January, 2011

William Jordan Jones (1933-2010)

William Jones

William Jones was one of the mathematicians who worked closely with me for 18 years to create software that I used in my design work. He died just before Christmas of 2010.

Bill's Last Christmas Card

Each year, Bill created a digital Christmas card that combined his love of mathematics and design, along with colorful illustrations and a bit of a mathematics lesson, all written in his uniquely wry, pun-filled style.

One way to glimpse how Bill thought (and to imagine how we worked together) is to read one of his Christmas cards. His final card, distributed in December of 2009, appears below. I hope you'll enjoy reading it, and that it gives you a sense of Bill's humor and thinking process.

Why Celtic Knots at Christmas?

Well, why knot? The decorative art of Celtic Knotwork pretty much dates from its heavy use in the Book of Kells, a painstaking transcription of the New Testament done around 800 AD. Hence the hard-C pronunciation of Celtic, except for the Boston NBA team, who possibly were granted a special dispensation by St. Patrick. Whatever, the Book is Ireland's great national treasure and is maintained in the Trinity College Library in Dublin. Apropos Christmas, the Books of Kells has the four Gospels with their Nativity good news. And the Knots? They are colorful and intriguing, plus they have a rigorous geometrical discipline to them but offer great creative variety, so working with them is a craft that can be carried out in many media. What's knot to like here? A page from The Books of Kells. ⇒ ⇒ ⇒ That's some serious doodling, dude!

Celtic Knots

A Celtic Knot pattern starts with a grid inside a grid inside a grid. The Primary grid on the right is 3 by 3 units determining the size of the knot elements. The Secondary grid in this color splits the Primary grid down the middle, and it's here that you can set up barriers that will divert the knot paths and create variations. The Tertiary grid in light blue is where you start the paths that make up the knots. Below you can see the first four segments of a Celtic Knot path being drawn on a 3 by 3 grid with no barriers:

Celtic Knots 2009 Page 1 of 8

On the next page are some more steps on the way to completing the knot:

The last one above is the complete knot, crudely drawn but hinting at the over and under discipline that gives Celtic Knots their 3-D look. To the right below is the finished knot with a few cosmetic enhancements. The grids are erased, the corners are beveled, the paths are drawn in two colors, and the over and under discipline is noticeable. Do you think you can figure out the rules for drawing paths? You start at any edge unit and draw a segment that's two ticks long in the tertiary grid just inside the edge. Then you shoot off away from that edge at a 45- degree angle as far as you can go. When you approach another edge, you straighten the path for two ticks, then "bounce off" at 45 degrees again. If you hit a corner unit, you follow around the corner and then "bounce off." You will eventually end up where you started, closing the path, and either you've drawn the whole knot in one path or you haven't. If you haven't, you find a unit with no path and start again.

The over and under discipline is what makes knots look like knots. All crossings are diagonal path segments meeting at right angles at the centers of horizontal or vertical segments of the primary grid. If it's a horizontal segment, the path with the / slant is on top. If vertical, the path with the \ slant is on top. It was surprising to me that this simple rule ignores the twists and turns and straightened paths caused by barriers but is sufficient to guarantee that every path goes alternately over and under the successive paths it crosses. Trace some of the paths on pages 5 and 6 to verify this.

The knot above happens to have three separate paths in it, all woven through each other. One of them is a big diamond with path segments in all eight outside units; it is shown here using different colors than those of the other two paths . If you were to render this knot manually as an artist, you might choose to make the diamond path a circle crossing the other two paths in roughly the same places.

Many Celtic Knot patterns, even large and complex ones, consist of a single path. If you start with 3 by 4 or 3 by 5 units in the primary grid, as shown in these examples, you will get a single path. Trace them and see.

Celtic Knots 2009 Page 2 of 8

Celtic Knot patterns are made more complex, varied, and interesting by the addition of barriers to the grid prior to tracing the knot. Barriers are line segments drawn on the secondary grid connecting centers of the units of the primary grid. When you draw paths, you treat barriers as if they were edges; you run your path parallel to them for two ticks and "bounce off" the other way. The path can also get trapped between

barriers and edges and have to straighten out for a while. The four barriers shown in the left figure above force the path shown to its right, with the finished version on the far right. This pattern is an attractive single path. It has pinwheel symmetry (quarter turn doesn't change the pattern) because the barrier pattern has pinwheel symmetry. In the examples to follow, maybe you can visualize where barriers were placed. The final patterns have the same kinds of symmetry that the barrier patterns have except for slight asymmetry at the crossings. Much of the creative scope of Celtic Knots comes from devising barrier patterns.

Here are four patterns with barriers (erased) on a 3 by 3 primary grid (also erased):

And here are some with barriers on a 3 by 4 primary grid:

Celtic Knots 2009 Page 3 of 8

Back in the last Millennium . . .

I investigated Celtic Knots more than ten years ago looking for interesting fabric patterns for Jhane Barnes. I made the software create the grid in the computer's memory, follow one of a number of symmetry-based schemes for setting up barriers, and trace the paths just like someone would with pencil and paper. The big difference was that you'd get an elaborate pattern in any size in a variety of colors instantly on a click of the mouse rather than take hours to draw one. The enhanced patterns you see here were produced that easily by the software.

It was fun developing this Celtic Knots generator circa 1999, and I have a cool Jhane Barnes shirt made with fabric it created, but if I did it today I'd take a completely different approach and provide a lot more user controls and variations. I would love to find a young kindred spirit, a nerd like me who is into Math, art, and programming to show how to do this.

So far all the Celtic Knots patterns you've seen have been closed, with no paths crossing the edges of the primary grid. Patterns can also be wrapped, allowing paths to escape all four edges. Below you see six copies of the wrapped pattern marked off in the lower right corner repeated in a 2 by 3 array. The paths flow seamlessly from any copy to its neighbors in all directions, enabling the knot pattern to ramble continuously over any size piece of fabric.

Celtic Knots 2009 Page 4 of 8

Knots to You

Some Celtic Knots patterns in various sizes and colors:

Celtic Knots 2009 Page 5 of 8

Celtic Knots 2009 Page 6 of 8

Try It!

Here's a grid with 64 Primary units in the shape of a Greek cross. Some symmetrical barriers have been set up in red The Celtic Knot pattern has been started with the green octagon in the middle, confined by four barriers. The 4-part knot pattern has a part mostly in the extremes of the cross and two parts forming an X and crossing the green part. The octagon shows slight breaks following the over and under discipline.

If you are game to try drawing the pattern, I suggest you make a few copies of this page first. It's easy to make mistakes.

Celtic Knots 2009 Page 7 of 8

Still Here?

Finally, if you've been patient with me and just in case the idea of creating your own Celtic Knot patterns really appeals to you, here's a grid of 10 by 10 Primary units to play with. You can make copies of the page, and you can draw heavy lines to mark off smaller areas to work in, like 3 by 10, 5 by 5, 5 by 10, etc.

You can invent your own barrier patterns, referring to the Greek Cross grid on Page 7 for the grid lines they use and noting that they connect centers of Primary units. It's OK to make barriers two or more units in length. If a barrier is a closed path, there will be a knot pattern inside the path and another outside it.