So here at Westminster we do a “Spooky Science” community activity thing each year around Halloween. I've run the following activity about spoooooooky non-orientable surfaces a couple of times, and people of many ages (from 6 to 66!) have fun with it. Also fun in classrooms!

Ingredience

You will need:

  • A bunch of strips of paper
    • A good size is to cut an 8.5” x 11” sheet of printer paper, or a 9” x 12” sheet of construction paper, into six strips.1
  • Tape
    • I guess you could use glue sticks but I low-key wouldn't. I guarantee someone will glue the wrong side and get sticky everywhere.
  • Scissors
  • Pens or markers

The activity

Make the loops

So we're going to make the strips into loops, but instead of taping them straight up, we're going to tape them with a half-twist. The easiest way to do this is:

  1. Bring the ends together like you are just going to make a regular ol' loop, like a link in a paper chain.
  2. Flip one end over, so that the other side of the paper is showing.
  3. Have a friend tape up the join. Do this on both sides.

The resulting object is called, if I am typing very carefully, a Möbius strip, Möbius band, or Möbius loop.

Why is this cool?

The interesting thing about a Möbius strip is that it is non-orientable. What does this fancy word mean? On a regular loop, you can assign an orientation, which is a consistent labeling of the inside and the outside. There's no way to do this on a Möbius strip. Therefore, a Möbius strip has only one side. (Here is where I say, “spooooooky,” and wiggle my fingers around.)

That is an absurd thing to say. But I can show you that it is true! Grab a pen and draw a dot on your Möbius loop. That dot is a little spider (spooooooky) and it is going to go on a journey around the world. The spooky spider is going to walk straight down the middle of the strip as far as it can. You're going to draw the journey of the spider with your pen.

Now is a good time to ask people to make a prediction. What do you think will happen? What would happen if this was just a regular loop?

What ends up happening, much to people's surprise, is that after a long journey, the spider ends up exactly back where it started. So the paper really does only have one side!

Scissors time

You have now drawn a line right down the middle of your loop. You have a pair of scissors. A lightbulb appears above your head and you grin maniacally as you have a great idea: let's cut down this line.

Now is another good time to ask people to make a prediction. What do you think will happen if you take your scissors and cut right down this line? What would happen if this was just a regular loop? Well obviously it would fall apart into two separate, narrower loops.

It's tricky to get started because of the weird shape of this object. A good way to do this is to pinch the paper into a little fold, right where you taped it. Then you can just snip a little slit with the very tips of the scissors. Unpinch it, stick the point of one scissors blade through, and you can cut pretty normally from there.

When you finish the cut, a complicated and surprising moment happens, where the strip does this funny hoppity-spring and falls open into one longer loop!

A new loop?!

This new loop is very weird. Fun things you could explore here:

  • It's even twistier than the original loop. How much twistier, exactly?
    • How many (half-)twists does this object have?
    • Count by un-taping your object, stretching it out, and carefully untwisting.
    • Surprisingly, the new loop has four half-twists – that is, two whole twists. ???
  • Does this new object have one side or two?
    • Surprisingly, two. ???
  • What happens if you cut down the middle of this object?
    • Surprisingly, it falls apart into two separate loops, but they are intertwined. ??!

Let's ask why

There is no creature more pathetic than a mathematician who has an observation but not an explanation. They snuffle around desperately, like dogs who smell a french fry but can't find it in the bag. “Where is it?! Whyyyy!!!”

  • Why does this thing have only one side?
    • How does the spooky spider get from the one side to the other? When exactly in its journey does this weird thing happen?
    • Try, before you tape up a strip, coloring just one side. If you made it into a regular loop, what's up with the colors? How is it different if you make it into a Möbius strip by doing the twist?
  • Why, when you cut this thing in half, does it stay in one piece?
    • Before you tape up a strip, draw a line down the middle. Color in one side of the line. If you made it into a regular loop, what's up with the colors? How is it different if you make it into a Möbius strip by doing the twist?
  • Why, when you cut this thing in half, does the resulting object have four half-twists?
    • What exactly happens, if you look at it real carefully, when you finish the cut?
    • Line it up so you're going to finish your cut right at the tape. Just before you finish, untape it, and then cut through. You're looking at four ends – like, the top strand left end, top strand right end, bottom strand left end, bottom strand right end, do you know what I mean? What colors are they? How come?

Variations

A common question that mathematicians ask is: What if we do this again, but a little different?

  • What if, instead of putting one half-twist in the band, we put two half-twists in the band?
    • How about three?
    • What numbers of twists make it so the object has one side, and what numbers of twists make it have two sides? (Odd numbers, and even numbers, respectively.)
    • Why is that?
  • What if, instead of tracing the side, we trace the edge?
    • Turns out there's only one of those, too.
  • What if, instead of the spider walking right down the middle, the spider walks along at the 1/3 mark?
    • Hold the paper up to the light so you can see what happened on, uh, “both sides.”
  • What if you cut along the 1/3 mark instead of halfway?
    • Is your new object orientable or no?
    • How many twists does your new object have?

Follow-ups

  • How might an object with just one side be useful in the real world?
    • They actually do this when they build big long conveyor belts. It is called belt turnover. Could be a fun research project.
  • How long exactly is the spider's journey?
    • Twice the length of the original strip of paper. Why?
  • I said some pretty specific things about the size of the strip, mostly for logistical reasons. But there are also mathematical reasons! If the strip is too wide, you can't get the half-twist taped down.
    • Why not?
    • What's the absolute widest you could possibly do? (You will have to crease the paper flat.)
    • Does your answer change if your paper is longer?
    • Okay, so what's the exact ratio of width to length that is the widest possible? (I think this question is accessible to a middle-school geometry class!)
  • What happens if you chain two strips together and then cut one of them down the middle?
    • (I literally do not know the answer to this question.)

  1. Good motor skills and estimation skills practice if you want to make kids do it. ↩