Author Topic: Lots and lots of Turk's Heads  (Read 1130 times)

clsn

  • Jr. Member
  • **
  • Posts: 16
Lots and lots of Turk's Heads
« on: April 10, 2018, 01:05:50 AM »
OK.  So I have the problem of not being a regular reader here or very much in touch with the user community, so I don't know if the stuff I've been working on (an awful lot) is original or not.  I thought I'd ask here, and also find out if there's interest in me sharing this.

I've been working a lot with Turk's Head Knots (THKs).  This, I understand, is not unusual.  Nor is tying THKs of arbitrary dimensions.  You can easily plot the over-under diagram of a knot on graph paper, or use programs like GridMaker to generate a nifty diagram that you can wrap around a mandrel, drive some pins, and off you go.  But I lack the patience for working with tools like that all the time, and I've been working on tying them in-hand (actually, around my hand).

I think I have found some math and some techniques for in-hand tying that might be original.  Certainly they are powerful.  Here is a very incomplete list of knots I can tie in-hand, with no guides or diagrams or references, just looping around my hand without much trouble:

  • all 3-lead knots, obviously (braiding)
  • 5x2, 5x3, 5x4(built from 3x2), 5x6(built from 3x4), 5x7, 5x8, 5x9(built from 3x5), etc: all 5-lead knots
  • 7x2, 7x3(built from 3x1), 7x4(built from 3x2, building the "other way"), 7x5, 7x6(built from 5x4), 7x8(built from 5x6), 7x9, 7x10(built from 3x4 the "other way"), 7x11(built from 3x5 the "other way"), 7x12, 7x13, etc: all 7-lead knots
  • 9x2, 9x4(built from 5x2), 9x5(built from 5x3), 9x7, 9x8(built from 7x6), 9x10(built from 7x8), 9x11, 9x13(build from 5x7), 9x14(build from 5x8), 9x16, 9x17(built from 7x13), etc: all 9-lead knots
  • 11x2, 11x3, 11x4(built from 5x2), 11x5(built from 7x3), 11x7(built from 5x3 the "other way"), 11x8, 11x9, 11x10(built from 9x8), 11x12(built from 9x10), 11x13, 11x14, ... (I'm in the middle of working on these)
  • All 2-bight knots (odd leads, of course), and thus anything that can be built therefrom

I can do a lot of the even-lead ones as well, just haven't spent as much time on them (and the 2-leads are just multiple overhand knots, the 4-leads build from the 2-leads, and the 6-leads build from the 4-leads.  When you hit 8 leads, it gets a little more complicated, but I have a lot of those also)

In the list above, knots which are tied by tying a smaller knot and "building" it are so noted.  Turns out there are two options when "building" a knot, so some are done by using the option less chosen.  I stopped listing the knots above when the bights started to exceed twice the leads, since at that point it's just a repeating pattern and they're all tied essentially the same.

These techniques are straightforward, in a sense, by which I mean it isn't like I have completely novel and idiosyncratic ways to tie each and every one of these knots.  They follow patterns and are tied by applying the same techniques with different starts.  They are teachable techniques.

I have some math to go along with these, to predict how things will build by means of the various methods, building, etc, at least some of which is mathematically proven (and more is provable if I'd bother).  I'm working on some possibilities; it is possible I'll be able to tie all possible THKs by these methods if some of them generalize in ways I'm suspecting.

So... Is all this old news to folks out there?  "Oh yeah, everyone knows how to do Lx(kL+3) knots, why didn't you just look at this site?"  Or am I onto something that people would be interested in hearing about?  I need all the encouragement I can get.  Thanks!
« Last Edit: April 10, 2018, 01:38:36 PM by clsn »

knotsaver

  • Sr. Member
  • *****
  • Posts: 281
Re: Lots and lots of Turk's Heads
« Reply #1 on: April 10, 2018, 07:05:36 PM »
... so I don't know if the stuff I've been working on (an awful lot) is original or not.  I thought I'd ask here, and also find out if there's interest in me sharing this.

Hi clsn,
if you want to know if the stuff is original or not you have to publish it! :) Please do, I'm interested and curious (especially for the "in hand" tecnique).
"Something" was already said by Ashley and by Schaake and others, but please go ahead! ;)
...my 2 cents of encouragement ;)

Ciao,
s.

clsn

  • Jr. Member
  • **
  • Posts: 16
Re: Lots and lots of Turk's Heads
« Reply #2 on: April 13, 2018, 07:17:47 PM »
Hey, thanks for the response!  That's mainly why I posted this, to see if anyone out there would even care.

The technique is all "in-hand".  This morning I tied, in-hand, a 17x12 THK.  Really.  In one try.  (start with a 3x2, double it (following behind), then once more (so you've tripled it), then "build" in the usual way, going around yet one more time (not adding a new ply to the whole knot like before, this is just a single time around, like you would to make a 5x4) and then just stay between the parallels like on a normal build, just going around until you've finally split them all.  In the book I will/would be writing, I'll explain in a lot better detail than that).

I've been talking to a photographer friend who is willing to help me make this happen.  I think this weekend we'll try to shoot a show-off (teaser?) video showing me tying some unreasonably large number and variety of THKs (with somewhat speeded-up video), just to get across the notion that it's possible to sit down with a piece of string and hands and nothing else and in a couple of minutes get a 7x9 THK or something (I do those on the train without looking now, as a fidget).  The putative book will take way longer; it isn't even properly organized in my mind, much less written.  I've started writing the text, but I'm really long-winded and I keep thinking of more stuff to say.  I've also been working on a diagram showing all the THKs that can be tied and how to get to each one.  Maybe I'll post a piece of that.

Meanwhile, here's another rapid-fire, no-pictures explanation for you to play with: to tie any 2-bight THK (obviously, odd number of leads): spiral up your hand (or fingers, or any foundation really) one or more times.  Then spiral down at the same "rate", so crossovers with your original pass are in a vertical line (this is easier done than said: it's probably the most obvious way to do things when you think "spiral up and spiral down")  All crossings are over (isn't that easy?)  When you get to the bottom, cross over the short end and start back up, next to your original strand, going up alongside it.  All crossing are still over, including one more over the strand you've been following when you get to the top.  Now all you have to do is come back down. There is at this point basically one obvious place to go back down, starting off with an under-crossing, and if you stay in the same spiral, you'll find that you're alternating over-crossings and under-crossings, "climbing ladders" down until you get to the bottom.  This can make for some fearsomely wide THKs, especially if you build them to 4 bights.

Any other ideas?  Are there any dedicated THK-fanatic forums?

Thanks!!

knotsaver

  • Sr. Member
  • *****
  • Posts: 281
Re: Lots and lots of Turk's Heads
« Reply #3 on: April 16, 2018, 07:10:27 AM »

The technique is all "in-hand".  This morning I tied, in-hand, a 17x12 THK.  Really.  In one try.  (start with a 3x2, double it (following behind), then once more (so you've tripled it), then "build" in the usual way, ...

...I miss something, I don't see how the plies become bights: do you have to "braid" them?

Quote
Meanwhile, here's another rapid-fire, no-pictures explanation for you to play with: to tie any 2-bight THK (obviously, odd number of leads): spiral up your hand (or fingers, or any foundation really) one or more times...

that is a good method (you can find it shown for instance in Cyrus Day's Art of knotting & splicing) but  I think the advantage is to tie it directly around an object.

I look forward to watching your video.  ;)

Ciao,
s.

clsn

  • Jr. Member
  • **
  • Posts: 16
Re: Lots and lots of Turk's Heads
« Reply #4 on: April 16, 2018, 09:46:20 PM »
So it turns out I can tie *ALL* possible THKs this way, without pins or landmarks etc.  Not that I've tried all infinity of them, but if my math is correct I at least can (theoretically) tie every possible one with 100(!) or fewer leads and bights  (I've known this for a while now).  I don't have a mathematical proof (yet) that I can get to all of them, but I suspect one could be found.  I know for absolute certain that I have tied every one with 10 or fewer.

I'm working on a description, and a short video demonstration of a few (speeded up) should be forthcoming, when my editor has tweaked it a little more.

I feel like someone (apart from just me) should be really really excited about this...
« Last Edit: May 12, 2018, 12:15:45 AM by clsn »

Wed

  • Sr. Member
  • *****
  • Posts: 312
Re: Lots and lots of Turk's Heads
« Reply #5 on: May 12, 2018, 01:25:56 PM »
I didn't respond the first time, because I don't do many TH knots in hand. I usually tie a THK when I need to cover something. Generally, fitting bight pins is just too helpful not to do.

I developed a way to tie arbitrarily sized THK:s http://igkt.net/sm/index.php?topic=4610.0

As I think of it, the pattern is: "over one, under one". https://en.wikipedia.org/wiki/Plain_weave
As such, the barber pole could fill in the truly square THK:s that can't be done with the traditional THK.

And then on to "over two, under two"-weaves, that are pine apple knots

Satin weaves can be made with barber poles with multiple interweaves: http://igkt.net/sm/index.php?topic=6129.0

After that THK:s can be combined with other knots at the ends. Either tied in or with the same cord(s). Not to mention interesting patterns in the THK weave itself. But by this time, a predrawn (printed) pattern is nearly necessary.
« Last Edit: May 12, 2018, 01:32:18 PM by Wed »

clsn

  • Jr. Member
  • **
  • Posts: 16
Re: Lots and lots of Turk's Heads
« Reply #6 on: May 13, 2018, 05:52:28 AM »
Those are pretty cool ideas!  You're coming at it from a completely different perspective than I am, and solving different problems (which does not make your solutions in any way inferior).  You're looking at "how can I cover this thingy here with a pretty THK?" and coming up with ways to make one that fits.  The method in your first link basically boils down to "guess at some number of bights and leads, 'draw' a grid using your first pass, then follow the grid."  This has distinct advantages over the more "scientific" method of deciding on a knot, drawing the grid, and wrapping that around your foam core or whatever.  You have more flexibility and can intuit the shape of the final knot better; you're not limited to the strict 45? grid normal graph paper (or most graphing programs) would limit you to.  I think Ashley also mentioned something about using a first pass (of different string) to form a "diagram" or map for a THK, but that was, again, after working out the numbers in advance.  Your way of using it essentially to decide the knot you're tying is clever.

From what I can tell, your second link involves THKs with multiple strands (as well as various different weave patterns), which is also pretty neat, and you get some great patterns that way.

I was sort of taking a more mathematical approach, not a practical one (I'm not a really practical person, it seems.)  I've probably tied literally thousands(?) of THKs by now... and untied 99% of them.  I carry string around and tie THKs around my hands on the train and at work, some of them huge and elaborate... and then untie them and start again.  So I'm not focussing on actually creating something for a purpose, it would seem.  I was thinking of a "purist's" subset of THKs: single-stranded, over-one-under-one, same number of bights on both sides, etc.  Or another way, in the form of a French braid (over-1-under-1) that goes around in a circle.  And I was thinking, how can I tie these?  My motivation was more theoretical and mathematical, I guess: trying to see what could be done to form these particular knots, of various dimensions, but not necessarily for a particular purpose.

Let me see if I can explain what I have, in extremely broad strokes:

Basically, there are certain classes of knots you tie more or less "directly": all one-bight knots, all two-bight knots, all two-lead knots (multiple overhand), and all three-lead knots (braiding around).  Also two more classes that involve some "tricks", but can similarly be tied sorta-directly: Nx(kN+2) and Nx(kN-2) knots.  I have methods for tying these.

Once you have a THK, there are ways to "build" it into a higher-order one.  They're pretty well-known, at least in simple cases, but I don't think anyone else has worked them all out in the detail I have.  Each knot can be built in two directions, plus there are sort of higher-order builds (not the same as building and then building again).  Suffice to say, it turns out that with the building techniques and the classes of knots described in the last paragraph, you can get to *everything*.  I have a little program that lets me make a pretty table of the knots, with each one showing the knot it builds from and how... Here, I'll upload a small example.  The real one will be bigger and with better detail, but you can see in this one, each knot shows in color above it the knot it's built from.  The ones labeled in purple are in the "base classes," others have a label naming a smaller THK, and the color indicates how to build.  It works out, you'll see.

Wed

  • Sr. Member
  • *****
  • Posts: 312
Re: Lots and lots of Turk's Heads
« Reply #7 on: May 13, 2018, 08:12:23 AM »
The method in your first link basically boils down to "guess at some number of bights and leads, 'draw' a grid using your first pass, then follow the grid."
That's right. After a while, it's an educated guess though. The main advantage of course, is that I don't care about tying intermediate knots to get to the target.

Quote
I think Ashley also mentioned something about using a first pass (of different string) to form a "diagram" or map for a THK, but that was, again, after working out the numbers in advance.  Your way of using it essentially to decide the knot you're tying is clever.
It will also show if the final knot will work, or if my initial guess was wrong and the loop closes before I'm back at the beginning. Add to that, a rough estimate of how much cord is needed.

Quote
From what I can tell, your second link involves THKs with multiple strands (as well as various different weave patterns), which is also pretty neat, and you get some great patterns that way.
The barberpole isn't strictly speaking a THK. However it vill give the same weave, pattern, braid ... appearance. It is also among the simplest of knots to tie. Until you use the same colour of cord on all interweaves.

Quote
I was sort of taking a more mathematical approach, not a practical one (I'm not a really practical person, it seems.)  I've probably tied literally thousands(?) of THKs by now... and untied 99% of them.  I carry string around and tie THKs around my hands on the train and at work, some of them huge and elaborate... and then untie them and start again. 
That much labour is what I call practical

Quote
My motivation was more theoretical and mathematical, I guess: trying to see what could be done to form these particular knots, of various dimensions, but not necessarily for a particular purpose.
Just as I don't see the point of the knot unless it comes to use, which brings out the beauty of it. And both of us are happy.

Quote
I feel like someone (apart from just me) should be really really excited about this...
And that is why I responded this time. I'm excited about this. Even if my motivation differ from yours. I enjoy finding a "crack" for a knot in order to make it easy to tie out of a simple set of rules.

clsn

  • Jr. Member
  • **
  • Posts: 16
Re: Lots and lots of Turk's Heads
« Reply #8 on: June 01, 2018, 04:54:27 PM »
All right!  Finally got a small video together, which doesn't so much show a lot, but it more shows that it *can* be done.  It's just me tying a few knots in fast motion.  Posted on a friend's YouTube; he did the videography and editing:

https://youtu.be/xu_0oqBU-kM

I should come up with ways to get this kind of thing done more quickly and simply.

Alas, the obsession is fading for me, but I think I need to force myself to stay with it and finish writing up the book or whatever so that this can be shared with the world, in case it doesn't already have it.  Thanks for responses to remind me that there are folks out there who care!

Trespidian

  • Newbie
  • *
  • Posts: 1
Re: Lots and lots of Turk's Heads
« Reply #9 on: June 05, 2018, 09:30:46 PM »
Looking forward to hearing more of this -- and maybe a slower tutorial type video :)