Knots, Part 2

In an earlier post, I talked about knots. And knots are entanglement, there is no doubt. They serve to bind, secure, to tie down, to hang up, and even to keep our shoes on.

In this post I will talk about knots as a way to entangle two threads. I will continue to use the planar method of showing knots, combined with precedence at crossover points. An over-under rule is used to keep the knots maximally entangled.

In addition, I will show how to draw knots using my drawing style, which is a little bit scratchboard-watercolor, a little bit woodcut, and a lot retro. You can find more of my style (and lots more knots) at my Pinterest artwork board.

The over-under rule characterizes one of the best ways to organize the making of a knot. In its simplest form, you can see a less confusing, more iconic representation.

This knot is a clover interleaved with a ring. The ancient name for the clover symbol is the Saint John's Arms. The clover is used to symbolize places of interest on a map, the command key on Macs, and cultural heritage monuments in Nordic countries, Estonia, and a few other places. This symbol has been around for at least 1500 years.

The other day while working on a complicated programming problem, and I drew such a clover absent-mindedly and realized suddenly that I could pass a ring through its loops, hence this figure. When you draw the clover as a knot, it is also called the Bowen knot.

It seemed like the simplest thing at the time. Then I tried to draw it in its current form: not so easy! After a few hours (off and on) with Painter yesterday I finally had this figure smoothed out in nice outlines. Today I shaded and colored it. Sure, maybe the purple is a bit much, but I like the simple forms and the way they intertwine.

After making this figure originally, I went back to my programming. But there was a nagging question in the back of my head. What was the simplest intertwined figure that had a twist in it? I had to think simple, so I drew an infinity as a twisted bit of rope.

Then I wondered how a ring might enter the picture. I tried one way and then it hit me: use the over-under rule.

This is the figure I ended up with. Now that's much simpler than the first, and iconic in its own way, I think. It could be a logo in an even simpler form. O-infinity? Well, there's nothing like a logo created for no particular reason!

But how are such knots created, really? Is there an easy way?

Start with a line drawing showing the paths of the two threads. This is how I started. I put them at an angle because I drew the oval first. This was a natural angle for me to draw it right-handed.

Then I turned the page and drew the infinity so that the oval passed through each of the figure-eight's loops.

It wasn't exactly symmetric. Though I do like symmetry, I like even more to make my drawings a bit imperfect to show that they are hand-drawn. If I were designing for a logo, though, I'm not sure I'd make the same choice.

Next I drew the figure again, but with an indication (by breaking the lines so they don't quite cross over each other) of which thread is on top and which crosses under.

Here is my first attempt.

But there is a basic flaw: if I were to grab the oval and pull it, it would easily come loose from the figure-eight! Needless to say this wasn't the knot I was looking for so I redrew it again using the tried-and-true over-under rule which states this: as you pass along a thread, it must pass first over and then under the other threads, alternating in succession.

Here is the result of redrawing it. As you can see, it has a much nicer integrity. It seems to be entangled properly.

So now I have a basic plan for the entanglement of the knot. Now I must plan to draw the knot using outlines for each thread. This means that each thread must really be two lines that are parallel to each other. I call this the schematic version.

I use the original line drawing as a guide and draw two lines parallel to the original line, one line on each side. Originally I worked in black ultra-fine Sharpie on thick 32# copy paper.

The wide lines drawing, as you can see, is getting a bit complicated. But fortunately I have a legend for which lines to draw in and which lines to erase: the second hidden-line diagram above.

I use this as a template so I can redraw the image, using only the new wide lines. With this I can create a hidden-line version of the wider knot. It is easy to accomplish this by placing the blank sheet over the original and using it as tracing paper.

Of course when I do this, I avoid drawing the centerline. This keeps the drawing simple. In this way, you can see that the centerline was a for-reference-only diagram for what follows.

Here is the wide hidden-line version. This one is much clearer and certainly much closer to what I was trying to create.

But it is a bit flat, like a road. And the crossings are really dimensionless.

I brought this into Painter and smoothed out the lines, making them a bit more consistent. Then I worked a bit of magic by using my woodcut style.

How do I do that?

I'm glad you asked! At each crossover, I draw three or four lines on the "under" sides of the crossover. Then I draw to create wedges of black that meet very close to the "over" lines. Finally I use a small white brush to sculpt the points of the wedges, making them very pointy.

This simulates what could be created using a V-shaped ductal tool with linoleum or wood.

Well, this process takes a bit of time. If you count, you can see I had to create about 40 wedges, sculpting each of them into a perfect line or curve. But I am patient.

Sometimes I widened the "under" lines to meet the outermost wedges. This makes a more natural-looking woodcut.

Finally, in Painter I use a gel layer and fill in color on top, filling in each area of the thread using a slightly different color.

This gives me the final result, a unified entanglement of two interesting threads! This result is quite similar to the scratchboard-watercolor look that I like. I used the same technique exactly to create the knot at the top of this post. In past posts, I have used this technique to create many illustrations, of course. I like this look because it's easy to print and it is good for creating logos.

For instance, if I take the plain wide line version and blacken the white background, I get a version that can be manipulated into a logo form. After that, I invert the colors of the image and that gives me a clean black logo on white. Then I use a layer in Screen mode to colorize the black segments of the threads.

Here is a logo version of the knot, expressed in colorful tones. But this won't do for O-infinity at all! It might easily be an O in purple and the figure-eight in navy blue. On black.

But that's not my idea of a good company name, so I will leave it like this!

There are plenty of styles for redrawing this knot that make interesting illustrations.

This one is not a knot, really. But it is an interesting redrawing of the figure.

This is called an inline treatment.

Remember the Neuland Inline font that was used for the movie Jurassic Park?

This figure can be used as the start of about 100 different illustrations, depending upon which crossings you want to black in or erase.

I tried several before I realized that it wasn't the direction I wanted to go with the logo.

Trial-and-error is often the way with creativity!

I have other knots I'd like to draw, but they certainly do take time! It's good to be drawing again.

Knots

Knots are entanglement. They serve to bind, to secure, to tie. But the complex bonds that they represent are more than just bonds and complexity. There is actually a science to them.

Their overlapping, interweaving character is what makes them secure. Because of it, they don't fall apart. If you take one strand, you can't pull it away because it's entangled in the knot. This is one simple definition of a knot: no one strand can be removed without untying it.

In mathematics, a knot is a loop. The loop is officially embedded in three-dimensional space, but you can also embed it in the plane, if you consider, in addition, the crossing.

The overhand knot shown above is actually a trefoil knot with one cut. It is crucial to realize that any knot can be untied if the strand is cut.

In the real world, knots may be composed of more than one loop as well. An instance of this is two loops interlocking. Or the small weave is another good example.

Knots are important to sailors because ropes tie all things on a ship. Sometimes, like in the figure eight bend knot which can connect two ends of rope, they are designed to be strong, and yet untied in an instance so the two ends of rope can be disconnected.

In mathematics, again, there is a notion of a prime knot. This is a knot consisting of a loop that is indecomposable into a simpler prime knot. Each prime knot is defined by the number and configuration of their crossings. Cut the loop once and you can untie it.

The trefoil knot is the only prime knot with three crossings. It is one of the prototypes for the Valknut (along with the Borromean rings), a knot from Norse mythology. In the mathematical world, it is the simplest prime knot.

A simple figure-eight may be twisted and turned into a simple loop, called the unknot.

If you cut it at any place, you end up with the standard overhand knot, which is a knot in the middle of a single strand, used to thicken a strand so it might stay in place, for instance. This is called a stopper knot. A better stopper knot is the figure eight knot. Over, under, around, and through.

Some knots are really just intertwining of multiple strands. But these are not prime knots. A typical example of this is the simple 3-braid. This consists of three strands that are interwoven.

Pippi Longstocking (Pippi Långstrump in Sweden) probably braids her red pigtails this way. As did the Viking women more than a thousand years ago on Gotland.

Given three fibers, pretty much everybody knows how to make a braid. But really a braid can be made out of any number of fibers. In fact, the copper wire shielding on coaxial cable is braided in a pattern that wraps around in a circle, so it represents a higher-order of braid with a cylindrical connectivity. This kind of braid is less embeddable in the plane than, say, the 3-braid shown here.

In some ways, braids are a degenerate form of a weaving. But knots aren't really.

Knots are interesting to tie from rope. They have several uses. For instance, you can make a loop at the end of a rope for securing the rope to an object. This is a common knot, the slip knot. It has the distinction for being less secure than other knots made for the same purpose. For instance, the noose (shown) is a more secure loop end.

This is because, when you make a slip knot, it is important to take care that the part that slips is not the loose end. Otherwise it will slip right out. As you can see here, you make the connected end the part that slips and when you pull on it, it will become tighter.

To a point. And then the loose end will eventually pull through. Which is why a real noose additionally secures the loose end in some way. Using a figure eight knot as a stopper knot on the loose end is a good way to make sure it won't slip through eventually.

This is a very simple utilitarian knot.

But if you really need a good stopper knot at the end of a rope, try the double overhand knot. This knot is a bit larger than the overhand knot, more secure than the figure eight knot, and it's also probably easier to untie when you need to.

The double overhand knot is like an overhand knot, but you pass the loose end through one more time than usual. Once you have made the knot, then pull it tight and it makes a tight ball at the end of your rope that is very secure. It looks like the knot shown here. It makes one of the best stopper knots. When you don't want a rope to pull through a hole, you use this one.

It's secure enough to use when climbing, for instance.

If you want to learn how to tie proper knots, you can use the animated knot site, or their animated knot how-to apps.

Cryptography?

Anyway, knots are fun, and they have a mathematical basis that is even useful for cryptography. Imagine a digital signature scheme using a braid group, for instance. You can come up with standard notations for the prime knots also. More complex knots can be found to be decomposed into connected sums of prime knots. Unfortunately this is hard, like factoring numbers. And this is another reason why knot theory is sometimes used for cryptography.

Knots form groups and groups can be the source of endless interest.

Knot theory has often been identified by physicists as a basis for the process of quantum entanglement and other natural processes that occur at the subatomic level.

It seems that there is more to knots than meets the eye!