4D Unknot

In four dimensions, you can’t tie your shoelaces — because 4D knots don’t work. Any 1D curve in 4D space can be continuously deformed to the unit circle, which is anĀ unknot.

The looping animation below demonstrates how to undo a trefoil knot in 4D, where rainbow colors code the 4th dimension. The animation pauses when curve segments appear to intersect, but the segments’ different colors reveal their separation in the fourth dimension.

2D is too small to allow complicated neural circuits, and 4D is too large to enable knots; perhaps unsurprisingly, we find ourselves in a 3D world.

Unknotting a trefoil knot in 4D, where rainbow colors code the 4th dimension

Unknotting a trefoil knot in 4D, where rainbow colors code the 4th dimension

About John F. Lindner

John F. Lindner was born in Sleepy Hollow, New York, and educated at the University of Vermont and Caltech. He is an emeritus professor of physics and astronomy at The College of Wooster and a visiting professor at North Carolina State University. He has enjoyed multiple yearlong sabbaticals at Georgia Tech, University of Portland, University of Hawai'i, and NCSU. His research interests include nonlinear dynamics, celestial mechanics, and neural networks.
This entry was posted in Mathematics. Bookmark the permalink.

1 Response to 4D Unknot

  1. Pingback: 8 Topological Knots

Leave a Reply

Your email address will not be published. Required fields are marked *