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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. 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. The Balinese cup trick, appearing in the Balinese candle dance, is a different illustration of the same mathematical idea. 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 rotation of vectors does not encompass all of the properties of the abstract model of rotations given by the rotation group. The property being illustrated in this game is formally referred to in mathematics as the "double covering of SO(3) by SU(2)". This abstract concept can be roughly sketched as follows. Rotations in three dimensions can be expressed as 3x3 matrices, a block of numbers, one each for x,y,z.

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Concepts associés (3)

3D rotation group

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique inverse rotation, and the identity map satisfies the definition of a rotation.

Tangloids

Spineur

vignette|Le cube peut tourner continument sans que les ficelles qui le retiennent s'emmêlent. Après un mouvement de 360°, la configuration a changé. Mais au bout de 720° on revient à la position initiale. Un cube "détaché" se comporte comme un vecteur ordinaire, le cube attaché comme un spineur. Formellement, un spineur est un élément d'un espace de représentation pour le groupe spinoriel.