Introduction: Moebius Knot

Picture of Moebius Knot

You know the Moebius strip, the most famous amusement that a paper strip is capable to offer. For those who don't know, when you glue the one end of a paper strip to the other end, and turn the one end by 180°, you get a strip with only one side and other super powers.

But did you know, this does also work in three dimensional space? An object with only one side? And the fun thing is, you can also build it out of paper! It just takes about 100 times longer than the smaller version.

Step 1: The Moebius Knot – What Is This Black Magic?

Picture of The Moebius Knot – What Is This Black Magic?

Couldn’t you just say a sphere does also have only one side? Not really, most objects have another side on the in-side, you just can't see it.

The basis of the moebius knot was a simple overhand knot (The knot you make when someone tells you to make a knot). But instead, the ends are open and directly connected to the outer shape, turning their inside out. A lot of math and stuff that made my brain push the envelope when I made the model.

For our model, this means, it has no inside. You can’t fill it with water, even if it wasn’t made out of paper. It has no mass, despite the shell. Or you can make two points anywhere on the model and connect them. This is the power of only one side.

As far as my mathematical understanding goes, this should be the simplest shape with only one side. A commenter referred to the "Klein Bottle", which has basically the same principle, but looks like a, well, bottle. And I have no idea if anyone actually calls it Moebius Knot, I totally made that up.

Step 2: Preparation

Oh look, I made a video.

Too fast? Okay, I will write down how to make one. Careful – It is not an easy model!

You need the following:

Thick paper: about 160 g/m2, or just two normal sheets glued together. White paper allows no faults, that’s why I used red paper.

A printer

Crafting stuff: scissors, glue, ...

The template from kamibox.de

Step 3: Pieces 1 – 10

Picture of Pieces 1 – 10

Print everything but the first page.

The model consists of 22 rings and a center piece. The first half of the 22 rings exactly looks like the second one. Math magic! The pieces have numbers and arrows that point in the direction where the next ring goes.

When you glue a ring together, there is always a small "step" in the height of the paper’s thickness, where the end of the ring is glued to the flap. I will call these steps "tiny edges" in the rest of the Instructable, because they are very important and I don’t know how else to call them.

Before you cut the shapes out, you should scratch the dotted folding lines with the back of your scissors, or in case you are square as I am, with a bone folder. Glue the ring together, you may want to pull it over the edge of the table before, then it gets a nice rounding. The printed side is on the inside (Oh, I said it. But don’t worry, the inside will become the same-side when the model is finished). Fold the flaps towards the center of the ring and attach ring (x) to ring (x-1). Glue the flap that is nearest to the arrow first, then the others. This way you make sure that you get the pieces lined correctly (pic. 3). Take care that the arrow does always point in the right direction and that the tiny edges line up very well.

Step 4: Piece 11

Picture of Piece 11

This piece is a bit different, because the flap is on the other side of the ring. This is because ring 11 and 12 will not enclose ring 1 or 22, they will be the pieces that are connected to the center piece. But don’t worry, just glue it to ring 10 like all the others, but take care that the arrow is still pointing in the right direction.

When you do everything correctly, the tiny edge faces to the other direction than the rest.

Step 5: Piece 22

Picture of Piece 22

What, already ring 22? Yes, we skip a few.

Ring 22 is exactly the same ring as ring 1. Glue it together and then attach it to the model like in picture 1. The edge of the paper should fit on the tiny edge where ring 1 sticks to itself. Turn it sideways and glue the spot on the second picture (same here).

Then glue the egde of the ring into the other ring, counter clockwise, starting from the point that you glued first. It should fit exactly in the "canal" where the tiny edges of the rings line up. With a piece of paper, you can provide it with glue (pic. 3).

In the end, it should look like the last 2 pictures.

Step 6: The Center Piece

Picture of The Center Piece

There are 2 more lines on the center piece that indicate where to fold it. Fold one with a valley fold and one with a mountain fold. Then attach the piece to the model, see the pictures. It doesn’t really matter on which side the flap goes, but for symmetry’s sake, I glued the flaps to different sides.

Step 7: Piece 12

Picture of Piece 12

As piece 12 is the same as piece 11, you glue it together like any other ring before and attach it to ring 11. This time be especially aware that the arrow points in the right direction.

Step 8: Pieces 13 – 22

Picture of Pieces 13 – 22

Starting with ring 13, you should glue the edge of ring 22 right in the tiny edge of the ring you are currently attaching. Your conscientious crafting will be rewarded when piece number 21 fits exactly in the gap.

And there you have it, now go out and play with it!

Comments

silverplato11 (author)2016-04-08

How long does this take to make?Does this need any skills or can anyone do it? Great idea:)

HappyJoyLuckStar-K (author)2015-01-09

Hi, quick Q. Does this turn out flexible or is it rigid? It looks like it would be almost springy and movable.

-K

It is completely rigid.

Void Schism (author)2015-01-06

I'm fairly sure this is topographically different to a Klein Bottle. They are both single sided, but the Klein Bottle is not truely reproducible in reality. The Klein Bottle is special in that it has no edges and the glass ones that have been made lack the surface passing through the neck.

azharz (author)2015-01-04

Finely Awesome!

hotchocdrop (author)2015-01-03

"Klein bottle"

kamibox.de (author)hotchocdrop2015-01-04

Thank you, I didn’t know that!

SphereX (author)2015-01-03

Cool! Would like to see a fibonacci torus.

belsey (author)SphereX2015-01-03

With 3D printing, everything is possible! http://www.shapeways.com/model/281324/fibonacci-mobius-loop.html?materialId=5

hardlydavidson (author)2015-01-03

all I can say is gulp! thanks for sharing. it's complicated and beautiful :)

belsey (author)2015-01-03

I've seen these Klein bottles in glass... but it never occurred to me to make one of paper! Beautiful!

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