Bridges Baltimore, July 2015

Möbius-Twisted Turk’s Head Knots

Carlo H. SéquinUniversity of California, Berkeley

Lorenzo Larrucea Sierra Valley Ranch, Calpine, CA

My Co-Author: Lorenzo Larrucea

Photo by Marry Williams Hyde

Some Bosals by Lorenzo Larrucea

Noseband portion of the hackamore headstall

Bosal Hackamore (Headgear)

Close-up of some Woven Parts

Rawhide

Latigo Leather

Turk’s Head Knots

p = 6, b = 7 p = 5, b = 11 p = 5, b = 4

p: parts (strands) b: bights (“bays”)

It should be woven from a single filament (“thong”).

Single-Thong Conditions

For a “plain” or “tabby” weave:

Parts and Bights must be mutually prime.

If they have a common divisor, knot will have more than one separate loop.

p = 6

b = 32

Yellow loop

Orange loop

The Challenge:

Use this kind of weaving to make a Möbius band!

Transfer weave pattern from one surface to the other.

This is single-sided !

New Constraints: 6, 31 are definitely mutually prime.

But if we take a 2-colored braid of p=6 and b=31and join its ends after a 180 flip,we obtain a nice color-consistent match,showing that this is a 2-thong Möbius band!

p = 6

b = 31

Yellow thong

Orange thong

After a lot of hard work . . .

. . . we figured out the rules:

Möbius Twist Modifies Rules: For a plain (“tabby”) woven braid, p must be even

to give an over-under pattern that looks the same on both sides.

Möbius twist reverses left-to-right pattern of strands; this must be compensated for:half a braid-period (p bights) also reverses strand pattern.

To make a 1-thong knot: Select mutually prime numbers; then subtract or add a half-period worth of bights (p/2).

Some # of bights that work for various braid widths:

2 parts: all even # of bights

4 parts: 1+2, 3±2, 5±2, 7±2 … all odd # of bights

6 parts: 2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40 . . .

8 parts: 1+4, 3+4, 5±4, 7±4 … all odd # of bights

Proof of Concept

Paper model of a 6-part, 16-bight Möbius knot.

Thong longitudinally follows a rainbow sequence to make it easily traceable (once around hue circle).

p = 6

b = 16

b = 2.5p + 1

How to Make a Woven Möbius Knot?

Now that we have figured out the math, how do we actually construct such a knot

out of a single thong of rawhide or leather ?

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What Do Cowboys Do? Follow instructions or learned rules:

Robert Woolery: “Braiding Rawhide Horse Tack”

Other Types of Instructions

A gizmo that may help:

Application to a Möbius Band ?

This does not work for the single-sided Möbius band!

If you tried to build the knot against a solid surface, then the braid would be encapsulated on both sides.

You could not access the woven band.

There is no room to remove any temporary pegs.

“Substitution Method” Make a braid from separate strands;

Twist and connect strands temporarily;

Thread a long thong, replacing temporary place-holders.

4-part, 15-bight

Front/Back Pattern for Woven Leather There is one other issue:

leather strips do have two different sides.

In a cylindrical braid you can have the shiny side on the outside, and the fuzzy side on the inside.

In a Möbius band both “sides” must look the same. Thus we need to see 50% of both sides. Here is a pattern that works:

More constraints for the p-b-math ? YES: p must be even.

Strands must not “flip sides” at junction !

Thong turns sides each time it travels around the loop.

A Nice Triply-Twisted Knot

6 parts

34 bights

Paper-Strip Knots Flip thong at braid edge, rather than bending it!

Thong always reverses side at the braid edge.

4-part, 21-bight 8-part, 31-bight

Let’s Try to Actually Weave One … Start with many shorter paper strips, suitably colored!

Then twist the braided ribbon and join its ends …

Needs careful pre-calculation of the positions of the differently colored strips.

Plenty of opportunities to make mistakes!

Result

8-part, 31-bight Möbius knot(in Art Exhibit)

Scalloped Paper Braid Models

Scallop type (suitable for leather): b = 1.5p +1rainbow colored thong, with grey/white backside.

p = 14, b = 22 p = 16, b = 25 p = 16, b = 25

Future Work

Explore different “Interweaves”

for instance, this type of twill weave,using an – over – over – under – under – pattern:

How does this affect the p-b-math on a Möbius band?

Twill-Woven Möbius Knot

Michel Since, http://charles.hamel.free.fr/knots-and-cordages/

An Even Bigger Challenge:

Can we weave a Klein bottle

with a single thong?

Reins Made by Lorenzo Larrucea

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