Erwin Schrödinger famously and presciently ascribed the vehicle transmitting the hereditary information underlying life to an 'aperiodic crystal'. We compare and contrast this, only later discovered to be stored in the linear biomolecule DNA, with the information-bearing, layered quasi-one-dimensional materials investigated by the emerging field of chaotic crystallography. Despite differences in functionality, the same information measures capture structure and novelty in both, suggesting an intimate coherence between the information character of biotic and abiotic matter-a broadly applicable physics of information. We review layered solids and consider three examples of how information- and computation-theoretic techniques are being applied to understand their structure. In particular, (i) we review recent efforts to apply new kinds of information measures to quantify disordered crystals; (ii) we discuss the structure of ice I in information-theoretic terms; and (iii) we recount recent investigations into the structure of tris(bicyclo[2.1.1]hexeno)benzene, showing how an information-theoretic analysis yields additional insight into its structure. We then illustrate a new Second Law of Thermodynamics that describes information processing in active low-dimensional materials, reviewing Maxwell's Demon and a new class of molecular devices that act as information catalysts. Lastly, we conclude by speculating on how these ideas from informational materials science may impact biology.

We review recent progress in applying information-theoretic and computation-theoretic measures to describe material structure that transcends previous methods based on exact geometric symmetries. We discuss the necessary theoretical background for this new toolset and show how the new techniques detect and describe novel material properties. We discuss how the approach relates to well known crystallographic practice and examine how it provides novel interpretations of familiar structures. Throughout, we concentrate on disordered materials that, while important, have received less attention both theoretically and experimentally than those with either periodic or aperiodic order.

Given a description of the stacking statistics of layered close-packed structures in the form of a hidden Markov model, analytical expressions are developed for the pairwise correlation functions between the layers. These may be calculated analytically as explicit functions of model parameters or the expressions may be used as a fast, accurate and efficient way to obtain numerical values. Several examples are presented, finding agreement with previous work as well as deriving new relations.

Localized waves in disordered one-dimensional materials have been studied for decades, including white-noise and correlated disorder, as well as quasi-periodic disorder. How these wave phenomena relate to those in crystalline (periodic ordered) materials---arguably the better understood setting---has been a mystery ever since Anderson discovered disorder-induced localization. Nonetheless, together these revolutionized materials science and technology and led to new physics far beyond the solid state. We introduce a broad family of structurally complex materials---chaotic crystals---that interpolate between these organizational extremes---systematically spanning periodic structures and random disorder. Within the family one can tune the degree of disorder to sweep through an intermediate structurally disordered region between two periodic lattices. This reveals new transport and localization phenomena reflected in a rich array of energy-dependent localization degree and density of states. In particular, strong localization is observed even with a very low degree of disorder. Moreover, markedly enhanced localization and delocalization coexist in a very narrow range of energies. Most notably, beyond the simply smoothed bands found in previous disorder studies, islands of transport emerge in band gaps and sharp band boundaries persist in the presence of substantial disorder. Finally, the family of materials comes with rather direct specifications of how to assemble the requisite material organizations.

In the previous paper of this series [D. P. Varn, G. S. Canright, and J. P. Crutchfield, Physical Review B, submitted] we detailed a procedure--epsilon-machine spectral reconstruction--to discover and analyze patterns and disorder in close-packed structures as revealed in x-ray diffraction spectra. We argued that this computational mechanics approach is more general than the current alternative theory, the fault model, and that it provides a unique characterization of the disorder present. We demonstrated the efficacy of computational mechanics on four prototype spectra, finding that it was able to recover a statistical description of the underlying modular-layer stacking using epsilon-machine representations. Here we use this procedure to analyze structure and disorder in four previously published zinc sulphide diffraction spectra. We selected zinc sulphide not only for the theoretical interest this material has attracted in an effort to develop an understanding of polytypism, but also because it displays solid-state phase transitions and experimental data is available.

In a recent publication [D. P. Varn, G. S. Canright, and J. P. Crutchfield, Phys. Rev. B {\bf 66}:17, 156 (2002)] we introduced a new technique for discovering and describing planar disorder in close-packed structures (CPSs) directly from their diffraction spectra. Here we provide the theoretical development behind those results, adapting computational mechanics to describe one-dimensional structure in materials. By way of contrast, we give a detailed analysis of the current alternative approach, the fault model (FM), and offer several criticisms. We then demonstrate that the computational mechanics description of the stacking sequence--in the form of an epsilon-machine--provides the minimal and unique description of the crystal, whether ordered, disordered, or some combination. We find that we can detect and describe any amount of disorder, as well as materials that are mixtures of various kinds of crystalline structure. Underlying this approach is a novel method for epsilon-machine reconstruction that uses correlation functions estimated from diffraction spectra, rather than sequences of microscopic configurations, as is typically used in other domains. The result is that the methods developed here can be adapted to a wide range of experimental systems in which spectroscopic data is available.