# How do Slinkies become tangled?

The following image describes the problem better than I can:

As you know, sometimes Slinkies can twist such that the direction of the coil can be reversed. However, though reversed, the coil still maintains its radius, number of turns, and shape. How is this possible? Intuitively, untwisting a slinky should increase the radius of curvature and decrease the number of turns.

I have read about inverting spheres and topological spheres with holes, and this problem seems as though it could be analyzed with similar methods. I am new to topology but if someone could point me in the right direction, either with a resource (textbook, paper) or an explanation that would be helpful.

• Note sure if this is really a math question, but spiral phone cords do this, too. – Thomas Andrews Jul 23 '13 at 18:14
• Funnily, I found this while I was on Quora. Not sure if it will be helpful, but the authors talk about number of ways of tangling of headphones. – Torsten Hĕrculĕ Cärlemän Jul 23 '13 at 18:17
• It's not about Slinkies, but this paper describes how a certain metal band can coil: Topology Explains Why Automobile Sunshades Fold Oddly – Chris Culter Jul 23 '13 at 21:00