# Fold and Cut, My Favorite Spaces, and Hook

Welcome to this week’s Math Munch!

Before you watch this video, think about this question: Do you think you could fold a piece paper so that you could cut a square out of it using exactly one straight cut? How about a triangle? Hexagon? Christmas tree shape??

Give it a try. Then watch this video:

Pattern for a very angular swan, by Erik Demaine

Surprised? As you may have seen in the video, using the “fold and cut” process you can make any shape with straight sides! Isn’t that crazy? I learned about this a few years ago, and now cutting weird shapes out of paper using just one cut is one of my favorite things to do.

The person who proved this amazing result is one of my favorite mathematicians, Erik Demaine. (You might remember him from our post a few years ago about origami mazes.) I think it’s really interesting that this idea that’s now a mathematical theorem appeared throughout history as a magic trick and a method for cutting out five-pointed stars to make American flags. Check out this website about the fold and cut problem to learn more about the history of the theorem, Demaine’s method for cutting out any straight-edged shape, and other related problems.

Evelyn wearing a Borromean ring cowl. Sweet!

I found out about this video from another favorite mathematician of mine, Evelyn Lamb. Evelyn writes a blog about math for Scientific American called Roots of Unity that’s really fun to read. Check it out if you get the chance!

She has a series of posts called “A Few of My Favorite Spaces” (cue Sound of Music song, “My Favorite Things”). Favorite spaces, you may ask? I’m not familiar with spaces plural. There’s more than just regular old 3D space? Yes, in fact there are! And if you read Evelyn’s blog you’ll learn about how mathematicians like to invent new spaces with bizarre properties– and sometime find out that what they thought was a completely new space actually resembles something very familiar.

Such as… The “house with two rooms.” As I understand it, this a box (“house”) with two floors and two tunnels in it– one punched from the top of the box and another from the bottom. The top tunnel lets you get from the roof of the house to the ground floor; the bottom tunnel lets you get from below the house to the second floor.

If you want to see someone making this crazy house in Minecraft and hear a much better explanation of what the house is like, here’s a video!

Ok, so what’s the point? Well, it turns out you can squish (just squish– no ripping or gluing) this house all the way down to a single point. This means that in topology (the type of math that involves a lot of squishing), the crazy tunnel house space is the same as the really boring space of just one point. I might want to live in a house with all these tunnels– but I definitely don’t want to live in a point. But in topology-world, they’re the same space. Huh.

To learn more about the house with two rooms (aka, point) and other crazy spaces, check out Evelyn’s blog!

Finally, speaking of squishing things down to a point, I want to show you a fun new game I found that involves a lot of squishing– Hook! Here’s a trailer video for the game:

You can find this game online at Kongregate. Enjoy!

Bon appetit!

### 24 responses »

1. Woah i never knew it was possible to do the full alphabet with one cut! Do you think it’s possible to do a whole word in one cut?

2. Woah! Never new it was possible to make the whole alphabet each letter wth one cut. Do you know if it’s possible to do a whole word in one cut?

• I don’t know! That’s a great question! If you try, we’d love pictures of the results. Thanks for the comment!

3. It was really cool to see math in minecraft. It really shows that math is everywere. Also, I think it would be cool to have somthing launch you up the bottom pipe

4. I watched the video about the two story house. Could you make a 3 story house instead of a 2 story house like this?

5. It’s really cool how the house stays the same size when it “squishes” down. Also that you can’t tell the other floor exists.

6. Wow! That is so cool! I never would have thought that you can make a 5 pointed star with just one cut! Do you know if it is possible to make any more complex shapes using this theorem?

7. I don’t think that my gears move in the same way as hers, but I am glad that some people’s do! I never knew that you could shapes- or even letters for that matter- with just one cut.

8. I think it’s cool that no matter how many sides a shape has you can fold it and cut it out with one cut

9. That is so interesting that you can make every letter of the alphabet with one cut! Do you think you could make a shape, such as a circle, that had a hole in them with just one cut?

10. I never heard about the Fold and Cut Theorem and probably would’ve never thought of it existing if I hadn’t seen this video. It’s really intriguing because it’s another way to see math in everything. Even though they’re simply pretty pieces of cutout paper, there is mathematical substance behind them. It also goes the opposite way by giving a hands on version to a theorem. Great project for any kind of math class.

11. The ability to cut out a complex shape in just one cut astonishes me. I had no idea that a theorem such as ” The fold and cut theorem” existed. The only theorems I ever knew of existing included physics and geometry. Lynda’s video inspires me to think out of the box more often and to search up easier ways to complete tasks.

12. This is crazy! i am actually learning how to do this . its not hard but its not easy.

13. I never knew that the cut and fold theory was a part of American history! It’s interesting that even in the 1770s people, such as Betsy Ross were using math tricks to make life more efficient. it’s also hard to believe that nearly any shape can be cut out with only one cut!

14. Wow I never knew that the cut and fold theory was part of American History! It’s so interesting that even in the 1770s people, such as Betsy Ross, used math tricks to make life more efficient. I also think it’s interested that nearly any shape can be cut out using only one cut!

15. It’s so interesting that the Cut and Fold Theory was part of American
History. It’s interesting that even back in the 1770s people, such as Betsy Ross, were using math tricks to make life more efficient. It’s crazy that nearly any shape can be cut out only using one cut.