Chiral polycubes

Figure 6.4 Mirror image pair of the smallest chiral polycubes.

For both two- and three-dimensional chirality, a formal degree of chirality has been introduced [54,55], based on a discretization of shape features using lattice animals and polycubes [240,243]. These definitions are based on chiral animals and chiral polycubes, for which chirality can be detected by simple algebraic means. 

The smallest chiral polycube has four cubes (as shown in Figure 6.4), and the number 3 in the denominator ensures that the den Ve of chirality takes values from the [0,1] interval. 

The chirality quantification technique proposed by Harary and Mezey [54,55] is motivated by the Resolution Based Similarity Measure (RBSM) approach used in more general molecular similarity analysis [243]. This method does not rely on a single reference object. Instead, it characterizes shape on any desired finite level of resolution by considering various A(J,n) parts of square lattices, called lattice animals or P(G,n) parts of cubic lattices called polycubes which can be inscribed within the two- or three-dimensional objects J or G, respectively. In the above... 

The actual determination of a set M for some chiral set T and the calculation of the volume v(T) are usually rather difficult problems (see some relevant comments in references [51-53,58,240,242]), and the same applies for superset N. However, within a RBSM framework, the analogous chirality measures given in terms of a discretization procedure using polycubes (or lattice animals in 2D) [240] do not require the explicit determination of a maximal volume (area) achiral subset M and the calculation of its exact volume (or area) v(M). 

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