How Efficiently Can One Untangle a Double-Twist? Waving is Believing!

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 69

This is the net price. Taxes to be calculated in checkout.


  1. [1]

    John H. Conway, Derek A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A. K. Peters, Natick, MA, 2003.

  2. [2]

    Charles Curtis, Linear Algebra, Springer, New York, 1993.

  3. [3]

    Antonio Martos de la Torre, Dirac’s Belt Trick for Spin 1/2 Particle,, accessed October 2, 2016.

  4. [4]

    Greg Egan, Dirac Belt Trick, ETS/21/21.html, accessed October 2, 2016.

  5. [5]

    Leonhard Euler, Formulae generales pro translatione quacunque corporum rigidorum (General formulas for the translation of arbitrary rigid bodies), E478, Novi Commentarii Academiae Scientiarum Petropolitanae 20 (1776), 189–207; http://eulerarch, accessed October 2, 2016.

  6. [6]

    Euler’s Rotation Theorem, Wikipedia,’s_rotation_theorem, accessed October 2, 2016.

  7. [7]

    Euler’s Theorem␣(Rotation), Citizendium,’s_theorem_(rotation), accessed October 2, 2016.

  8. [8]

    Richard P. Feynman, Steven Weinberg, Elementary Particles and the Laws of Physics: The 1986 Dirac Memorial Lectures; Lecture Notes Compiled by Richard MacKenzie and Paul Doust, Cambridge University Press, New York, Cambridge, 1987.

  9. [9]

    George Francis, Louis Kauffman, Air on the Dirac strings, in Mathematical Legacy of Wilhelm Magnus, Contemporary Mathematics 169 (1994), 261–276.

  10. [10]

    George Francis et al, Air on the Dirac Strings, movie, University of Illinois, Chicago, 1993,,,, accessed October 2, 2016.

  11. [11]

    Alfred Gray, Elsa Abbena, Simon Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, Chapman & Hall CRC, Boca Raton, FL, 2006.

  12. [12]

    Jean Gallier, Geometric Methods and Applications: For Computer Science and Engineering, Springer, New York, 2001.

  13. [13]

    Andrew J. Hanson, Visualizing Quaternions, Elsevier, 2006.

  14. [14]

    John Hart, George Francis, Louis Kauffman, Visualizing quaternion rotation, ACM Transactions on Graphics 13 (1994), 256–276.

  15. [15]

    Morris Hirsch, Differential Topology, Springer, New York, 1997.

  16. [16]

    Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, W. H. Freeman, San Francisco, 1973.

  17. [17]

    Orientation Entanglement, Wikipedia,, accessed October 2, 2016.

  18. [18]

    Robert A. (Bob) Palais, Bob Palais’ Belt Trick, Plate Trick, Tangle Trick Links,,, http://, accessed October 2, 2016.

  19. [19]

    David Pengelley, Daniel Ramras, The double-tipping nullhomotopy,, accessed October 2, 2016. Also at

  20. [20]

    Roger Penrose, Wolfgang Rindler, Spinors and Space-Time: Volume 1, Cambridge University Press, 1984.

  21. [21]

    Donnelly Phillips, Mae Markowski, Joseph Frias, Mark Boahen, Virtual Reality Experience of Nullhomotopy in SO(3,R) , Oculus Rift App, George Mason University Experimental Geometry Lab,, accessed October 2, 2016.

  22. [22]

    Plate Trick, Wikipedia,, accessed October 2, 2016.

  23. [23]

    Quaternions and Spatial Rotation, Wikipedia, http://en.wikipedia. org/wiki/Quaternions_and_spatial_rotation, accessed October 2, 2016.

  24. [24]

    Tari Piring - Indonesian Candle Dance,, accessed October 2, 2016.

Download references

Author information

Correspondence to David Pengelley.

Additional information

An erratum to this article is available at

Electronic Supplementary Material

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pengelley, D., Ramras, D. How Efficiently Can One Untangle a Double-Twist? Waving is Believing!. Math Intelligencer 39, 27–40 (2017).

Download citation