Tangloids

Tangloids is a mathematical game for two players created by Piet Hein to model the calculus of spinors.

Tangloids apparatus

A description of the game appeared in the book "Martin Gardner's New Mathematical Diversions from Scientific American" by Martin Gardner from 1996 in a section on the mathematics of braiding.[1][2][3]

Two flat blocks of wood each pierced with three small holes are joined with three parallel strings. Each player holds one of the blocks of wood. The first player holds one block of wood still, while the other player rotates the other block of wood for two full revolutions. The plane of rotation is perpendicular to the strings when not tangled. The strings now overlap each other. Then the first player tries to untangle the strings without rotating either piece of wood. Only translations (moving the pieces without rotating) are allowed. Afterwards, the players reverse roles; whoever can untangle the strings fastest is the winner. Try it with only one revolution. The strings are of course overlapping again but they can not be untangled without rotating one of the two wooden blocks.

Mathematical purpose

This game serves to clarify the notion that rotations in space have properties that cannot be intuitively explained by considering only the rotation of a single rigid object in space. Specifically, considering the rotation of vectors and derived quantities (i.e., tensors of higher order via tensor multiplication) does not provide for all the properties of rotations as a more abstract concept. The extra information in representation theory of groups is provided by the spinor representations. These are objects defined in mathematical terms that do transform under the given rotation group (see group theory) but however their properties cannot be visualized with the idea of rotating a rigid object. These extra features are provided for in this game with the presence of strings.

The pedagogical aim is to show that rotations have extra consequences when one considers properties of the object being subjected to them in relation with its surroundings or space itself. Without trying to make a direct analogy one can be convinced of the importance of considering these extra properties following the rationale implied by this game: an object is defined here consisting of two rods and strings connecting them. Applying a rotation means here rotating one of the two rods 360 degrees. The rod returns in the same place as before the rotation, thus we say that it transforms as a vector under rotations in three-dimensional space (i.e. under the special orthogonal group of dimension 3). We are not saying any more here other than that if you turn a full circle around your self you will end up where you were before. However, the object as we defined it being the two rods and string is not in the same state as before, the strings are entangled and cannot be unentangled without applying again a rotation in any part of the system. If we rotate the rod again in the same directions so that it will have completed a 720 degrees rotation in total the strings can be untangled without rotating any part (e.g. by "sliding" the rods and/or stretching the strings). We then say it transforms as a spinor. This is actually how an electron behaves and we say it is a spin-1/2 particle.

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See also

References
  1. Piet Hein, www.piethein.com, downloaded 13-12-2011
  2. Scientific American excerpt of the book by M. Gardner: Martin Gardner's New Mathematical Diversions from Scientific American, Simon and Schuster, 1996, ISBN 978-0-671-20989-6
  3. M. Gardner: Sphere Packing, Lewis Carroll, and Reversi: Martin Gardner's New Mathematical Diversions Archived 2012-04-06 at the Wayback Machine, Cambridge University Press, September, 2009, ISBN 978-0-521-75607-5
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