**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 sorta 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?

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

https://plus.google.com/u/0/114982179961753756261/posts/VdiBx4jz3U1

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

https://blogs.ams.org/visualinsight/2015/02/01/pentagon-decagon-packing/

#geometry

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- oh, the lone green line was result of laziness, and i thought, it adds a artist quirk, and jolts people's attention.

but i didn't realize, it can actually be interpreted mathematically as 2 groups of distinct straight lines. i.e. the negative sloped ones always go over the positive sloped ones!47w - +Xah Lee - I don't like artistic quirks in mathematical art... i like the pattern to be perfectly systematic, not quirky. It's funny: I hadn't noticed that
*just one*ribbon was green. I automatically assumed all ribbons with the same slope had the same color! If you can make that one green ribbon the same color as the rest, I'll try to return the favor someday.47w - 47w
- the coloring of the path seems to have destroyed the stars.47w
- +Xah Lee - I like it! If you want a free electronic copy of
*Gauge Fields, Knots and Gravity*, I can email you one.47w - 47w

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