Helix irrational

When the time dependence takes the form of a periodic perturbation of some parameter, we speak of this as periodic forcing.19 The response will obviously not be a steady state, but can be periodic, quasi-periodic, or chaotic. If the response is periodic, it may be with a period that is a multiple of the period of forcing. It is quasi-periodic if the response winds itself onto the cylinder in a helix whose pitch is an irrational multiple of the forcing period, so that it is never quite truly periodic. An example20 of a forced system is the Gray-Scott autocatalator with the feed concentration sinusoidally perturbed ... 

It is of interest to note that one may change the translation lattice of Fig. 5.3 by replacing the translation lattice vector c with the molecular helix lattice, keeping the translation symmetries a and This would lead to a match of the molecular helix symmetry with the crystal symmetry and even for irrational helices, a crystal stracture symmetry would be recognized. In fact, a whole set of new lattices can be generated replacing all three translation symmetry operations by helix symmetry operations [5]. Since a 1 1/1 hehx has a translational symmetry, this new space lattice description with helices would contain the traditional crystallography as a special case. 

Here a is the radius. Is this helix left- or right-handed Write down the parameterization of its enantiomer. The symmetry of a helix is based on a screw axis, which corresponds to a translation in t. It is composed of a translation along the z-direction with a concomitant rotation in the xy-plane. Now decorate the helix with atoms at points tk/a = 2nk/m, where k and m are integers. Determine the screw symmetry of this molecular helix. If n/m is irrational, the helix is noncommensurate. Will it stiU have a symmetry in this case ... 

Note that a uniform sign change of t would leave the right-handed helix unchanged. For the discrete helix, the screw symmetry consists of a translation in the z-direction over a distance InaIm in combination with a rotation around the z-axis over an angle Innlm.lfmis, irrational, the helix will not be periodic, and the screw symmetry is lost. 

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