Symmetric Bends: How to Join Two Lengths of Cord
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: OC Is there a OCybestOCO bend and, if so, what is it?OCOMost of the well-known bends happen to be symmetric OCo that is, the two constituent cords within the bend have the same geometric shape and size, and interrelationship with the other. Such OCysymmetric bendsOCO 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 OCytriply symmetricOCO bends. Full details, including beautiful colour pictures, are given of the OCybest 60OCO known symmetric bends, many of which were created by these methods of invention.This work will appeal to many OCo mathematicians as well as non-mathematicians interested in beautiful and useful knots."
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Chapter 1 INTRODUCTION SUMMARY
THE ELEMENTARY SYMMETRIC BENDS
GEOMETRY PLANAR REPRESENTATIONS
Chapter 4 TOPOLOGICAL CONSIDERATIONS AND A THEOREM
Chapter 5 PRACTICAL CONSIDERATIONS TRIPLE SYMMETRY
SIXTY SYMMETRIC BENDS
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alternative knot Asher Ashley Ashley’s aspect Symmetry aspect bend diagram CARRICK bend central inversion centre point Chapter Col(B colour interchange colour plate consider COW hitch crossover curve dark and light deﬁned diagram in Figure end designations example ﬁnal ﬁrst four ends free ends geometric hence hitch invention knot diagrams Knotting Matters lengths of cord loop loose loosened Mandeville marginal knots mathematical knot mirror congruent mirror image mixed bends mixed e-diagrams mixed lanyard bend non-rewoven Odd triple Odd orthogonal outline OVERHAND knot overhand SB’s pairs practical properties pure lanyard bend Q SB Q SB’s REEF knot Rev(B reverse invariant rightover rotation Section 5.4 SHEET bend speciﬁc spliced square lattice standing and free standing axis standing ends symmetric bends symmetric diagrams Symmetry aspect Symmetry symmetry diagonal tensioned THIEF knot topological transformation topologically equivalent triple Odd triple triple symmetry aspect triply symmetric tying method types walk WHATNOT yields