Exploring Spaces 2: Twisted Taping

elliot evans
April 24 2024

Twisted Strip

In the previous "Exploring Spaces", I made a torus out of paper.

This time I taped the edges of a long band of paper into a twisted strip.

I drew a blue line on one side and a red line on the other side. After I twisted and taped, the lines connected into one long line!

Möbius Strip

This twisted strip is usually called a Möbius Strip. Its named after a person. I want to avoid using names of mathematical objects based on the names of people in this series.

Interactive Taping Diagram

Here's a diagram showing how to tape the paper strip to itself. Tape the two horizontal edges so that the arrow heads are aligned with eachother to get a twist!

The diagram is interactive, so please try exploring this twisted space by moving the circle around and through the blue arrows.

Twisted in Two Directions

Now I want to show you a space where both the horizontal and vertical edges are taped in a twisted way.

Torus taping diagram
First though, for comparison, here's a torus:

Real Projective Plane
taping diagram
Now here's the doubly twisted space. It is called the "Real Projective Plane".

Unlike the torus, the arrows on opposite edges are pointed in different directions.

When I started playing around with this, I was surprised by how different the space feels; I actually thought I taped it incorrectly.

The Same Space, But Round

This space is also a Real Projective Plane. It is connected in the same way as the rectangle.

Unfortunately, you cannot really make a paper model of the Real Projective Plane. It doesn't embed in 3D space. If you cut slits in the paper so that the paper can pass through itself you can though. Check out the link above to see what a 3D model of that looks like.

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