Just for fun, I doodled some knots and made a video.
Actually, this video started as a lesson on counting strands in a rectangular Celtic knot, but I got distracted by the colors, and my lesson turned into a doodle. It happens.
So here is what I MEANT to say in the video before I wandered off into swirly knot land. The first knot I drew, called an "unknot," is a 1 by 1 square knot with 1 component (strand). The second one is 2 units wide by 2 units high and it has 2 components. The third is 3 by 3 with 3 components, and the fourth is 4 by 4 with 4 components. So, you might have noticed a pattern here. If you have a square knot of this sort, and it is p units wide by p units high, it will have p components.
The next one I drew was rectangular, 2 by 5, and it has 1 component. There's a theorem that says
that the number of components in a p by q knotwork panel is given by the greatest common divisor gcd(p, q). So applying this to the rectangular knot, gcd (2, 5) is 1, and the theorem works. If we apply this to square knots, when the panel is square, we get p = q and the gcd(p, p) = p.
Then I got bored with my lesson so I started doodling. Sorry, but I don't know any good math to go with the rest of the doodles.
I made this video with Doceri and lots of coffee. Too much coffee.
Nice music and you make it look so easy. I always struggle at the end points.
ReplyDeleteOne of the reasons why I wanted to learn cubic right angle weave was to be able to do lace tatting like patterns; seeing this I think Celtic knots would be a good choice too.
The music is Bach. These videos always make it look easy because I'm playing them back at 1% of the time it took me to make them. ;)
ReplyDeleteYou can definitely translate RAW or Cubic RAW into knot designs. I haven't tried it, but I'm sure it would work. I'd love to see it if you make something.
I was disappointed in cubic raw because the holes in the beads showed. On the other hand, I really like the depth of the beadwork. It might just work for Celtic Knots because only the top side would be showing. Now I am going to have to get out my test pieces and take a look and think about it.
ReplyDeleteI REALLY don't like seeing ANY thread in my beadwork. It's one of my basic rules about a quality design. That's why, when I do cubic RAW (I like to call it box stitch, it's just shorter), I always fill in the spaces with more beads. For examples, see here: http://www.etsy.com/listing/70382644/3-intersecting-rectangles-for-the-math
ReplyDeleteand here: http://www.etsy.com/listing/69907401/octahedreal-orb-beaded-bead-for-the-math It works with flat RAW also.
Do you think we might be distantly related? :)
ReplyDeleteI don't know if filling up the spaces would work so well for what I am envisioning, but adding edging would.
I have way to much to do to even think about a new design. (I bought some beautiful slab turquoise last year that I still haven't gotten around to.) I am not sure having to much to do will stop me; after all, I learned cubic raw a last year for just such a piece. Maybe that 2 x 2 would be a good start.
We might be, if you go back far enough :)
ReplyDeleteI love Celtic Knots, they are so beautiful, but I never thought of them as mathematical. Just for fun, I bead embroidered a knot that I traced from a celtic knot book and transferred to some felt, it was fairly intricate on a 5x7 piece of material. Only two colors, the knot and the background. It was so meditative to embroider that knot, like walking a labyrinth.
ReplyDeleteKnot theory is a whole branch of mathematics, under the subject of topology. You could spend the whole rest of your life studying knots from a mathematical perspective. I like your analogy to a labyrinth, which reminds me of a paper I once read on the mathematics of mazes. All good stuff. Got a photo of your embroidery? I'd love to see it if you can post a link.
ReplyDeleteI finally had the free time and the sunshine to photograph my Celtic knot loomed necklace. I mentioned you and your friend Erin. You can see the pictures on my blog at this post:
ReplyDeletehttp://kjsbeadaciousbeads.blogspot.com/2011/05/celtic-knot-loomed-necklace.html
Thanks KJ. I left my comments there.
ReplyDelete