Helicoid Bonnet transformation

We have referred above to the promise of the Bonnet transformation in describing actual martensitic transformations. As a generalization of the catenoid/helicoid... 

The helicoid is the only minimal surface built up entirely of straight lines (a ruled surface) and the catenoid is the only minimal surface of revolution. These surfaces are related through the Bonnet transformation that will be discussed later. 

The distinct sequence of deformations is characterised mathematically by the variation in the Bonnet angle d in eqs. (18). The simplest example of this Bonnet transformation is the bending of the catenoid into the helicoid. (Fig. 1.19). 

The general features of the Bonnet transformation can be seen in the simplest example, namely the isometry between the catenoid and the helicoid (Fig. 1.19). Under the action of the transformation, each point on the surface traces an ellipse in space, centred at the origin. If the Cartesian coordinates of identical points on adjoint surfaces are (x,y,z) and (x",y",z"), the coordinates of an associate surface, characterised by a Bonnet angle of 0 are ... 

The effect of the Bonnet transformation on IPMS is to transform the lattice of catenoidal channels - characteristic of IPMS - into helicoidal strips, through a screw operation on the whole surface. For example, the channels in both the P- and D-surfaces are transformed into spiral tunnels in the g)u oid. Due to its intermediate Bonnet angle with respect to the P- and D-surfaces, it lacks straight lines (2-fold axes) and mirror planes. The labyrinths on both sides are enantiomorphic one labyrinth is left-handed and the other right-handed (Fig. 1.21). 

The main reaison for this remcirkable property of the DNA molecule is its general shape. The double helix sits on a helicoid, and therefore it shares the properties of that surface. The most important of these is the way the helicoid can be deformed, via the Bonnet transformation. 

Still another important property of die Bonnet transformation is that it imposes simultaneity on the system. In a true, mathematiced. Bonnet transformation, all points on the surface move in unison. In the DNA molecule, which is only a good approximation of the helicoid, the partial folding of the molecule at one location will lead to the imposition of a siiiular structure at nearby sites. 

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