Structural complexity of minerals: information storage and processing in the mineral world

Abstract Structural complexity of minerals is characterized using information contents of their crystal structures calculated according to the modified Shannon formula. The crystal structure is considered as a message consisting of atoms classified into equivalence classes according to their distribution over crystallographic orbits (Wyckoff sites). The proposed complexity measures combine both size- and symmetry-sensitive aspects of crystal structures. Information-based complexity parameters have been calculated for 3949 structure reports on minerals extracted from the Inorganic Crystal Structure Database. According to the total structural information content, IG, total , mineral structures can be classified into very simple (0-20 bits), simple (20-100 bits), intermediate (100-500 bits), complex (500-1000 bits), and very complex (> 1000 bits). The average information content for mineral structures is calculated as 228(6) bits per structure and 3.23(2) bits per atom. Twenty most complex mineral structures are (IG, total in bits): paulingite (6766.998), fantappieite (5948.330), sacrofanite (5317.353), mendeleevite-(Ce) (3398.878), bouazzerite (3035.201), megacyclite (2950.928), vandendriesscheite (2835.307), giuseppetite (2723.097), stilpnomelane (2483.819), stavelotite-(La) (2411.498), rogermitchellite (2320.653), parsettensite (2309.820), apjohnite (2305.361), antigorite (m = 17 polysome) (2250.397), tounkite (2187.799), tschoertnerite (2132.228), farneseite (2094.012), kircherite (2052.539), bannisterite (2031.017), and mutinaite (2025.067). The following complexity-generating mechanisms have been recognized: modularity, misfit relationships between structure elements, and presence of nanoscale units (clusters or tubules). Structural complexity should be distiguished from topological complexity. Structural complexity increases with decreasing temperature and increasing pressure, though at ultra-high pressures, the situation may be different. Quantitative complexity measures can be used to investigate evolution of information in the course of global and local geological processes involving formation and transformation of crystalline phases. The information-based complexity measures can also be used to estimate the ‘ease of crystallization’ from the viewpoint of simplexity principle proposed by J.R. Goldsmith (1953) for understanding of formation of simple and complex mineral phases under both natural and laboratory conditions. According to the proposed quantitative approach, the crystal structure can be viewed as a reservoir of information encoded in its complexity. Complex structures store more information than simple ones. As erasure of information is always associated with dissipation of energy, information stored in crystal structures of minerals must have an important influence upon natural processes. As every process can be viewed as a communication channel, the mineralogical history of our planet on any scale is a story of accumulation, storage, transmission and processing of structural information.

[1]  A. P. Shevchenko,et al.  TOPOS3.2: a new version of the program package for multipurpose crystal- chemical analysis , 2000 .

[2]  M. Hawthorne.  THE CRVSTAL STRUCTURE OF GEMINITE , Cu 2 * { AsOrOH ) ( H 2 Ol , A HETEROPOLYHEDRAL SHEET STRUCTURE , 2006 .

[3]  Hazen,et al.  Review Paper. Mineral evolution , 2008 .

[4]  S. Krivovichev.  Information-based measures of structural complexity: application to fluorite-related structures , 2012, Structural Chemistry.

[5]  T. Armbruster,et al.  Stavelotite-(La), a new lanthanum-manganese-sorosilicate mineral from the Stavelot Massif, Belgium , 2005 .

[6]  M. Buerger,et al.  Serendibite, a complicated, new, inorganic crystal structure. , 1974, Proceedings of the National Academy of Sciences of the United States of America.

[7]  E. Lutz,et al.  Experimental verification of Landauer’s principle linking information and thermodynamics , 2012, Nature.

[8]  W. E. Brown,et al.  XO4n- ion hydration. The crystal structure of magnesium phosphate docosahydrate , 1978 .

[9]  U. Kolitsch.  The crystal structures of kidwellite and ‘laubmannite’, two complex fibrous iron phosphates , 2004, Mineralogical Magazine.

[10]  A. Putnis,et al.  The symmetry of vesuvianite , 1993 .

[11]  U. Kolb,et al.  Towards automated diffraction tomography. Part II--Cell parameter determination. , 2008, Ultramicroscopy.

[12]  U. Kolb,et al.  The structure of charoite, (K,Sr,Ba,Mn)15-16(Ca,Na)32[(Si70(O,OH)180)](OH,F)4.0‧nH2O, solved by conventional and automated electron diffraction , 2010, Mineralogical Magazine.

[13]  M. Catti,et al.  A case of polytypism in hydrated oxysalts: the crystal structure of Mg3(PO4)2 · 22 H2O-II , 1981 .

[14]  T. Encrenaz,et al.  Global Mineralogical and Aqueous Mars History Derived from OMEGA/Mars Express Data , 2006, Science.

[15]  S. Krivovichev,et al.  Crystal chemistry of lead oxide chlorides. I. Crystal structures of synthetic mendipite, Pb3O2Cl2, and synthetic damaraite, Pb3O2(OH)Cl , 2001 .

[16]  G. Giester,et al.  Tschortnerite, a copper-bearing zeolite from the Bellberg volcano, Eifel, Germany , 1998 .

[17]  D. Peacor,et al.  Transmission Electron Microscope Observations of Illite Polytypism , 1991 .

[18]  Susan L. Brantley,et al.  Frontiers in exploration of the critical zone , 2005 .

[19]  M E J Newman,et al.  Modularity and community structure in networks. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[20]  S. Krivovichev,et al.  Where are genes in paulingite? Mathematical principles of formation of inorganic materials on the atomic level , 2008 .

[21]  A. Kolmogorov.  Three approaches to the quantitative definition of information , 1968 .

[22]  THE CRYSTAL STRUCTURE AND HYDROGEN BONDING OF MAGNESIUM SULFATE HEXAHYDRATE , 1964 .

[23]  F. Liebau,et al.  Structural Chemistry of Silicates: Structure, Bonding, and Classification , 1985 .

[24]  T. Armbruster,et al.  Relationship among metamorphic grade, vesuvianite ”rod polytypism”, and vesuvianite composition , 2006 .

[25]  Y. Takéuchi,et al.  Electron-density distributions of three orthopyroxenes, Mg2Si2O6, Co2Si2O6, and Fe2Si2O6 , 1982 .

[26]  P. Moore,et al.  Crystal structure of steenstrupine: A rod structure of unusual complexity , 1983 .

[27]  F. Hawthorne,et al.  Paravinogradovite, (Na, □)2 [(Ti4+, Fe3+)4 {Si2 O6}2 {Si3 al O10} (OH)4] H2O, a new mineral species from the Khibina Alkaline Massif, Kola Peninsula, Russia: Description and crystal structure , 2003 .

[28]  I. Sunagawa,et al.  Hexagonal CaAl2Si2O8 in a high temperature solution; metastable crystallization and transformation to anorthite , 1995 .

[29]  P. Burns,et al.  A NEW RARE-EARTH-ELEMENT URANYL CARBONATE SHEET IN THE STRUCTURE OF BIJVOETITE-(Y) , 2000 .

[30]  W. H. Baur,et al.  Salt hydrates. IX. The comparison of the crystal structure of magnesium sulfate pentahydrate with copper sulfate pentahydrate and magnesium chromate pentahydrate , 1972 .

[31]  J. Paasiv́irta,et al.  The Crystal Structure of Mg3(PO4)2. , 1968 .

[32]  W. H. Baur.  On the crystal chemistry of salt hydrates. II. A neutron diffraction study of MgSO4.4H2O , 1964 .

[33]  F. Hawthorne.  Towards a structural classification of minerals , 1984 .

[34]  M. Epple,et al.  Hydrated barium aluminosilicates, BaAl 2 Si 2 O 8 .n H 2 O, and their relations to cymrite and hexacelsian , 1992 .

[35]  G. Artioli,et al.  Quantitative determination of chrysotile asbestos in bulk materials by combined Rietveld and RIR methods , 1995, Powder Diffraction.

[36]  Sergey V. Krivovichev,et al.  Algorithmic crystal chemistry: A cellular automata approach , 2012 .

[37]  C. HlwrrroRNE,et al.  The crystal structure of spangolite, a complex copper sulfate sheet mineral , 1993 .

[38]  K. Campbell,et al.  Mineralogical and textural changes accompanying ageing of silica sinter , 2000 .

[39]  R. Landauer.  The physical nature of information , 1996 .

[40]  A. Mackay.  On complexity , 2001 .

[41]  F. Hawthorne.  Structural aspects of oxide and oxysalt crystals , 1994 .

[42]  H. T. Evans.  The crystal structures of low chalcocite and djurleite , 1979 .

[43]  M. Gazzano,et al.  Tubular-shaped stoichiometric chrysotile nanocrystals. , 2004, Chemistry.

[44]  M. M. Boldyreva,et al.  PAUTOVITE, CsFe2S3, A NEW MINERAL SPECIES FROM THE LOVOZERO ALKALINE COMPLEX, KOLA PENINSULA, RUSSIA , 2005 .

[45]  DaNrnr.  Natural kalsilite, KAlSiO4, with P31c symmetry: Crystal structure and twinning , 2007 .

[46]  D. Pedron,et al.  Evidence of dmisteinbergite (hexagonal form of CaAl2Si2O8) in pseudotachylyte: A tool to constrain the thermal history of a seismic event , 2010 .

[47]  F. Hawthorne.  Structural hierarchy in M[6]T[4]ϕn minerals , 1990 .

[48]  G. Capitani,et al.  The modulated crystal structure of antigorite: The m = 17 polysome , 2004 .

[49]  T. Schleid,et al.  Lanthanido ammonium cations [NM4]9+ as main structural features in lanthanide(III) nitride chalcogenides and their derivatives , 2008 .

[50]  Robert M. Hazen,et al.  Evolution of uranium and thorium minerals , 2009 .

[51]  PnrBn C. BunNs.  A new uranyl oxide hydrate sheet in vandendriesscheite : Implications for mineral paragenesis and the corrosion of spent nuclear fuel , 2007 .

[52]  D. Peacor,et al.  Hyttsjöite, a new, complex layered plumbosilicate with unique tetrahedral sheets from Långban, Sweden , 1996 .

[53]  B. Velde,et al.  Illite: Origins, Evolution and Metamorphism , 2010 .

[54]  T. Armbruster,et al.  Mineralogy and crystal structure of bouazzerite from Bou Azzer, Anti-Atlas, Morocco: Bi-As-Fe nanoclusters containing Fe3+ in trigonal prismatic coordination , 2007 .

[55]  F. Hawthorne,et al.  The crystal structure and crystal chemistry of mendeleevite-(Ce), (Cs,⃞)6(⃞,Cs)6(⃞,K)6(REE,Ca,⃞)30(Si70O175)(H2O,OH,F,⃞)35, a potential microporous material , 2011, Mineralogical Magazine.

[56]  B. W. Evans.  The Serpentinite Multisystem Revisited: Chrysotile Is Metastable , 2004 .

[57]  Giovanni Ferraris,et al.  Crystallography of Modular Materials , 2004 .

[58]  J. Zemann,et al.  The crystal structure of caratiite , 1984, Mineralogical Magazine.

[59]  K. Sahl.  Zur Kristallstruktur von Lanarkit, Pb2O(SO4) , 1970 .

[60]  M. Jansen,et al.  Structural relationships between cations and alloys; an equivalence between oxidation and pressure. , 2002, Acta crystallographica. Section B, Structural science.

[61]  S. Menchetti,et al.  The halotrichite group: the crystal structure of apjohnite , 1976, Mineralogical Magazine.

[62]  D. Y. Pushcharovsky,et al.  The mineralogy and the origin of deep geospheres: A review , 2012 .

[63]  F. Hawthorne,et al.  THE CHEMISTRY OF VESUVIANITE , 1992 .

[64]  V. Trommsdorff,et al.  Antigorite polysomatism: behaviour during progressive metamorphism , 1987 .

[65]  Anthony L. Spek,et al.  Journal of , 1993 .

[66]  M. Sebastiani,et al.  Kircherite, a new mineral of the cancrinite-sodalite group with a 36-layer stacking sequence: Occurrence and crystal structure , 2010 .

[67]  Jason Eugene Stephens,et al.  Phase transitions in K2Cr2O7 and structural redeterminations of phase II. , 2004, Acta crystallographica. Section B, Structural science.

[68]  Jacek Klinowski,et al.  Towards a grammar of inorganic structure , 1984 .

[69]  P. Moore,et al.  The remarkable langbanite structure type; crystal structure, chemical crystallography, and relation to some other cation close-packed structures , 1991 .

[70]  B. Andersson,et al.  Decomposition and ordering in Fe1−xO , 1975 .

[71]  S. Krivovichev,et al.  Rickturnerite, Pb7O4[Mg(OH)4](OH)Cl3, a complex new lead oxychloride mineral , 2012, Mineralogical Magazine.

[72]  C. Burnham,et al.  A COMPREHENSIVE STRUCTURE.MODEL FOR VESUVIANITE: SYMMETRV VARIATIONS AND CRYSTAL GROWTH , 1992 .

[73]  Franco Nori,et al.  Colloquium: The physics of Maxwell's demon and information , 2007, 0707.3400.

[74]  S. Lloyd.  Programming the Universe: A Quantum Computer Scientist Takes on the Cosmos , 2006 .

[75]  Joshua J. Golden,et al.  Mercury (Hg) mineral evolution: A mineralogical record of supercontinent assembly, changing ocean geochemistry, and the emerging terrestrial biosphere , 2012 .

[76]  R. Wirth,et al.  Antigorite Pressure and temperature dependence of polysomatism and water content , 2001 .

[77]  Peter C. Burns,et al.  U6+ MINERALS AND INORGANIC COMPOUNDS: INSIGHTS INTO AN EXPANDED STRUCTURAL HIERARCHY OF CRYSTAL STRUCTURES , 2005 .

[78]  P. Shen,et al.  Kumdykolite, an orthorhombic polymorph of albite, from the Kokchetav ultrahigh-pressure massif, Kazakhstan , 2008 .

[79]  A. Navrotsky,et al.  Thermochemistry of microporous and mesoporous materials. , 2009, Chemical reviews.

[80]  S. Borisov,et al.  Oxocentered polycationic complexes — An alternative approach to crystal-chemical investigation of the structure of natural and synthetic mercury oxosalts , 2000 .

[81]  P. Bonazzi,et al.  Natural kalsilite, KAlSiO4, with P31c symmetry: Crystal structure and twinning , 1997 .

[82]  Gregory J. Chaitin,et al.  Algorithmic Information Theory , 1987, IBM J. Res. Dev..

[83]  David Olson,et al.  Atlas of Zeolite Framework Types , 2007 .

[84]  S. Krivovichev,et al.  Chloroxiphite Pb3CuO2(OH)2Cl2: structure refinement and description in terms of oxocentred OPb4 tetrahedra , 2008, Mineralogical Magazine.

[85]  T. Armbruster,et al.  Synchrotron X-ray diffraction study of the structure of shafranovskite, K2Na3(Mn,Fe,Na)4[Si9(O,OH)27](OH)2⋅nH2O, a rare manganese phyllosilicate from the Kola peninsula, Russia , 2004 .

[86]  A. DuNc.  Metastability in serpentine-olivine equilibria , 2022 .

[87]  F. Hawthorne,et al.  PARAERSHOVITE, Na3K3Fe3+2(Si4O10OH)2(OH)2(H2O)4, A NEW MINERAL SPECIES FROM THE KHIBINA ALKALINE MASSIF, KOLA PENINSULA, RUSSIA: DESCRIPTION AND CRYSTAL STRUCTURE , 2010 .

[88]  A. Navrotsky.  Energetic clues to pathways to biomineralization: precursors, clusters, and nanoparticles. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[89]  U. Kolb,et al.  Essential features of the polytypic charoite-96 structure compared to charoite-90 , 2011, Mineralogical Magazine.

[90]  John W. Morse,et al.  Ostwald Processes and Mineral Paragenesis in Sediments , 1988, American Journal of Science.

[91]  Prrnn C. BunNs,et al.  The crystal structure of sinkankasite , a complex heteropolyhedral sheet mineral , 2007 .

[92]  P. Burns,et al.  Structure and hydrogen bonding in preobrazhenskite, a complex heteropolyhedral borate , 1994 .

[93]  D. Bonchev.  On the complexity of directed biological networks , 2003, SAR and QSAR in environmental research.

[94]  M. Sano,et al.  Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality , 2010 .

[95]  P. Buseck,et al.  Revised structure models for antigorite: An HRTEM study , 2002 .

[96]  E. Bonaccorsi.  The crystal structure of giuseppettite, the 16-layer member of the cancrinite–sodalite group , 2004 .

[97]  Mark E. Davis,et al.  SAXS and USAXS Investigation on Nanometer-Scaled Precursors in Organic-Mediated Zeolite Crystallization from Gelating Systems , 1999 .

[98]  B. N. Taylor,et al.  The 1986 CODATA Recommended Values Of the Fundamental Physical Constants , 1987, Journal of Research of the National Bureau of Standards.

[99]  M. Mellini.  The crystal structure of lizardite 1T: hydrogen bonds and polytypism , 1982 .

[100]  Peter C. Burns,et al.  The crystal chemistry of uranium , 1999 .

[101]  S. Krivovichev.  Actinyl Compounds with Hexavalent Elements (S, Cr, Se, Mo) – Structural Diversity, Nanoscale Chemistry, and Cellular Automata Modeling , 2010 .

[102]  Th. Hahn,et al.  Nomenclature and generation of three-periodic nets: the vector method , 1984 .

[103]  W. Klee.  Crystallographic nets and their quotient graphs , 2004 .

[104]  R. Bodnar,et al.  An Occurrence of Metastable Cristobalite in High-Pressure Garnet Granulite , 1997, Science.

[105]  G. Capitani,et al.  The crystal structure of a second antigorite polysome (m = 16), by single-crystal synchrotron diffraction , 2006 .

[106]  D. Peacor,et al.  Refinement of the nepheline structure at several temperatures , 1970 .

[107]  N. Chatterjee,et al.  Synthesis, structure, thermodynamic properties, and stability relations of K-cymrite, K[AlSi3O8]·H2O , 1997 .

[108]  Matthias Dehmer,et al.  A history of graph entropy measures , 2011, Inf. Sci..

[109]  E. Abe,et al.  A structural model for charoite , 2009, Mineralogical Magazine.

[110]  P. Burns.  Nanoscale uranium-based cage clusters inspired by uranium mineralogy , 2011, Mineralogical Magazine.

[111]  F. Bellatreccia,et al.  Farneseite, a new mineral of the cancrinite - sodalite group with a 14-layer stacking sequence: occurrence and crystal structure , 2005 .

[112]  A. Knoll,et al.  Needs and opportunities in mineral evolution research , 2011 .

[113]  Charles H. Bennett,et al.  Notes on Landauer's Principle, Reversible Computation, and Maxwell's Demon , 2002, physics/0210005.

[114]  S. Merlino,et al.  Crystal structure of afghanite, the eight-layer member of the cancrinite-group; evidence for long-range Si, Al ordering , 1996 .

[115]  N. Bansal,et al.  Crystal growth kinetics in BaOAl2O32SiO2 and SrOAl2O32SiO2 glasses , 1996 .

[116]  N. Chukanov,et al.  Cattiite, Mg 3 (PO 4 ) 2 ·22H 2 O, a new mineral from Zhelezny Mine (Kovdor Massif, Kola Peninsula, Russia) , 2002 .

[117]  F. Hawthorne,et al.  The role of fluorine in vesuvianite; a crystal-structure study , 1992 .

[118]  S. Zones,et al.  Unraveling the Perplexing Structure of the Zeolite SSZ-57 , 2011, Science.

[119]  P. Moore,et al.  Pinakiolite, Mg2Mn3+O2,[BO3]; warwickite, Mg(Mg0.5Ti0.5)O[BO3]; Wightmanite, Mg5(O)(OH)5[BO3]·nH2O: Crystal chemistry of complex 3 Å Wallpaper Structures , 1974 .

[120]  P. Orlandi,et al.  Marinellite, a new feldspathoid of the cancrinite-sodalite group , 2003 .

[121]  S. Saxena,et al.  Volumetric properties and phase relations of silica — thermodynamic assessment , 2001 .

[122]  F. Hawthorne,et al.  Sveinbergeite, Ca(Fe62+Fe3+)Ti2(Si4O12)2O2(OH)5(H2O)4, a new astrophyllite-group mineral from the Larvik Plutonic Complex, Oslo Region, Norway: description and crystal structure , 2011, Mineralogical Magazine.

[123]  F. Hawthorne,et al.  Excess Y-group cations in the crystal structure of vesuvianite , 1994 .

[124]  D. Veblen.  Polysomatism and polysomatic series : a review and applications , 1991 .

[125]  Daniel J. Graham.  Chemical Thermodynamics and Information Theory with Applications , 2011 .

[126]  P. Burns.  A new uranyl oxide hydrate sheet in vandendriesscheite: Implications for mineral paragenesis and the corrosion of spent nuclear fuel , 1997 .

[127]  D. Veblen,et al.  New biopyriboles from Chester, Vermont; II, The crystal chemistry of jimthompsonite, clinojimthompsonite, and chesterite, and the amphibole-mica reaction , 1978 .

[128]  Topological analysis of crystal structures , 1983 .

[129]  James Odell,et al.  Between order and chaos , 2011, Nature Physics.

[130]  S. Merlino,et al.  Polysomatic approach in the crystal chemical study of minerals , 1997 .

[131]  E. Estevez-Rams,et al.  On the concept of long range order in solids: The use of algorithmic complexity , 2009 .

[132]  Jordi Rius,et al.  A large-cavity zeolite with wide pore windows and potential as an oil refining catalyst , 2002, Nature.

[133]  A. McDonald,et al.  ROGERMITCHELLITE, Na12(Sr,Na)24Ba4Zr26Si78(B,Si)12O246(OH)24·18H2O, A NEW MINERAL SPECIES FROM MONT SAINT-HILAIRE, QUEBEC: DESCRIPTION, STRUCTURE DETERMINATION AND RELATIONSHIP WITH HFSE-BEARING CYCLOSILICATES , 2010 .

[134]  Stephen Wolfram,et al.  A New Kind of Science , 2003, Artificial Life.

[135]  B. Andersson,et al.  Decomposition and ordering in Fe1-xO , 1977 .

[136]  B. Velde.  Experimental determination of muscovite polymorph stabilities , 1965 .

[137]  Plul,et al.  Hematolite : a complex dense-packed sheet structure , 2022 .

[138]  E. Potter,et al.  Crystal structures of CsFe2S3 and RbFe2S3: synthetic analogs of rasvumite KFe2S3 , 2004 .

[139]  P. Shen,et al.  Kokchetavite: a new potassium-feldspar polymorph from the Kokchetav ultrahigh-pressure terrane , 2004 .

[140]  A. Caticha.  Information and Entropy , 2007, 0710.1068.

[141]  B. Taylor,et al.  CODATA recommended values of the fundamental physical constants: 2006 | NIST , 2007, 0801.0028.

[142]  E. Makovicky,et al.  Refinement of the crystal structure of nuffieldite, Pb 2 Cu (sub 1.4) Bi (sub 0.4) Bi (sub 0.4) Sb (sub 0.2) )Bi 2 S 7 ; structural relationship and genesis of complex lead sulfosalt structures , 1997 .

[143]  P. Moore,et al.  The crystal structure of philolithite, a trellis-like open framework based on cubic closestpacking of anions , 2000 .

[144]  A. A. Kashaev,et al.  Crystal structure of a new representative of the cancrinite group with a 12-layer stacking sequence of tetrahedral rings , 2004 .

[145]  Phase transitions in K 2 Cr 2 O 7 and structural redeterminations of pha , 2004 .

[146]  O. Mentré,et al.  Bi3+/M2+ oxyphosphate: a continuous series of polycationic species from the 1D single chain to the 2D planes. Part 1: From HREM images to crystal-structure deduction. , 2006, Inorganic chemistry.

[147]  S. Krivovichev,et al.  THE CRYSTAL STRUCTURE OF SVYATOSLAVITE AND EVOLUTION OF COMPLEXITY DURING CRYSTALLIZATION OF A CaAl2Si2O8 MELT: A STRUCTURAL AUTOMATA DESCRIPTION , 2012 .

[148]  O. Mentré,et al.  Bi3+/M2+ oxyphosphate: a continuous series of polycationic species from the 1D single chain to the 2D planes. Part 2: Crystal structure of three original structural types showing a combination of new ribbonlike polycations. , 2006, Inorganic chemistry.

[149]  H. Fjellvåg,et al.  The synthesis and crystal structure of a hydrated magnesium phosphate Mg3(PO4)2.4H2O , 2001 .

[150]  S. Filatov,et al.  The crystal structure of klyuchevskite, K3Cu3(Fe,Al)O2(SO4)4, a new mineral from Kamchatka volcanic sublimates , 1992, Mineralogical Magazine.

[151]  L. Glebsky.  MEASURES OF -COMPLEXITY , 2005 .

[152]  N. D. Samotoin,et al.  KINETICS AND MECHANISM OF HYDROTHERMAL CRYSTALLIZATION OF 2M1 MUSCOVITE: AN EXPERIMENTAL STUDY , 1985 .

[153]  V. Blatov,et al.  Computer simulation of the self-assembly of paulingite crystal structure from suprapolyhedral nanocluster precursors K6, K16, and K20 , 2011 .

[154]  S. Krivovichev,et al.  Topological complexity of crystal structures: quantitative approach. , 2012, Acta crystallographica. Section A, Foundations of crystallography.

[155]  P. McMillan,et al.  An experimental and thermodynamic study of cymrite and celsian stability in the system BaO-Al 2 O 3 -SiO 2 -H 2 O , 1992 .

[156]  S. Krivovichev,et al.  Crystal structures and cellular automata. , 2004, Acta crystallographica. Section A, Foundations of crystallography.

[157]  C. Adami,et al.  Evolution of Biological Complexity , 2000, Proc. Natl. Acad. Sci. USA.

[158]  U. Kolb,et al.  Towards automated diffraction tomography: part I--data acquisition. , 2007, Ultramicroscopy.

[159]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[160]  J. Wheeler.  Information, physics, quantum: the search for links , 1999 .

[161]  James B. Thompson.  Biopyriboles and polysomatic series , 1978 .

[162]  J. R. Goldsmith.  A "Simplexity Principle" and Its Relation to "Ease" of Crystallization , 1953, The Journal of Geology.

[163]  P. Burns.  A new complex sheet of uranyl polyhedra in the structure of wölsendorfite , 1999 .

[164]  Filatov,et al.  Metal arrays in structural units based on anion-centered metal tetrahedra. , 1999, Acta crystallographica. Section B, Structural science.

[165]  M. F. Márquez-Zavalía,et al.  THE CRYSTAL STRUCTURE OF ALFREDSTELZNERITE, Ca4 (H2O)4 [B4O4(OH)6]4 (H2O)15, A COMPLEX HYDROXY-HYDRATED CALCIUM BORATE MINERAL , 2010 .

[166]  M. Catti,et al.  A case of polytypism in hydrated oxysalts: the crystal structure of Mg3(P04)2. 22 H20-II , 1981 .

[167]  L. Pauling.  THE PRINCIPLES DETERMINING THE STRUCTURE OF COMPLEX IONIC CRYSTALS , 1929 .

[168]  Charles H. Bennett,et al.  The thermodynamics of computation—a review , 1982 .

[169]  Claude E. Shannon,et al.  A Mathematical Theory of Communications , 1948 .

[170]  F. Hawthorne.  Graphical enumeration of polyhedral clusters , 1983 .

[171]  N. Rashevsky.  Life, information theory, and topology , 1955 .

[172]  Y. Takéuchi,et al.  The crystal structure of hexagonal CaAl2Si2O8 , 1959 .

[173]  M. O'keeffe,et al.  An alternative approach to non-molecular crystal structures with emphasis on the arrangements of cations , 1985 .

[174]  S. Krivovichev.  Structural and topological complexity of zeolites: An information-theoretic analysis , 2013 .

[175]  S. Guggenheim,et al.  Chapter 17. CRYSTAL CHEMISTRY, CLASSIFICATION, AND IDENTIFICATION OF MODULATED LAYER SILICATES , 1988 .

[176]  V. Chevrier,et al.  Mineralogy and evolution of the surface of Mars: A review , 2007 .

[177]  Todd A. Ehlers,et al.  REVIEWS IN MINERALOGY AND GEOCHEMISTRY , 2005 .

[178]  S. Samson,et al.  Crystal Structure of the Zeolite Paulingite , 1966, Science.

[179]  S. Merlino,et al.  Modular Microporous Minerals: Cancrinite-Davyne Group and C-S-H Phases , 2004 .

[180]  D. Bonchev,et al.  Complexity in chemistry, biology, and ecology , 2005 .

[181]  A. Rex,et al.  Maxwell's demon 2: entropy, classical and quantum information, computing , 2002 .

[182]  S. Filatov,et al.  Types of cationic complexes based on oxocentred tetrahedra [OM4] in the crystal structures of inorganic compounds , 1998 .

[183]  W. H. Baur.  On the crystal chemistry of salt hydrates. IV. The refinement of the crystal structure of MgSO4.7H2O (epsomite) , 1964 .

[184]  F. Hawthorne,et al.  The crystal structure of geminite, Cu (super 2+) (AsO 3 OH)(H 2 O), a heteropolyhedral sheet structure , 1995 .

[185]  A. Kremenović,et al.  Thermally induced phase transformations of Ca-exchanged LTA and FAU zeolite frameworks: Rietveld refinement of the hexagonal CaAl2Si2O8 diphyllosilicate structure , 1996 .

[186]  P. Heaney,et al.  The Crystal Structure of Bannisterite , 1992 .

[187]  W. E. Brown,et al.  Crystal structures of bobierrite and synthetic Mg 3 (PO 4 ) 2 .8H 2 O , 1986 .

[188]  D. Modell,et al.  The Structure of Vesuvianite Ca10Al4(Mg,Fe)2Si9O34(OH)4 , 1931 .

[189]  T. Armbruster,et al.  International Mineralogical Association, Commission on New Minerals and Mineral Names: Definition of a mineral , 1980, Mineralogical Magazine.

[190]  S. Krivovichev,et al.  Particular topological complexity of lead oxide blocks in Pb31O22X18 (X = Br, Cl). , 2006, Inorganic chemistry.

[191]  P. Rentzeperis,et al.  The crystal structure of the anhydrous magnesium sulphate , 1958 .

[192]  F. Hawthorne,et al.  The crystal structure of kombatite, Pb 14 (VO 4 ) 2 O 9 Cl 4 , a complex heteropolyhedral sheet mineral , 1994 .

[193]  S. Merlino,et al.  The crystal structure of liottite, a six-layer member of the cancrinite group , 1996 .

[194]  Alan L. Mackay,et al.  Cellular automata and local order in the structural chemistry of the lovozerite group minerals , 2010 .

[195]  S. Guggenheim,et al.  Effect of temperature on the structures of lizardite-1T and lizardite-2H , 1998 .

[196]  Evandro Agazzi,et al.  What is Complexity , 2002 .

[197]  F. C. HewrnonNr.  Towards a structural classification of minerals: The vIMIvTrOo minerals , 1985 .

[198]  W. Bragg.  XXV. The Structure of Silicates , 1930 .

[199]  F. Hawthorne.  Structural hierarchy in VI M x III T y phi z minerals , 1986 .

[200]  G. Vezzalini,et al.  Crystal structure of the zeolite mutinaite, the natural analog of ZSM-5 , 1997 .

[201]  F. Hawthorne.  Structure and chemistry of phosphate minerals , 1998, Mineralogical Magazine.

[202]  Hongwei Ma,et al.  Letter. Determination of the crystal structure of sanderite, MgSO4·2H2O, by X-ray powder diffraction and the charge flipping method , 2009 .

[203]  M. Sebastiani,et al.  Fantappièite, a new mineral of the cancrinite-sodalite group with a 33-layer stacking sequence: Occurrence and crystal structure , 2010 .

[204]  S. Krivovichev,et al.  PUNKARUAIVITE, LiTi2[Si4O11(OH)](OH)2·H2O, A NEW MINERAL SPECIES FROM HYDROTHERMAL ASSEMBLAGES, KHIBINY AND LOVOZERO ALKALINE MASSIFS, KOLA PENINSULA, RUSSIA , 2010 .

[205]  S. Krivovichev,et al.  Armbrusterite, K5Na6Mn3+Mn2+ 14[Si9O22]4(OH)10·4H2O, a new Mn hydrous heterophyllosilicate from the Khibiny alkaline massif, Kola Peninsula, Russia , 2007 .

[206]  N. Hirano,et al.  Mineralogical and textural variation of silica minerals in hydrothermal flow-through experiments: Implications for quartz vein formation , 2010 .

[207]  F. Hawthorne,et al.  The incorporation of boron into the vesuvianite structure , 1994 .

[208]  Marc Van Houteghem,et al.  UV-Raman and 29Si NMR Spectroscopy Investigation of the Nature of Silicate Oligomers Formed by Acid Catalyzed Hydrolysis and Polycondensation of Tetramethylorthosilicate , 2011 .

[209]  C. Wan,et al.  Structural chemistry of copper and zinc minerals. VI. Bayldonite, (Cu,Zn)3Pb(AsO4)2(OH)2: a complex layer structure , 1979 .

[210]  E. Bonaccorsi,et al.  The crystal structure of sacrofanite, the 74 Å phase of the cancrinite–sodalite supergroup , 2012 .

[211]  Why Are the Structures of Some Solids So Complex but Others So Simple , 1994 .

[212]  V. Blatov,et al.  Geometric and topological analysis of zeolite crystal structures by the tiling method: The model of structure of Na,K-paulingite (PAU) Na82K72[Al154Si518O1344] · wH2O , 2011 .

[213]  Peul,et al.  Hyalotekite , a complex lead borosilicate : its crystal structure and the lone-pair effect of Pb ( II ) , 2007 .

[214]  E. Trucco.  A note on the information content of graphs , 1956 .

[215]  N. Chukanov,et al.  Model of the crystal structure of biachellaite as a new 30-layer member of the cancrinite group , 2008 .

[216]  S. Krivovichev,et al.  Tubular chains in the structures of natural and synthetic silicates , 2011 .

[217]  S. Krivovichev,et al.  The Crystal Chemistry of Sulfate Minerals , 2000 .

[218]  John Scales Avery,et al.  Information theory and evolution , 2003 .

[219]  F. Hawthorne,et al.  The crystal structure of curetonite, a complex heteropolyhedral sheet mineral , 1994 .

[220]  W. Bragg.  The Structure of Silicates. , 1930, Nature.

[221]  Mill Johannes G.A. Van,et al.  Transmission Of Information , 1961 .

[222]  E. Galli,et al.  Crystal structure refinement of aluminian lizardite-2H2 , 1997 .

[223]  J. Löns,et al.  Strukturverfeinerung von Sodalith, Na8Si6Al6O24Cl2 , 1967 .

[224]  I. Hassan,et al.  The crystal structure of basic cancrinite, ideally Na 8 [Al 6 Si 6 O 24 ](OH) 2 .3H 2 O , 1991 .

[225]  A. Spek,et al.  Crystallization and Characterization of a New Magnesium Sulfate Hydrate MgSO4·11H2O , 2007 .

[226]  H. Sørensen,et al.  The hyper-agpaitic stage in the evolution of the Ilímaussaq alkaline complex, South Greenland , 2001 .

[227]  F. Cámara,et al.  Thermoelasticity and high-T behaviour of anthophyllite , 2011, PCM 2011.

[228]  S. Merlino,et al.  THE CRYSTAL STRUCTURE OF FRANZINITE, THE TEN-LAYER MINERAL OF THE CANCRINITE GROUP , 2000 .

[229]  H. F. Shurvell,et al.  Letter: Meridianiite: A new mineral species observed on Earth and predicted to exist on Mars , 2007 .

[230]  S. Krivovichev,et al.  Structural principles for minerals and inorganic compounds containing anion-centered tetrahedra , 1999 .

[231]  Gregory J. Chaitin,et al.  On the Simplicity and Speed of Programs for Computing Infinite Sets of Natural Numbers , 1969, J. ACM.

[232]  A. K. Pant.  A reconsideration of the crystal structure of β-Na2Si2O5 , 1968 .

[233]  A. Baronnet,et al.  Ab-initio synthesis and TEM confirmation of antigorite in the system MgO-SiO2-H2O , 1997 .

[234]  S. Uehara.  TEM and XRD study of antigorite superstructures , 1998 .

[235]  F. Hawthorne.  Towards a structural classification of minerals; the VI M IV T 2 phi n minerals , 1985 .

[236]  S. Krivovichev,et al.  Minerals and synthetic Pb(II) compounds with oxocentered tetrahedra: review and classification , 2008 .