torsion

Torsion describes a condition wherein an independent set of n unison vectors
{u1, u2, ..., un} (<uvector.htm>) defines a non-epimophic (epimorphic.htm>)
periodicity block, because of the existence a torsion element, meaning an
interval which is not the product

u1^e1 u2^e2 ... un^en

of the set of unison vectors raised to (positive, negative or zero) integral
powers, but some integer power of which is. An example would be a block defined
by 648/625 and 2048/2025;
here 81/80 is not a product of these commas, but
(81/80)^2 = (648/625) (2048/2025)^(-1).


Torsion may be tested by forming the n by n+1 matrix whose rows correspond to
the unison vectors, and calculating the
gcd(<<http://mathworld.wolfram.com/GreatestCommonDivisor.html>>)
of the minors
(<<http://mathworld.wolfram.com/Minor.html>>)
of the matrix. If the rows are linearly independent but the gcd is not one, we
have torsion.

The term comes from mathematical usage, see
<<http://mathworld.wolfram.com/TorsionGroup.html>>.