the two constituent cords within the bend have the same geometric shape and size, and interrelationship with the other. Such 'symmetric bends' have great beauty, especially when the two cords bear different colours. Moreover, they have the practical advantage o