How efficiently can one untangle a double-twist? Waving is believing!
Abstract
It has long been known to mathematicians and physicists that while a full rotation in three-dimensional Euclidean space causes tangling, two rotations can be untangled. Formally, an untangling is a based nullhomotopy of the double-twist loop in the special orthogonal group of rotations. We study a particularly simple, geometrically defined untangling procedure, leading to new conclusions regarding the minimum possible complexity of untanglings. We animate and analyze how our untangling operates on frames in 3-space, and teach readers in a video how to wave the nullhomotopy with their hands.
Keywords
Cite
@article{arxiv.1610.04680,
title = {How efficiently can one untangle a double-twist? Waving is believing!},
author = {David Pengelley and Daniel Ramras},
journal= {arXiv preprint arXiv:1610.04680},
year = {2018}
}
Comments
To appear in The Mathematical Intelligencer. For supplemental videos, see http://www.math.iupui.edu/~dramras/double-tip.html , or https://www.youtube.com/playlist?list=PLAfnEXvHU52ldJaOye-8kZV_C1CjxGx2C . For a supplemental virtual reality experience, see http://meglab.wikidot.com/visualization