Paradoxical Triangular Braid

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This triangular braid with three links colored red, yellow, and blue was designed by Koos Verhoeff and realized as mathematical art in three types of wood. The paradox comes about as follows. If there is no red link, then the blue link is completely on top of the yellow link. If there is no yellow link, then the red is completely on top of the blue. Finally, if there is no blue link, then the yellow link is completely on top of the red. How can it be that yellow is above red, red is above blue, and blue is above yellow?


In topology, a knot is an embedding of a circle in 3D (with no loose ends). A link is a collection of knots that may be intertwined.

This arrangement is an example of a Brunnian link, which cannot be taken apart without cutting a link, but when one link is removed, the others come apart without cutting. The Borromean rings are the simplest Brunnian links.


Contributed by: Tom Verhoeff  (August 2013)
(Eindhoven University of Technology)
Open content licensed under CC BY-NC-SA



See [1] for details on the design of this triangular braid. For a video, see


[1] T. Verhoeff, "3D Turtle Geometry: Artwork, Theory, Program Equivalence and Symmetry," International Journal of Arts and Technology, 3(2/3), 2010 pp. 288–319.

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