Posted on Leave a comment

Triknots

Our first knot puzzle was based on Celtic knot designs that begin with a square lattice, with diagonals representing the rope of the knot. The Celtic artist selectively replaced intersections with turns to create variations and symmetries, and refined the intersections to show the rope weaving over and under.

Contructing rectangular Celtic knots

We wondered what it would look like if we used a grid of triangles. We settled on the idea of using triangular tiles that have two ropes passing through each edge. Unlike the Celtic knot, this means you can connect any two tiles, and the ropes will always meet each other.

a single triknot tile

There are seven patterns on the triangular tiles. Each tile can be characterized by where the ropes enter and exit. Can you figure out why we have seven patterns?

the seven triknot tiles

With your tiles, you can explore some of the ideas of knot theory, which studies closed loops. You can close loops using any of the edge pieces.

triknot edges

The simplest closed loop is called the unknot. You can make one with two edge tiles:

an unknot

Below are two unknots. One of them has a twist. It is still an unknot because you can untwist it to have zero crossings.

If a knot has fewer than three crossings, it is equivalent to an unknot. The simplest knot that is not an unknot is the trefoil.

the trefoil, the only knot with three crossings.

You can use your knots to explore symmetries. When the trefoil is rotated 120 degrees, it looks the same. After three such rotations, you are back to where you started. So we say the knot has 3-fold symmetry.
M.C. Escher creatively used color to reduce symmetry. He called this anti-symmetry. Looking only at the patterns on the tiles, ignoring color, this knot has 6-fold symmetry.

If you consider color, what is the rotational symmetry?

Things to think about…

• Can you explain why the knots you create always alternate going over and under?

• How many kinds of rotational symmetry are possible?

• Why are seven tiles needed to characterize the ways the ropes can enter and exit?

• If we had used square tiles with two ropes on each edge, how many different tiles would we need to show all possible connections between edges?

• Which knots in the Knot Atlas can be made with the triknot tiles?

Posted on

A shared hobby created

I’ve had a few careers, all of them motivated by curiosity. Flying airplanes, writing software, financial engineering all provided a creative outlet, but as jobs go, that creativity was necessarily narrowly focused. I always enjoyed teaching, and figured I might take it up when I had explored enough other careers. To my surprise, I discovered teaching high school was the job that really indulged my creative side in the most general way. I taught high school math, physics, and computer science. Teaching computer science in particular enabled me to explore a trove of interesting problems to solve with the students. We wrote games in Scratch, constructed enormous structures in Minecraft using Python and Javascript, and sketched dynamic and interactive visualizations with Processing (and later p5.js).

Processing was especially inspiring. I used its pdf library to algorithmically generate drawings. It was this, combined with a visit to Ann Arbor’s MakerWorks, that ultimately led to Cherry Arbor Design.

MakerWorks has an array of maker tools, but I was drawn to the laser cutter, because I could see how the drawings created in Processing could be turned into precise wood or acrylic representations.  When I brought my creations home, Heidi was immediately intrigued with the possibilities. We started having our date nights at MakerWorks (yep, we’re nerds), creating earrings and other small items from thin cherry and maple boards, colorful acrylic, and Baltic birch plywood. Eventually, so that we could have unlimited access, we decided to buy a laser cutter. I still do most of my work with Processing and pf.js, while Heidi prefers Affinity Designer, an alternative to Adobe Illustrator.

Today, Heidi and I spend much of our time together making things, and further developing our design skills. We are all about nonstop learning, so at the moment, we’re taking a PhotoShop class at the local community college.