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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 ?
?
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
?