Tangloids
Tangloids is a mathematical game for two players created by Piet Hein to model the calculus of spinors.
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.
Then the first player tries to untangle the strings without rotating either piece of wood.
Afterwards, the players reverse roles; whoever can untangle the strings fastest is the winner.
The strings are of course overlapping again but they can not be untangled without rotating one of the two wooden blocks.
The anti-twister mechanism is a device intended to avoid such orientation entanglements.
A mathematical interpretation of these ideas can be found in the article on quaternions and spatial rotation.
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.
The property being illustrated in this game is formally referred to in mathematics as the "double covering of SO(3) by SU(2)".
In small neighborhoods, this collection of nearby points resembles Euclidean space.
In fact, it resembles three-dimensional Euclidean space, as there are three different possible directions for infinitesimal rotations: x, y and z.
This properly describes the structure of the rotation group in small neighborhoods.
Systems that behave like Euclidean space on the small scale, but possibly have a more complicated global structure are called manifolds.
At the completion of this 360 degree journey, one has not arrived back home, but rather instead at the polar opposite point.
And one is stuck there -- one can't actually get back to where one started until one makes another, a second journey of 360 degrees.
The structure of this abstract space, of a 3-sphere with polar opposites identified, is quite weird.
One can try to imagine taking a balloon, letting all the air out, then gluing together polar opposite points.
If attempted in real life, one soon discovers it can't be done globally.
Locally, for any small patch, one can accomplish the flip-and-glue steps; one just can't do this globally.
, the circle, and attempt to glue together polar opposites; one still gets a failed mess.
The best one can do is to draw straight lines through the origin, and then declare, by fiat, that the polar opposites are the same point.
The so-called "double covering" refers to the idea that this gluing-together of polar opposites can be undone.
This can be explained relatively simply, although it does require the introduction of some mathematical notation.
This is the result of applying an ordinary, "common sense" rotation to
The tangeloid game is meant to illustrate that a 360 degree rotation takes one on a path from
The notion of double-covering used here is a generic phenomenon, described by covering maps.
Covering maps are in turn a special case of fiber bundles.
The classification of covering maps is done via homotopy theory; in this case, the formal expression of double-covering is to say that the fundamental group is
In this sense, the rotation group provides the doorway, the key to the kingdom of vast tracts of higher mathematics.
