1. Bridges Baltimore, July 2015 Möbius-Twisted Turk’s Head Knots Carlo H. Séquin University of California, Berkeley Lorenzo Larrucea Sierra Valley Ranch, Calpine, CA

2. My Co-Author: Lorenzo Larrucea Photo by Marry Williams Hyde

3. Some Bosals by Lorenzo Larrucea Noseband portion of the hackamore headstall

4. Bosal Hackamore (Headgear)

5. Close-up of some Woven Parts u Rawhide u Latigo Leather

6. 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”).

7. Single-Thong Conditions u For a “plain” or “tabby” weave: u Parts and Bights must be mutually prime. u If they have a common divisor, knot will have more than one separate loop. p = 6 b = 32 Yellow loop Orange loop

8. The Challenge: u Use this kind of weaving to make a Möbius band! u Transfer weave pattern from one surface to the other. This is single-sided !

9. New Constraints: u 6, 31 are definitely mutually prime. u But if we take a 2-colored braid of p=6 and b=31 and 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

10. After a lot of hard work...... we figured out the rules:

11. Möbius Twist Modifies Rules: u For a plain (“tabby”) woven braid, p must be even to give an over-under pattern that looks the same on both sides. u Möbius twist reverses left-to-right pattern of strands; this must be compensated for: half a braid-period (p bights) also reverses strand pattern. u To make a 1-thong knot: Select mutually prime numbers; then subtract or add a half-period worth of bights (p/2). u Some # of bights that work for various braid widths: l 2 parts: all even # of bights l 4 parts: 1+2, 3±2, 5±2, 7±2 …  all odd # of bights l 6 parts: 2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40... l 8 parts: 1+4, 3+4, 5±4, 7±4 …  all odd # of bights

12. Proof of Concept u Paper model of a 6-part, 16-bight Möbius knot. u Thong longitudinally follows a rainbow sequence to make it easily traceable ( once around hue circle ). p = 6 b = 16 b = 2.5p + 1

13. 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 ? ?

14. What Do Cowboys Do? u Follow instructions or learned rules: Robert Woolery: “Braiding Rawhide Horse Tack”

15. Other Types of Instructions u A gizmo that may help:

16. Application to a Möbius Band ?  This does not work for the single-sided Möbius band! u If you tried to build the knot against a solid surface, then the braid would be encapsulated on both sides. u You could not access the woven band. u There is no room to remove any temporary pegs.

17. “Substitution Method” u Make a braid from separate strands; u Twist and connect strands temporarily; u Thread a long thong, replacing temporary place-holders. 4-part, 15-bight

18. Front/Back Pattern for Woven Leather u There is one other issue: leather strips do have two different sides. u In a cylindrical braid you can have the shiny side on the outside, and the fuzzy side on the inside. u 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 ? u YES: p must be even. u Strands must not “flip sides” at junction ! u Thong turns sides each time it travels around the loop.

19. A Nice Triply-Twisted Knot u 6 parts u 34 bights

20. Paper-Strip Knots u Flip thong at braid edge, rather than bending it! u Thong always reverses side at the braid edge. 4-part, 21-bight 8-part, 31-bight

21. Let’s Try to Actually Weave One … u Start with many shorter paper strips, suitably colored! u Then twist the braided ribbon and join its ends … u Needs careful pre-calculation of the positions of the differently colored strips.  Plenty of opportunities to make mistakes!

22. Result u 8-part, 31-bight Möbius knot (in Art Exhibit)

23. Scalloped Paper Braid Models u Scallop type (suitable for leather): b = 1.5p +1 rainbow colored thong, with grey/white backside. p = 14, b = 22p = 16, b = 25

24. Future Work u Explore different “Interweaves” u 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?

25. Twill-Woven Möbius Knot Michel Since, http://charles.hamel.free.fr/knots-and-cordages/

26. An Even Bigger Challenge: Can we weave a Klein bottle with a single thong?

27. Reins Made by Lorenzo Larrucea ?