# Coasts, Clueless Puzzles, and Beach Math Art

Ah, summertime. If it’s as hot where you are as it is here in New York, I bet this beach looks great to you, too. A huge expanse of beach all to myself sounds wonderful… And that makes me wonder – how much coastline is there in the whole world?

Interestingly, the length of the world’s coastline is very much up for debate. Just check out this Wikipedia page on coastlines, and you’ll notice that while the CIA calculates the total coastline of the world to be 356,000 kilometers, the World Resources Institute measures it to be 1,634,701! What???

Measuring the length of a coastline isn’t as simple as it might seem, because of something called the Coastline Paradox. This paradox states that as the ruler you use to measure a coastline gets shorter, the length of the coastline gets longer – so that if you used very, very tiny ruler, a coastline could be infinitely long! This excellent video by Veritasium explains the problem very well:

As Vertitasium says, many coastlines are fractals, like the Koch snowflake shown at left – never-ending, infinitely complex patterns that are created by repeating a simple process over and over again. In this case, that simple process is the waves crashing against the shore and wearing away the sand and rock. If coastlines can be infinitely long when you measure them with the tiniest of rulers, how to geographers measure coastline? By choosing a unit of measurement, making some approximations, and deciding what is worth ignoring! And, sometimes, agreeing to disagree.

Need something to read at the beach, and maybe something puzzle-y to ponder? Check out this interesting article by four mathematicians and computer scientists, including James Henle, a professor in Massachusetts. They’ve invented a Sudoku-like puzzle they call a “Clueless Puzzle,” because, unlike Sudoku, their puzzle never gives any number clues.

How does this work? These puzzles use shapes instead of numbers to provide clues. Here’s an example from the paper: Place the numbers 1 through 6 in the cells of the figure at right so that no digit appears more than once in a row or column AND so that the numbers in each region add to the same sum. The paper not only walks you through the solution to this problem, but also talks about how the mathematicians came up with the idea for the puzzles and studied them mathematically. It’s very interesting – I recommend you read it!

Finally, if you’re not much of a beach reader, maybe you’d like to make some geometrically-inspired beach art! Check out this land art by artist Andy Goldsworthy:

Or make one of these!

Happy summer, and bon appetit!

### 29 responses »

1. Pingback: CHOMP | Big Honkin' WordPress

2. The Australia coastline video was very interesting. I had no idea that if you measured a coastline you could have 2 different results because of what you used to measure it.

3. This video was amazing. It was interesting how he could make a animal like creature that can become independent like a person.

4. I thought the video on the infinite coastline was really interesting. I also got to learn about Koch’s snowflake, which I thought was really cool.

5. The Australian coastline seems like it will forever remain an unsolved puzzle. If you measure the coastline with a measure tape which is flexible, can’t you get a more accurate measurement of the coastline because you can get into the nooks and crannies?

• Hey Fifi. You watched the video and so did I. And I totally agree. See ya!

6. Theo Jansen’s Sandbeast animals are amazing! I believe in the future, his ideas and his inventions will be used by the movie industry as part of special effects.

7. So you would have a different length of coastline if you measured it with a yardstick and a meter stick? I’m kind of confused. How would that work. The Austrailian coastline is beautiful though. You’re so lucky to have been there. I’ve noticed that you have a lot of posts about fractals. Why is that? I’m glad you posted this post though because this is the only one that I understand. So a fractal is just a triangle with bunch of other triangles added onto it? Wouldn’t that equal infinity though? Thanks for posting this video.

• Hey Lilian what is up? I liked the video to it was super cool . And yes I totally agree. See ya later. Lily that’s not me. Wink, Wink! 😉

8. I learned that you can use math to make living and walking wood figures my question is how could he figure out how to make a stick robot thing use wind power and walk like that

• I have no idea, but I’d imagine a very long time. They are incredibly intricate.

9. I love the technical complexity in Theo Jansen’s creatures, but I have a question for him. What does he consider himself? A architect, maybe an engineer? Also does he have names for the creatures and did he design them after anything?

10. I found it very interesting to find examples of fractals on Australia’s coastline. Are there any other fractals that could be displayed on land?

11. I really wonder what was going through his head (Theo Jansen) when he thought of this. Because whatever it was, it was brilliant. Who knew you could actually use math to make something artificially come alive.

12. Amazing creation. But what I would like to know is what’s PDC? And how long do you predict it will take for your creatures to walk on their own along with the wind?

13. It’s hard to imagine that Australia’s coast line is somewhat infinite! So, I guess that the coastline is similar to fractals, but I don’t see exactly how they relate, because fractals you’re just adding (for example) triangle onto triangles, how is that similar to the Australian coastline?

14. I never knew coastlines were fractals or that the length of the coast changes based on the unit used. I always thought that you could actually measure coastal areas. Also, I thought that fact that it is infinite is mind boggling.

15. That is amazing. How did he make them all out of PVC? and what happens when they run into the waves? is there some type of sensor that can guide it?

16. This video and idea blew my mind. I never even thought that coastline distance was ever up for debate, but here I am learning that we’re never going to have a definite answer for any coastline. I understand how the paradox works with the fractals and all, but there is something I don’t understand. why do different sources use such drastically different measuring sizes? why can’t everyone just decided on one specific length of measuring tool, and then measure only during high tide or something? I feel like theres alt of unnecessary discrepancy between coastline distance. I loved the video, I learned about fractals and the coast paradox, and the koch snowflake. It still is a little confusing that there can be a defined size and shape of an object, but the perimeter can be endlessly big or small.

17. I found the Coastline Paradox video very interesting. I didn’t know that with different measuring tools you would get different answers for the coastline. But does that mean that there isn’t an answer that is a hundred percent correct?