Tangloids
Encyclopedia
Tangloids is a mathematical game
for two players created by Piet Hein
to model the calculus of spinor
s.
Two flat blocks of wood each pierced with three tiny 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 one defined by 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 (sliding the pieces) 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 overlaping again but they can not be untagled without rotating one of the two wooden blocks.
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 rigid object in space. Specifically considering the rotation of vectors and derived quantities (i.e. tensor
s 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 paedagogical 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 streching 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.
Mathematical game
A mathematical game is a multiplayer game whose rules, strategies, and outcomes can be studied and explained by mathematics. Examples of such games are Tic-tac-toe and Dots and Boxes, to name a couple. On the surface, a game need not seem mathematical or complicated to still be a mathematical game...
for two players created by Piet Hein
Piet Hein (Denmark)
Piet Hein was a Danish scientist, mathematician, inventor, designer, author, and poet, often writing under the Old Norse pseudonym "Kumbel" meaning "tombstone"...
to model the calculus of spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
s.
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 rigid object in space. Specifically considering the rotation of vectors and derived quantities (i.e. tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s 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
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
representations. These are objects defined in mathematical terms that do transform under the given rotation group (see group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
) 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 paedagogical 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
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
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 streching 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.