Four-Dimensional Paper Constructions After Möbius, Klein and Boy

The following sections are included:

• Introduction: If You Start on Earth and Travel Straight Up into the Universe, Where Will You End Up?
• About the Author
• Acknowledgements
• Contents
• Prologue: Time Machine and Teleportation

The following sections are included:

• ℝ⁴
• Getting Out in Four-Dimensional Space

The following sections are included:

• Let’s Construct Möbius Band
• Create a Möbius Band with Paper and Scissors

We construct a new figure by the following process. Begin with a square ABCD including the boundary and the body. Draw arrows on the edges like those shown in Figure 3.1.

Glue together the segments AB and DC so that the arrows coincide as in Figure 3.2. The points A and D meet, and the points B and C meet. Mathematically, we refer to points (respectively, segments) “meeting” as points (respectively, segments) being identified…

In this part, we use a paper construction to help familiarize the reader with four-dimensional space. In order to see objects in ℝ⁴, you only need a little bit of imagination!

It contains the following sections:

• Two Circles and a Disc
• You Can Do It in Four-Dimensional Space
• How to Draw Four Dimensional Space
• Put and Move Objects in Four-Dimensional Space
• The Hopf Link and Four-Dimensional Space
• A Construction using Paper, Wire, and Fire

The following sections are included:

• It Is Not a Torus
• The Klein Bottle and ℝ³
• The Klein Bottle and ℝ⁴
• How to Construct Klein Bottle
• Constructing the Torus

The following sections are included:

• One or Two Sides
• Non-orientability
• Caution!
• Boundary
• Klein Bottle = 2× (Möbius Band)

The following sections are included:

• It Is Not the Torus or Klein Bottle
• The Two-Dimensional Real Projective Space ℝP² and Four-Dimensional Space
• Möbius Band and a Disc
• Make Your Boy Surface
• Prove
• Five-Dimensional Space

We now answer a question posed in the Introduction: There is a possibility that our universe has the following property. If you leave Earth in a space ship and travel in one direction away from Earth, you will ever end up back at Earth again. Figure 8.1 is a sci-fi depiction of this scenario. Under this assumption, what is the shape of our universe?

The following sections are included:

• Movies
• Further Reading
• Index
• Epilogue: King of Inventors in the 27th Century