For my September 2017 diary, go here.

Diary — October 2017

John Baez

October 1, 2017

Mystery of the gray ribbons

Abdelaziz Nait Merzouk has done it yet again: he's created a mathematical work of art! This one is a traditional Islamic tiling pattern that flirts with the impossible... namely, 5-fold symmetry. See all the small green 5-pointed stars?

The most exciting feature is one you might not notice at first. It's the gray ribbons! Follow one with your eye and see where it goes. What does it do?

If you followed it forever, would it loop around back to where it started?

I don't know, so this makes a nice puzzle. Let's do it systematically.

In this picture you can see a lot of purple stars.

Puzzle 1. How many points does each purple star have?

Next to each purple star are a bunch of 5-pointed stars with light green points. I'll call these green stars.

There are also some more complicated things where two green stars overlap, sharing 2 points. I'll call these twin stars.

Puzzle 2. How many points of each purple star end in a green star?

Puzzle 3. How many points of each purple star end in a twin star?

If you look carefully, all the designs are formed by gray ribbons. And that's where things get really interesting. What happens to a gray ribbon as you follow it along? It's hard to say because the picture isn't big enough to see. But you can figure it out anyway.

When a gray ribbon goes through a green star an into a purple star, it turns either left or right and pops out.

Then the gray ribbon continues until it hits another purple star, and the story goes on. So we can keep track of its progress like this:

LRLRLLRLR....

...unless it hits a twin star!

When hits a twin star, it makes a slight turn either left or right. In this case let's write a lower-case "l" or "r". It then quickly reaches a purple star. It goes in, and as usual it turns either left or right and pops out.

So, we get a sequence sort of like this:

RRLRlRLRLLLRrRRLl....

I'm just making this one up, it probably ain't exactly right.

Puzzle 4. What's the pattern of this sequence?

I believe it's the same for every gray ribbon that hits a purple star. Some gray ribbons just go along straight lines, minding their own business. But let's ignore these.

Puzzle 5. If we follow a gray ribbon that hits a purple star for long enough, do we get back where we started? Is the answer the same for every gray ribbon?

For more of Abdelaziz Nait Merzouk's tiling patterns, go here.

The twin stars look like 'defects', but they're inevitable. Greg Egan and I explained the math here:

You can see answers to the puzzles in the comments on my G+ post. Xah Lee colored the ribbons in a way that shows the different kinds:

For my November 2017 diary, go here.


© 2017 John Baez
baez@math.removethis.ucr.andthis.edu

home