Quicktime Movies of Bob Palais demonstrating the Belt Trick and the Plate Trick

Bob Palais' Java and Flash visualizations of mathematical interpretations of the belt trick, the plate trick, and the tangle trick.

The belt trick and the plate trick are related to the fact the although the path in the three-dimensional rotation group from the identity to itself corresponding to a single rotation about a fixed axis cannot be continuously deformed to the constant path at the identity while leaving the endpoints fixed, but the path corresponding to two rotations about that axis can. More generally, any rotation of an object in space corresponding to an even number of rotations about a fixed axis can be continuously deformed to any other with the same initial and final configuration, and similarly for odd numbers, but odd and even numbers of rotations about an axis cannot be deformed to each other. The corresponding fact about plane rotations is that every path in the plane rotation group from the identity to itself can be deformed to exactly one standard path corresponding to some integer number of counterclockwise rotations about the the fixed center of the rotation. The axis in the belt and plate tricks cannot stay fixed during the rotation, or we would be able to deform two rotations in the plane to zero, and that is not possible. The `deformation classes' of plane rotations form a group called the fundamental group of plane rotations, isomorphic to the group of integers. The belt and plate tricks are manifestations of the fact that the fundamental group of three dimensional rotations is isomorphic to the group of remainders of integers, mod 2, obeying the addition rule for even and odd integers.

The first animation is a java applet (my first and only) that attempts to take the `translation' out of the tricks, and view the deformation in terms of rotations of a cube that leave its center fixed. A set of cubes are rotated simultaneously in time from an initial reference configuration to itself, with the leftmost cube undergoing two full turns about the vertical axis, and the rightmost cube staying fixed. The intermediate cubes are discrete slices of a continuous deformation of the leftmost path to the rightmost constant path. The corresponding paths all lie in a plane under a correspondence between three-dimensional rotations and three-dimensonal space plus the point at infinity, and are drawn below the cubes. An amusing fact about the animation is that it is implemented by rotating the graphics viewpoint around a single fixed cube in different ways, so what looks like twelve cubes undergoing different rotations is really just a single cube being viewed from differently changing perspectives simultaneously side-by side!

The Belt Trick and Plate Trick as Rotations of a Cube

The next pair of links are static and show snapshots of the orientation of the cubes in the previous animation with time shown horizontally left to right, and the deformation that was from left to right in the previous animation is done from top to bottom here, and the position of the cube relative to the axis, and perspective have been changed as well. This is done to make the correspondence with the plate trick more natural, so that it is the top face of the top `hand' cubes remain flat as they rotate twice about the vertical axis, and the bottom `shoulder' cubes remain fixed. In this representation, the straight `arms' at the left and right ends of the plate trick also represent the fixed common initial and final configurations of the belt, which morphs from twice twisted belt at the top to the flat belt at the bottom. This visualization may be the closest the the technical mathematical description of the deformations as a `homotopy', a mapping from the product of intervals [0,1]x[0,1] into the three-dimensional rotation group, SO(3), where one interval parametrizes each path from the identity to itself, and the other interval parametrizes the deformation.

The next Flash animation simply shows simultaneous vertical and horizontal slices of the `homotopy square' shown in the previous link, where the resolution can be increased or decreased by the user.

The Orientation Entanglement Trick is described and drawn in the classic tome on General Relativity, Gravitation, by Misner, Thorne, and Wheeler, a book as weighty as its subject. Here is a quicktime Tangle Trick Movie. The next link extends the belt and plate trick homotopies to the tangle trick homotopy by following the plate trick by itself in reverse, or alternatively, reflecting the plate trick across one end of the belt. This corresponds to the fact that when the center of the originally flat belt (or set of strings) is twisted twice about the vertical axis, the twists in the belt between one end and the center are reversed as we pass the center and continue to the opposite end of the belt. As we point out, the tangle trick will not work physically if two twists with the same orientation were put into the belt to begin with. This would correspond to the case of two plate tricks or belt tricks are repeated periodically instead of reversed or reflected! This involves some higher topology, and points out a subtle distinction between the mathematics and the physical demonstrations.

Orientation Entanglement Trick (`Tangle Trick') Static Homotopy

Finally, here are some links and references for other animations and discussions of the belt trick, the plate trick, and the tangle trick.

A beautiful early Java animation of the Dirac belt trick is one of many beautiful and mathematically authentic mathematical applets created by award-winning science fiction author Greg Egan.

Another excellent animation has appeared on Dror Bar-Natan's website highlighting sophisticated visualizations by many of his students. Although there was a discrepancy between the posted parametrization and the animation by Matthew Song that was posted (noted on the current page), it was nonetheless very nicely presented, and the intended parametrization was imported and implemented in the Surfaces section of 3D-XplorMath, that can be downloaded and viewed in anaglyph 3D as a Java or Mac application. In contrast to the animations above ( including Greg Egan's) these animations show belt tricks with ends of the belt that move relative to each other (though their relative orientation remains the same as it must.) I would like to systematically (mathematically) modify my Flash applets so that the physical translation of the belt performs a belt trick with fixed ends, as is shown in the quicktime movie and Greg Egan's applet. (Greg Egan's wrote me that his centerline motion was not obtained directly from the homotopy, but constructed ad hoc, which if anything makes it even more impressive.) If the belt is originally straight, to pass over its end it must be stretched, so arc-length cannot be preserved as it is in Dror Bar-Natan's version. If the arc-length is to be preserved, the belt must be initially `slack', as the non-stretch belt is in the quicktime move.

A novel yet simple and familiar way to realize the group SU(2) of unit quaternions (as well as the three-dimensional rotation group and general quaternions) will appear soon in the online math journal LOCI. It realizes SU(2) as equivalence classes of ordered pairs of unit vectors in three-space, in the same way that we commonly view the translation and plane rotation groups as a geometric composition on equivalence classes of ordered pairs of points in space or unit vectors in the plane. This view of SU(2) as a quotient of a product of two-spheres leads to a simple interpretation of the belt and plate trick homotopies by reducing the simple connectivity of SU(2) to that of one of the two-spheres, and the non-simple connectedness of SO(3) to the disconnectedness of the zero-sphere, {-1,+1}.

A very nice article on the geometric and algebraic aspects of the belt and plate tricks and generalizations involving plaiting of braids, "Touching the Z2 in Three-Dimensional Rotations" by Vesna Stojanoska and Orlin Stoytchev appears in the December 2008 issue of Math Magazine.

A 1993 movie `Air on the Dirac Strings' shows many belt-like tricks, though one end is rotating relative to the other in many of the iterations, and the connection with the basic mathematical facts are not elaborated upon.