pretty pictures of geometry

When I was first exploring this grid pattern for drawing knots on, I made a lot of things where I generated a random knot design and drew it on the grid with various minor modifications like straightening out lines that squiggle back and forth, making certain corners pointy instead of rectangular, and other things that are harder to describe in words. Then I would scan my hand-drawn version and color it on the computer. Here is an example:

At some point I ended up making a painting using this exact same knot. I put some paint on the canvas, let it dry, and then semi-randomly added some more paint and “traced” the knot by smearing the paint together along the strings. I tried to make it mostly possible to tell the different strings apart but the green string in the lower left corner definitely isn’t as distinguishable as in the digitally colored one.

Also, remember this one where I turned the knot inside out and then tried to make a straightedge and compass version but it got too weird and I had to make some stars on the computer? I ended up with this messy thing that I didn’t know what to do with

and here it is in a painting!

Knots are turning out to be a rather common theme in this blog, and in my artistic life generally. As I was writing the pretzels post, I remembered another notable theme in my artistic life: things that disappear. Not only do I enjoy using artistic media that are intrinsically impermanent, like pretzels, I’ve noticed I put just as much, if not more effort into creations that will necessarily be destroyed just days or hours later.

I made the colorful knots on these Easter eggs using a wax crayon, which doesn’t leave any visible mark until after the drawing is finished and the egg is dipped into the dye. I drew the white parts of each knot on the undyed egg, added a light coat of dye, let it dry, and then added a few more lines to the knot and thickened the existing ones, repeating this process until the egg was covered in a multicolored knot design on a dark background. While it is perhaps a testament to my obsession with knots that I can sort of draw them freehand on a non-euclidean surface with an invisible crayon, this remains an error-prone process and likely something I never would have tried in the first place were it not for the impermanent artistic medium.

Even better than things that disappear, are things that return naturally to their original state. Pretzels will be eaten, the dye on the eggs will be thrown away with the shells, but the lake will put the sand back exactly where I found it, ready for a new sandcastle.

I make sandcastles pretty much whenever I’m near wet sand, and I am so obsessed with knot designs that they have infected even my sandcastles, which don’t make much sense as a medium for knot drawing.

Above, we see two sandcastles connected by single a road, which takes a rather indirect route, crisscrossing back along itself, over the bridges and through the tunnels, almost to its starting point before looping around to finally lead towards its destination. There’s something else familiar from the pretzels post here: as I was drawing the longer knot diagrams in the sequence containing the pretzel and figure-8 knot, I recognized the pattern from this - the road connecting the castles is the next element in the sequence!

I completely forgot this blog existed for about 2 years until I was trying to explain the whole history of my knot drawing adventures to someone and then remembered I actually wrote some of this down before, with illustrations! And then I logged in again and found a post about sandcastles that I apparently forgot to actually post, so that one will be next...

Anyway here are a bunch of pen drawings I’ve made using the circley grid paper. I’m pretty sure these are in the order that I made them, so you can see how my decorative use of the not-knot-related grid-circles, and eventually also added straight lines, has been evolving.

With the innovative new technique of printing the grid in such a light color that it doesn’t show up when I scan the knots, I have eliminated the tracing step without altering any other part of the process, so the knots are appearing faster and faster.

I traced this design onto the grid using my computer, no paper involved, to make this:

which I copied and pasted under four rotations, connected those asymmetrically, and then repeated the process using different rotations and then symmetrical connections, and colored the result:

I created the one below in a more organic way, drawing the knot on a paper grid in various orientations and connecting them in whatever way made sense locally. This one also wraps around like a torus.

I made this one in a relatively unplanned way, starting in the center and then finding additional points to make the design locally symmetrical around until I ran into the edges of the printed grid and just attached nearby points together.

I made this one using a similar strategy of working my way outward from a point and doing similar kinds of symmetrical things, but then wrapped the design around both edges like a torus instead of closing it up. I also had more time to mess with the colors so I had some more fun with decorative shapes in the background and color gradients along the strings, at the expense of being able to immediately tell how many strings there are.

If you’re wondering, there are 8 strings:

the one with vertical blue twisty parts, and an identically structured one with horizontal orange twisty parts

the one with light gray corners between the pink octograms, and its rotated version which has the yellowish-brown corners

Uncolored, this design has two-way rotational symmetry, and, if the crossings are ignored, mirror symmetry across the lines y=x and y=-x. Each symmetry provides a way of imagining how the loose ends could be connected together, and they result in different colorings.

reflecting across y=x creates 12 different loops:

reflecting across y=-x creates 9:

rotation also creates 9 loops, but some of them are different:

I only had to invert this one once to get a shape I liked! (I also made the artistic choice to take out crossings that didn’t affect the knot/link structure, which is why that’s different in the inverted one.)

My black watercolor pencil is getting extremely short compared to most of the other colors 😢

I was originally saving this post for when I had a nice straightedge-and-compass version of this knot. I constructed one, but it was really complicated and I ended up doing part of it on my computer because there were several regular polygons with odd numbers of sides, which are tedious and error-prone, and the construction lines don’t turn into nice decorations like they did for that other one with the octagons. It looks like I’m never getting around to doing one in the same style, so here is the original mess:

This happened when my dad decided to try making our own soft pretzels. He invited my brother and I to help roll the dough into snakes and twist the snakes into shapes. I had been reading about knot theory and decided that since the dough was sticky and I could kind of squish the ends together, it would be a good artistic medium for representing mathematical knots - closed curves in space that do not intersect themselves. 

My dad made some normal pretzel shapes, I got out the above book and made a trefoil, 4-knot, and the second 5-knot, and my brother made another trefoil, a karate belt, and a noose.

I noticed that the 4-knot could be constructed by taking one end of a normal pretzel and putting it underneath the middle of the snake instead of on top, and then connecting the two ends. 

A year later, we were making pretzels again. That time, I had recently gone rock climbing and learned how to use the belay system for the first time, so I took the opportunity to practice my figure-8 knot. (We’re no longer speaking in terms of mathematical knots, since the ends of the string aren’t connected.)

I noticed a different kind of similarity with the normal pretzel (on the far left in the photo below, alongside several figure-8 knots). My first impression was that the figure-8 knot was somehow an extension of the pretzel, like there was just one more of something that changed the symmetry from bilateral to rotational (if we ignore which string is on top in the crossings).

Both of them are instances of a more general braid-like pattern that loops back through itself at either end, leaving two loose string ends.

Also, you may notice that the “normal pretzel” in this batch is different than the “normal pretzels” from the first batch. The first ones had the loose ends cross twice, forming a 4-knot when connected. The second, simpler, “normal pretzel” turns into a trefoil when its loose ends are attached.

My dorm had a Paint Night, and I was given a little rectangular canvas to paint on, many colors, and not very much time, so I estimated the dimensions to enter into the random knot generator and went straight to the canvas, no grid or measurements or planning. So it is imprecise, but colorful. There was only one thread (I left out the unlinked unknot in the middle because I couldn’t think of something interesting to do with it in the space and time available) which gave me plenty of length to gradually cycle through the whole rainbow.

In the previous post I took a small random knot and drew it on a grid and then turned it inside out four times, ending up with this design

which I used as inspiration for this completely freehand watercolor one (like I didn’t even use a pencil first, I just started coloring)

and, after much trial and error, this straightedge-and-compass one.

I got some new pens which are very tiny so now my knots are suddenly much more detailed. It is exciting.

I was digging around in my closet and came across a box of shapes that I apparently never took pictures of until now, so here are some more shapes.

Tiny one!

I also remembered that when I was first learning how to make things out of Sonobe units, all the shapes I had seen, and all the shapes I had put together by extrapolating from those, contained a multiple of three units. For awhile I thought maybe it had to be that way, but then I found a shape made of four units. Here it is next to one with only three units:

I was fascinated by the way the four units connected together, which was somewhat different than what I was used to in the shapes with a multiple of three. I started trying to combine the different techniques together. The corners of triangular one below are connected like the pointy parts of the four-shape, but the inverted triangle part in the middle is connected like the multiple-of-three-shapes. The weird crescent-shaped thing at the bottom has some pointy parts and some triangley parts as well.

I don’t remember exactly why I made these, but the three bigger ones appear to be vaguely inspired by this new way of connecting the units together. I think I had already made the one on the left and then realized it was what would happen if I extended the pattern of those other ones down to just three-way symmetry.

I wanted to see how many of the pointy things I could put in a circle, and it seems like the limit is about five or six, since the one with six wouldn’t quite stay together on the other side.