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  • LEVELS
    of being a Totality can be the corresponding graduality of the strength of the interactions at that lower level in virtue of which the parts are more or less strongly held together and in this way form higher or lower degree Totalities Are the interacting parts of the thing the true Substances And do these parts in turn consist of interacting parts in virtue of which those latter parts are the true Substances Are there consequently only genuine Substances to be found at one or another lowest level the ultimate inviolable level and not at any higher level A possible Solution Morphogenetic Fields Earlier I have stated that the cause of high level phenomena must be sought in the relevant interactions at a lower level This complies with the view of modern Natural Science Nevertheless sometimes doubts are cast on this view also by practitioners of that Science So for example by SHELDRAKE in his well known theory of morphogenetic fields Let me show an example leading to such a theory Proteins are long chain molecules which curl up and fold into complex forms Each protein species i e each individual of such a species folds itself in a specific way and in the same specific way on every occasion resulting in a typical with repect to that species three dimensional form the form of the individual protein molecules Precisely by having this particular form such a protein can execute its determined function within an organism The folding of such a molecule is a complicated process and could easily go wrong Nevertheless it always happens flawlessly This is explained by assuming that the form in question resulting from the folding process contains a minimum of energy in virtue of which precisely that particular form is the most stable energetically amongst a very large number of alternative ways of folding with other resulting forms but with higher energies In order to visualize all this suppose we have a marble in a bowl If one applies a knock to such a marble it will climb the wall of the bowl but eventually return to the bottom where it has the lowest potential energy It is a stable system On the other hand a marble on top of a bowl which is turned upside down embodies an unstable system with high potential energy When a small disturbance is applied to the marble it will roll downwards transforming its potential energy into kinetic energy and this latter will finally be dissipated in virtue of friction and impact on the ground Now the marble finds itself in an energy minimum and embodies a stable system But returning to proteins again it seems to turn out that within the total arsenal of with respect to any particular protein geometrically possible forms there are many such forms with the same energy or with only slightly different energies Having in mind this it is unexplainable that a certain particular species of protein always folds itself in the same way after it was unrolled in the laboratory The minimum energy state of a protein molecule is dependent on the particular chemical composition of such a molecule and this means in the case of proteins the particular set of aminoacids and their sequence Hence this chemical composition evidently does not embody all the information necessary for the particular way of folding which always takes place all the more because of the fact that there are proteins which are very similar in their three dimensional form in virtue of an almost identical way of folding but which differ strongly in their chemical constitution On the basis of all this but also of many other phenomena SHELDRAKE accordingly assumes that besides the condition of minimum energy there must be something else which governs the protein in its folding But what is that something else According to SHELDRAKE it is an immanent field under which influence the folding protein is going to find itself a so called Morphogenetic Field See SHELDRAKE R 1981 A new science of life and also SHELDRAKE R 1988 The present of the past Expressed differently such a folding protein tunes in to a definite chreode an abstract furrow as we could call it which leads the folding process to a definite end state This tuning is related to the vibrational nature of the molecule and the field Such a field is accordingly interpreted by him as formal cause structural cause It is clear that in this way we come close to something like a substantial form because the field is immanent Hence an emergent phenomenon would then perhaps not completely be reducible to lower levels These lower levels could be just the material cause of it while the morphogenetic field would represent the formal cause A Metaphysical Interpretation once more In In VII Met lectio 17 nr 1672 1680 St Thomas following Aristotle considers the character of being a Totality in order to figure out in what way Substance is cause and principle A true Totality IS not its elements The elements constitute the matter A Totality still contains something beyond its elements What would that be This something beyond could itself be either again an element or consisting of elements Both possibilities lead to an infinite regress In nr 1678 he proposes that this something beyond is the Substance of the thing however only in those things that are indeed true Substances This Substance or nature is not an element but a formal principle Here the Commentary on Book VII of the Metaphysics of Aristotle ends It ended up with an assessment of Substance what Substance is and where we can find it an assessment not along the via praedicationis i e through research into the way of predication but along the via physicorum i e through a causal consideration and one with respect to the nature of being a Totality We could also say along strict metaphysical lines in contrast to logical lines So in the text Substance or whatness is interpreted as not identical with the elements of the Whole That Whole is accordingly interpreted as a holistic Whole And thus the following applies Whatness is not the same as the Elements Hence a summation of the elements of which a Whole is composed does not according to the text give us the whatness of this Whole it only gives us its material cause This we could call its low level whatness The true whatness i e the Substance then is the whatness of the Whole at a high level So the text already gives a hint for recognizing a distinction between levels A possible thematic evaluation of what has been stated in Book VII of Aristoteles Metaphysics with respect to Substance could read as follows Essence Substance Substance is the individual thing minus designated matter materia signata and minus the Accidents Essence accordingly is forma totius The intrinsic causes of Essence are matter and form So the form is the formal cause of the Essence and as such the form is the whatness of the Substance concerned Matter is a component of the Essence of sensible things because matter is essential for sensible things as sensible A specific distinction i e a distinction with respect to content between sensible things and non sensible things is grounded in their respective forms not in their matter Matter takes care of the ontological distinction between sensible and non sensible things So the form is the whatness and as such it is the whatness of the thing at a high level i e as it is at a high structural level of the thing In virtue of matter the sensible thing has the ability to change because it then is catallel i e it is ontologically composed of two parts alleles and its matter could be informed by another form which replaces the original form In this way the matter is a fixed point of reference with respect to which forms can be replaced by other forms and this is a change i e a change of the thing The matter of sensible things does not partake in the whatness of the thing at a high level but is the whatness of the thing at a low level If the high level whatness is not specifically dependent on the low level whatness i e on the qualities of the elements thus if this high level whatness the form just needs in order to exist one or another apropriate material substrate thus not a specific substrate then this high level whatness is medium independent Such a high level whatness could exist in several diverse appropriate substrates This appropriate substrate is not Prime Matter but consists of the elements of which the Whole is composed Prime matter is just a principle of the possibility of radical change See the Essay on Medium independency for further implications It is however still to be investigated to what extent the high level whatness partakes in the elements With respect to content it cannot be completely separate from them But it could nevertheless be the case that a whatness A partakes in elements b and c but that such a whatness A can also result from the elements d and e however definitely not from whatever elements i e not from just any element whatsoever So the whatness of these elements does play a role but perhaps not a specific one but a generic one The whatness of the elements in the proposed view does not just like that enter the high level whatness of the thing The whatnesses of the elements are appropriate to be exchanged for the whatness of the Whole when we ascend from the lower level to the highest level See for such a rise from lower to higher level the Fuga Formicalis Ant fugue of HOFSTADTER D 1979 in his famous book Gödel Escher Bach Insofar as we recover the whatness of the elements in the Whole they are the whatness not anymore of the elements but of the Whole because the elements are only virtually present in the Whole We see this theme continually recurring The solution of the problem of the generation of an own specific whatness and even an own Identity in the sense of self being out of entities which are each for themselves with respect to content different from the mentioned whatness of the Whole directly implies our orientation with respect to the interpretation of Substance and especially of the substantial form namely whether the whatness is perhaps already implicitly present in the elements and how it finally becomes explicit in the generation of the Whole The distinction within a being of structural levels offers it is true a number of clues for a solution of the problem concerned but is not going to represent a final solution The matter presented in this way is still too vague We will see that the recent Theory of Non linear Dissipative Dynamical Systems which is intensively studied in our days can tell us more about just how such a Totality intrinsic being is actually generated from more primitive elements In this respect it especially concerns the evolutionary systems which I earlier have called process structures See the Essay Historical Individuum Here and now Individuum Section Process structures Parts of the Definition and Parts of the Thing In Chapter 2 Leonina edition line 84 163 of the De Ente et Essentia St Thomas discusses the integral parts of a Substance and in what way these parts appear in the Definition The further determination of the genus to one or another species is accomplished by the specific difference which is taken from the form The form provides the intelligible content of the difference The genus however contains implicitly potentially that something which is expressed in the possible differences And this is why the genus can be predicated of the species This also applies to a higher genus with respect to a lower genus which results from a further determination by a difference in this case not a specific difference but a generic one So for instance the genus body with respect to the lower genus animal which is given by St Thomas as an example However body can also be considered as part of an animal and as such it cannot with is be predicated of animal Something similar applies to animal with respect to man The Soul as Part of an Animal The Soul as Difference How should we precisely evaluate the view the alternative not exclusive possibility that body here not as a genus is a part of animal The conventional answer reads An animal consists of its body and its animal soul likewise in humans What precisely is this soul It is clearly not a material thing or part thereof to which you could point with your finger Hence part in this context means something else And if we accept on the basis of current theory that Life on Earth has originated some 3 x 10 9 years ago from inorganic materials then perhaps we will not assume that the animal soul is one or another immaterial but nevertheless really existing part of the animal because then we would have to ascribe a dual origin of animals And what then should be the origin of that immaterial part NOTE 4 The assumption of the existence of an immaterial part turns the matter at hand into a formidable problem and this is enough to be receptive to attempts to relate the whole animal with one type of origin As long however a satisfying solution is not at hand we should keep open both options The matter should have the form of an ongoing but friendly discussion Let me propose a possible solution about what the animal soul really is According to me animal soul is a description of an animal at a high level in contradistinction to for example a description of an animal on the basis of its chemical constitution which is a decription at a low level The low level description is a reductionistic one i e an analytical description while the high level description is a holistic one i e a synthetic description What exactly is it in Reality that corresponds to these descriptions The low level description doesn t exclude anything which could be indicated in the animal at least it pretends so In this sense it is complete and is used in Natural Science The high level description refers according to me to phenomena of the animal that it is true originate from those entities which are explicitly mentioned by the low level description but as such are not present in the low level description We already met them as emergent phenomena which are he result of interaction and morphological and functional integration of demonstrable parts in and of the animal Sensitive soul animal soul is such an emergent phenomenon If such a phenomenon is continually present and if it refers to the whole animal as such then we can interpret this phenomenon as the high level whatness of the animal and thus as the substantial Form of the animal i e as the Form of that thing which we indicate with the term animal as first intention Because this Form is the result of the integration of all the parts of the animal we can express this form by means of a difference and such a difference can be predicated directly of a species or of an individual a dog is sensitive and also Fido is sensitive The intelligible content of sensitive is taken from the Form i e the substantial Form In this way the Form is not a material integral part of the animal which indeed the heart is and also not an immaterial integral part of the animal something like an immaterial animal soul An animal consists it is true of integral parts for example its organs extremities etc but not of body and animal soul For parts are visible at one or another lower level while the soul is not This implies that the body isn t a part either And so it must be the Whole Again the Form is the result of the integration of parts it is not itdself a part And integration of parts means the in formation of matter Animal as Part Animal as Genus In Chapter 2 of De Ente et Essentia line 150 163 St Thomas discusses animal figuring as an example which sometimes can be assessed as part sometimes as genus If animal only refers to a thing that can perceive and that can move by itself and if animal does not include in its meaning a possible further perfection completion then animal must be taken as part If it does include although implicitly the possible further perfections then animal is a genus Of course it is clear that if we omit something from that what we call animal and consequently do not include into the meaning of animal then animal is just a part but only because we take away something conceptually namely the further perfections while these further perfections are present in every actually existing animal In Scholastic texts it is often suggested that the only further perfection completion of animal is to be rational i e the ability to think in a rational way resulting in a rational animal a human being But animal means sensitive organism and this can be further determined perfected completed in many ways One of these ways is to be rational For the other perfections we don t have existing names The Apes and Man have common ancestors who probably have looked quite similar for us at least to the recent Apes That ancestral species has evolved further into two or more different directions one branch leading to recent Man by acquiring rationality one or more other branches in the

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  • Continuation of Philosopghy
    individual And we do this in the following way The simplest geometric body geometrically expressing the ideal intrinsic symmetry of that given organismic individual and also geometrically expressing the number and arrangements of that individual s antimers is the promorph or ideal stereometric basic form of that given organismic individual We will see that the Promorph generally digs a little deeper into the global structure of a given organismic individual then Symmetry alone does After having expounded the Promorphological System of Basic Forms i e systematically presented all the possible types of promorphs we will apply this Promorphology to crystals as well This demanded a lot of theorizing because crystals do not as such possess antimers which fact tends to forbid considering crystals in a promorphological way It turned out however that crystals have something like a hidden promorphology and we succeeded to reveal it All this is quite a lot about symmetry Two dimensional crystals Group Theory as prelude to the study of symmetry in three dimensional crystals Group theoretic consideration of two dimensional periodic patterns for its own sake or also as prelude to 3 dimensional crystals and still in progress Organic Tectology subordinated individuals Organic Promorphology Stereometric basic forms in organisms and finally Inorganic Promorphology Stereometric basic forms in crystals The reader can find all this to begin with in the last series Special Series of First Part of Website present website and the rest in Second Part of Website As such then the reader studies one of the most important and general properties of intrinsic beings beings generated by certain dynamical systems and as such directly implying these properties which are aspects of the total spatial structure of such a being So what read next So this whole prospect as outlined above is very special in contrast to general indeed because it elaborates on just a few aspects of spatial structure emerging from one or another dynamical system when this system generates an intrinsic being It consists of a whole lot of crystallography biology and mathematics and the reader might become impatient as to when if ever the website continues the more general metaphysical approach of its very beginning largely First Part of Website where philosophical issues like substance form matter substrate individuality essence were pursued So I can well imagine that the reader wishes to see a continuance of p h i l o s o p h y i e a further elaboration of the Theory of Being rather then the extensive treatment of symmetry involving Crystallography Group Theory Promorphology etc The documents on these subjects could however also be studied for their subject s own sake and intrinsic interest and importance totally apart from philosophy Continuing Philosophy Well for such a continuance of philosophy here almost exclusively natural philosophy the website has much to offer The philosophically minded reader could skip the whole Second Part of Website i e skip all the detailed treatments about symmetry and promorph and directly proceed further with

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  • Crystals revisited
    insofar as only those coincidence operations are involved in which at least one point of the object remains in place during the execution of such a coincidence operation Examples are Reflection and Rotation in contradistinction to a translation linear shift in which not any point remains in place So a Point Group of an object describes the total of non translative symmetry operations that the object allows NOTE 6 With respect to Crystals there are only 32 Point Groups at all possible the 32 Crystal Classes Space Group The Space Group also describes the symmetry of an object i e describes the total of symmetry operations allowed by the object but now including all the executable translative coincidence operations with respect to that object as well In the case of a translative coincidence operation not any point remains in place all the points are displaced but the object when considered infinitely extended in space will coincide with itself An example is a linear shift of a regular point pattern in a certain direction and along a certain distance If we imagine a crystal to be indefinitely extended in space then crystals exhibit several translational symmetries meaning that when we subject it to certain translational operations then the result is a coincidence of the internal crystal pattern before and after the operation This translative symmetry aspect is however only microscopically demonstrable because the translation distances i e the repeat distances of atoms are very small NOTE 7 For crystals there are in principle 230 Space Groups possible If we subtract the translative symmetries from a Space Group then we end up with the corresponding Point Group The Space Group and by implication also the Point Group all by itself consequently does not directly describe the complete structure of the crystal lattice but exclusively its symmetry namely the complete symmetry of the crystal lattice considered to be provided with motifs motifs representing the point symmetry of the constituent atomic configurations that are repeated throughout the crystal The Space Group does not describe the constituent atoms but it does describe the total lattice symmetry caused by them So these atoms in the form of the Chemical Composition of the crystal cause in contradistintion to describe this symmetry but not only the crystal s symmetry they also cause the complete structure including the relevant distances between the lattice points of the crystal lattice NOTE 8 But the conditions during growth like temperature and pressure are also determining which implies that one chemical substance can nevertheless give rise to several different crystal forms Implicit and Explicit Description The characterization of a crystal i e the potentially complete description of all per se features with respect to that crystal can be done by means of its Chemical Composition Space Group Such a description is to be sure complete but not wholly explicit As has been said the Space Group exclusively specifies the complete symmetry aspect of the relevant lattice as it is when provided with motifs and it does so explicitly It does however not describe the different absolute distances between the lattice points or between the atoms for that matter neither does it describe the absolute angles between the relevant chemical bonds within the crystal The distances and angles are implied by the crystal s chemical composition So the Space Group does not describe structure The distances are directly dependent on the chemical composition namely on the action radii of the relevant atoms but are by mentioning just the chemical composition only implicitly given A completely explicit description of all per se features of the crystal should specify the crystal lattice under consideration quantitatively and then indicate which lattice points are occupied by or associated with which atoms ions or molecules The significatum that to which reference is made of both types of descriptions taken together is the fixed kernel of the crystal its actual structure a kernel which doesn t change during the growth of the crystal So it is concurrent with its Dynamical Law The Dynamical Law is the most implicit and so most compact characterization of the crystal by means of a very implicit description of all per se features of the crystal These per se features are the per se determinations and these are implied by the Dynamical Law in the sense of generated They are however not identical to the Dynamical Law The latter is the relevant crystallization law prevailing during the origin and growth of the crystal A precise formulation of the Dynamical Law of one or another crystal is generally not directly possible but the importance of the introduction of this concept consists in its emphasizing the regular course of real processes and this means that such a process is deterministic and in principle repeatable So the concept of Dynamical Law emphasizes the fact that the crystal is a product a stadium or state of a dynamical system and as such it denotes the Essence of the crystal Thermodynamic Conditions Also the external conditions of pressure and temperature p t thus the thermodynamic conditions are co determining the crystal structure which is generated under those conditions These thermodynamic conditions relate to energetic circumstances The crystal structure itself represents an energy state in which the whole of constituent particles in a crystal finds itself It is a state of lowest energy Thus Calcium Carbonate CaCO 3 under certain p t conditions crystallizes in the form of the mineral Calcite with Point Group 3 2 m and Space Group R3 2 c NOTE 9 while under different p t conditions it crystallizes in the form of the mineral Aragonite with Point Group 2 m 2 m 2 m and Space Group P2 1 n 2 1 m 2 1 a STRÜBEL 1977 p 424 BURZLAFF ZIMMEREMANN 1977 Kristallographie Band I Symmetrielehre pp 171 So within the one p t condition range the one Dynamical Law is triggered into action while within the other p t condition range the other Dynamical Law is triggered So a crystallization law Dynamical Law has a limited range of validity i e it only operates within a certain range of external conditions This range of external conditions wholly relates to the energy condition of the relevant particles taking part in the crystallization Thus the Dynamical Law is partly contained in these p t conditions but these p t conditions in turn are contained in the particles themselves Here including their state of motion Temperature and pressure are statistical quantities which can be reduced to the energy conditions of the individual particles and with it to their state of motion So the Dynamical Law remains immanent with respect to the particles involved in the crystallization and in this respect the p t conditions are not external A same chemical composition can accordingly be connected with different Space Groups and even already with different Crystal Classes Point Groups when we have to do with different p t conditions So the compound Al 2 SiO 5 Aluminum Silicate occurs in three different mineral species and thus in three different crystalline forms Mineral Chemical Formula Crystal Class Point Group Space Group Andalusite Al 2 SiO 5 2 m 2 m 2 m P 2 1 n 2 1 n 2 m Sillimanite Al 2 SiO 5 2 m 2 m 2 m P 2 1 n 2 1 m 2 1 a Disthene Kyanite Al 2 SiO 5 1 P 1 See STRÜBEL 1977 p 217 Here we see the necessity of including the Space Group component in the per se description of a crystal We also see that the mineral Aragonite chemical composition Calcium Carbonate and the mineral Sillimanite chemical composition Aluminum Silicate have the same Space Group and even the same Point Group so also the Chemical Composition component is indispensable for the per se description In chemically similar chemical compounds we still find differences in their crystal structure even when the symmetry of the lattices is exactly identical These differences consist in for example length proportions between the a axis and the c axis STRÜBEL 1977 p 173 Mineral Chemical Formula a c Crystal Class Point Group Space Group Calcite CaCO 3 1 0 854 3 2 m R3 2 c Rhodochrosite MnCO 3 1 0 818 3 2 m R3 2 c Siderite FeCO 3 1 0 814 3 2 m R3 2 c Smithsonite ZnCO 3 1 0 806 3 2 m R3 2 c Magnesite MgCO 3 1 0 811 3 2 m R3 2 c Here we see the necessity of including the Chemical Composition component into the the per se description Chemical Composition Space group reflects accordingly the Dynamical Law which operated during the growth of the crystal The concept of Dynamical Law has been related to the the per se characterization of a crystal in the form of Space Group Chemical Composition However one has synthesized crystals the so called Pseudo Crystals in which the 5 fold symmetry forbidden for crystals NOTE 10 nevertheless occurs In organisms this 5 fold symmetry is wide spread This implies that the inner structure resulting in them is not periodic in its nature STEWART GOLUBITSKY 1993 Fearful Symmetry Is God a geometer p 95 6 and BALL 1994 Designing the Molecular World Chemistry at the Frontier pp 122 Figure 3 A Pseudo Crystal These can have shapes that reflect the forbidden symmetries of the atomic structure In this picture the pseudo crystal has a dodecahedral shape After BALL 1994 This discovery could perhaps imply that the current Group Theoretical Paradigm expressed with the concepts of Point Group and Space Group is not necessarily all there is to say about crystal structure STEWART GOLUBITSKY 1993 p 96 Looking again to the aforementioned system of Al 2 SiO 5 we see that it is from a chemical perspective one system which can give rise to several mineral phases Andalusite Sillimanite and Disthene Each for themselves these phases are stable in a definite region of the Triple Point Diagram STRÜBEL 1977 p 217 which has two axes the temperature axis and the pressure axis So here we have the one system of Al 2 SiO 5 with three mineral phases and only at the Triple point all three mineral phases can co exist Of course not in a stable way Hoewever within our Totality Consideration we have to do with three systems because in such a consideration the crystals are central Accordingly in this case we have to do with three Dynamical Laws crystallization laws and their respective domains of validity are given in the Triple Point Diagram Aggregates The fact that minerals can form strange types of aggregates or weak Totalities can be illustrated by three examples Opal Wavellite and Rutilated Quartz Opal SiO 2 n H 2 O Although Opal is essentially amorphous it was demonstrated that it nevertheless has an ordered structure It is however not a crystal structure with atoms in a regular 3 dimensional array but consists of closely packed silica spheres in hexagonal and or cubic closest packing Air or water occupy the spaces between the spheres Figure 4 Scanning electron micrograph of an OPAL with chalky appearance showing hexagonal packing of silica spheres diameter of spheres approximately 3000 Angstrom 1 Angstrom 0 00000001 cm Because of the weak bonding between the spheres they are completely intact Here we have an example of an aggregate wich is almost a Totality although the pattern is not very elaborate it is just a dense packing of weakly bonded spheres wich are themselves probably more stronger Totalities consisting of SiO 2 Water After Hurlbut Klein 1977 Manual of Mineralogy In ordinary Opal the regions of equally sized silica spheres with uniform packing are small or non existent but in precious Opal large regions are constituted of regularly packed spheres of the same size The diameter of the spheres varies from one Opal sort to the other and ranges from 1500 Angstrom to 3000 Angstrom 1 Angstrom 10 8 cm HURLBUT KLEIN 1977 Manual of Mineralogy p 419 Opal can according to me best be interpreted as an aggregate albeit not situated at the very aggregate end of the scale Aggregate Continuum An Opal does not as it seems have an intrinsic interface with the environment the surroundings of the Totality or pseudo Totality yet another ground for interpreting it as an aggregate Wavellite Al 3 PO 4 2 OH 3 5 H 2 O Some minerals preferably form structured crystal aggregates for example Wavellite HURLBUT KLEIN 1977 p 333 STRÜBEL 1977 p 343 In the case of Wavellite individual crystals i e free crystals are rare Usually it occurs in the form of radiating spherulitic and globular aggregates Figure 5 Wavellite from Dünsberg near Giessen Germany After STRÜBEL 1977 Wavellite crystallizes in the Orthorhombic Crystal System in the Crystal Class Point Group 2 m 2 m 2 m and has as Space Group Pcmn abbreviated symbol So in the case of Wavellite we have to do with ordered aggregates of crystals having a more or less intrinsic spherical interface with the environment and accordingly they are as such be it weakly expressed Totalities and each for themselves they are a Totality of a higher scale order than is the individual Wavellite crystal However it could be that the structuring of the aggregate has external causes although the fact that Wavellite usually occurs in the form of such aggregates i e seldom in the form of individual free crystals speaks against this Radiating crystal aggregates are also formed in Zeolites for example Natrolite and also in Limonite Malachite and Antimonite Rutilated Quartz Rutile TiO 2 Quartz SiO 2 Also crystals within crystals do occur in Nature for example Rutilated Quartz This is Quartz which has fine needles of Rutile penetrating it Figure 6 Rutilated Quartz Brazil After HURLBUT KLEIN 1977 This we should interpret as an aggregate although a more or less weak form of it In this case it is an aggregate of two different crystal species Quartz and Rutile Pseudomorphosis It sometimes occurs in Nature that crystals dissolve in a special way the original chemical substance will be replaced by another material while the form of the original crystal is preserved So it can happen that Barite BaSO 4 dissolves because of changed pressure and temperature conditions in salty water The BaSO 4 is abducted and in its place Quartz SiO 2 is precipitated With it the outer form of the Barite crystals is preserved One calls this phenomenon a pseudomorphosis of Quartz to Barite Here we have an entity having a definite and specific boundary but this boundary is not related to its content The Quartz is formed within a special cavity which is however per accidens with respect to it Paramorphosis Paramorphosis is a phenomenon analogous to pseudomorphosis but in this case there is no change in chemical substance What changes is the inner structure while the outer form remains the same For example when high Quartz cools until below 573 degrees Celcius trigonal low Quartz is formed which could however preserve the outer hexagonal form of the high Quartz The transformation of high Quartz into low Quartz and vice versa with or without accompanying paramorphosis is a so called displacive polymorphic reaction See Figure 7 It consists in an internal adjustment which is very small and requiring little energy The structure is generally left completely intact and no bonds between ions must be broken only a slight displacement of atoms or ions and readjustment of bond angles kinking between ions is needed This type of transformation occurs instantaneously and is reversible The difference between the two forms of Quartz is expressed by their Space Groups low Quartz P3 2 21 high Quartz P6 2 22 The structural arrangement in the low temperature form is slightly less symmetric and somewhat more dense than that of the high temperature form HURLBUT KLEIN 1977 p 151 So this transformation is a very slight one indeed Nevertheless it results in a different Space Group Consequently in our ontological interpretation of crystalline chemical substances the change is a substantial change because the Chemical Composition Space Group is changed in virtue of the change in one of its components So even in the case when this process is moreover accompanied by paramorphosis i e the preservation of the outer form it is a change from one Substance in the metaphysical sense into another Figure 7 Lattices of low Quartz bottom and high Quartz top Projection of the lattices in the direction of the c axis When paramorphosis occurs the outer form of high Quartz is preserved after its transformation into low Quartz All the 6 fold screw axes 6 4 of high Quartz become 3 fold screw axes 3 2 after its transformation into low Quartz The black disks represent Silicon Si atoms The unit cell is outlined by dashed lines Its vertices are lattice points Compare with Figure 9 After STRÜBEL 1977 Formation of Crystal Lattices When the atoms in a melt or in a solution do not move too fast anymore when say the temperature falls they re going to attract each other strongly However when they in virtue of this attracion are coming too close to each other short range repelling forces come into being The system always strives to be in a lowest possible potential energy condition of those atoms i e of the constituents of that crystallization system While subjected to an attractive force for the atoms to be far from each other means that they then have a high potential energy which in the case of the atoms approaching each other will be transformed into kinetic energy i e energy of motion So on approaching each other the potential energy of the relevant atoms decreases But while subjected to a repelling force for the atoms to be nearer to each other means that they have potential energy When they then

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  • Outer Form of Crystals
    symbol for this Class is 4mm The Hermann Mauguin symbol of a Crystal Class consists of one two or three parts that relate to the following directions For the Crystal Systems and the crystallographic axes i e a axis b axis etc mentioned in the table see below Crystal System First Part Second Part Third Part Monoclinic Ortho axis Orthorhombic a axis b axis c axis Tetragonal c axis a and b axis bisectors between a and b axes Hexagonal c axis a b and d axis bisectors betw a b and d axes Isometric a b and c axis body diagonals plane diagonals Regarding the Isometric System Body diagonals are lines perpendicular to the octahedral faces while plane diagonals are lines that are parallel to one of the cube faces Regarding the Monoclinic System The Ortho axis is the crystallographic axis of the Monoclinic System that is perpendicular to the two other monoclinic crystallographic axes Regarding the Triclinic System It consists of only two Classes having respectively 1 and 1 as their Hermann Mauguin symbols Let us treat another example The symbol 4 m 2 m 2 m refers to a tetragonal crystal i e a crystal belonging to the Tetragonal Crystal System 4 also occurs in the isometric Classes but the symbols for such crystals are directly recognized by the 3 at the second part of the symbol This Class has a 4 fold rotation axis along the c axis and a mirror plane perpendicular to it so 4 m comes as the first part of the symbol It has moreover 2 fold rotation axes along the a and b axes each of them having a mirror plane perpendicular to it so 2 m will figure as the second part of the symbol there are two such rotation axes coupled with a mirror plane but because they are equivalent they are represented in the symbol by only one 2 m Further the Class has two more 2 fold rotation axes along the bisectors of the a and b axes and a mirror plane perpendicular to each one So they are also represented by 2 m which however will figure as the third part of the Hermann Maugin symbol of the Class Sometimes the Hermann Maugin symbols are used in an abbreviated form namely in those cases where some symmetry elements are implied by others The implied ones are then omitted if there is no possibility of confusion On this website we will however always use the complete symbols When we come to treat of the 32 Crystal Classes these symbols will be used and along with it better understood Because of several similarities obtaining between certain Crystal Classes the 32 Classes can be grouped into several collections the Crystal Systems although the Crystal Classes are completely independent There are six such Crystal Systems They will be individually expounded in subsequent Essays The CUBIC system also called the Isometric System The TETRAGONAL system The ORTHORHOMBIC system The MONOCLINIC system The TRICLINIC system The HEXAGONAL system Every Crystal System comprises a number of Crystal Classes Figure 3 The Six Crystal Systems After HOLDEN MORRRISON Crystals and Crystal Growing 1982 In the figure we can see that there are six styles of crystalline architecture each employing a different sort of building block And every crystal belongs to one of these six crystal systems Lines are drawn in these models to assist in visualizing which angles are right angles and which are not In the CUBIC block all angles are right angles and all sides are equal In the TETRAGONAL block all angles are right angles but there are two different lengths of side In the ORTHORHOMBIC block all angles are right angles but there are three different lengths of side The MONOCLINIC block is like the orthorhombic block pushed so eight of its angles are no longer right angles In the TRICLINIC block there is no right angle and there are three different lengths of side The HEXAGONAL block is a hexagonal prism with right angles between its vertical sides and its top and bottom faces After HOLDEN MORRISON So the Crystal Systems are based on the outer form of the building block of the crystal When considering just its outer form the building block has the same symmetry as the highest symmetrical class of the relevant Crystal System The building block is repeated in three directions and in this way conceptually builds up the crystal The Method of describing Crystal Faces and Basic Forms The morphology of crystals especially the orientation of its faces can be described by relating all this to an appropriate set of axes These axes are chosen in such a way that each one of them is parallel to a certain conspicuous crystal edge for cubic crystals for example the axes are chosen such that they correspond to three edges of the cube that are perpendicular to each other In this case we then obtain three equivalent axes that are perpendicular to each other For other types of crystals some other sets of axes are appropriate Such axes are called the crystallographic axes In the figure below we see an example of a set of three crystallographic axes We also see two polyhedra two different double pyramids bipyramids For both of them their upper right face is marked with a gray color Figure 4 A set of three crystallographic axes and two bipyramids with their geometrical centers coinciding with the origin of the system of axes Of each bipyramid one of their faces is marked with a grade of gray The faces cut off certain pieces of the axes Here the vertical axis is called the Z axis the axis that is more or less pointing to the observer is called the X axis while the one that is aligned East West is called the Y axis Each axis is divided into two parts by the origin of this set of axis and those parts are interpreted as negative or positive as indicated in the figure A face of a polyhedron and so also a crystal can cut off a segment from either all three axes or from two of them and then being parallel to a third axis or from only one of them and then being parallel to the two other axes In the above figure each face cuts off a piece from all three axes i e it is not parallel to any one of them Remark In the different Crystal Systems different axial systems are used involving different angles and sometimes more than three axes Parameter ratio If we consider a face of a crystal in its relation with a set of crystallographic axes like the one pictured above we can measure the cut off pieces intercepts of those axes pieces cut off from those axes by that particular face We shall denote the piece cut off from the X axis as a the piece cut off from the Y axis as b and the piece cut off from the Z axis as c Because parallel faces are crystallographically equivalent they have exactly the same orientation the absolute lengths of the cut off pieces are not important What is important to describe the orientation of the face is their ratio So we set the length of one of them b equal to 1 and determine the lengths of the other pieces the cut off piece of the X axis a and the cut off piece of the Z axis c relative with respect to b In this way we obtain with respect to the above set of axes three numbers and their ratio we call the parameter ratio of that particular crystal face Description of crystal faces and the generation of basic Forms To describe each face of a crystal species belonging to a certain Crystal Class thus possessing a certain symmetry content we first of all choose a conspicuous face conspicuous by say some physical properties like luster which cuts off all three axes at a finite distance from the origin of the axes system We will call such a face the unit face of the particular crystal class We measure the parameter ratio of that face Then we measure the parameter ratios of all other faces of the same crystal species Next we relate the latter measurement results to those of the unit face If we denote the parameter ratio of this unit face as a b c then we can express the other faces in the same way but now provided with numbers in front of each letter i e coefficients to express their orientation relative to the unit face and thus relative to the crystallographic axes In crystals these numbers turn out always to be rational numbers i e numbers like 1 2 1 3 1 3 5 etc and not irrational numbers like the square root of 2 or whatever number with a floating point like 2 54728 or 0 445106 etc Now if we take the gray shaded face of the smaller bipyramid in Figure 4 as unit face with respect to this particular type of axis system we can describe all the faces depicted in Figure 4 We will denote the unit face by a b c All other faces of the smaller bipyramid can be symbolized in the same way because they are assumed to involve cut off pieces intercepts of the same respective lengths as the corresponding ones of the unit face except for their signs or So the face just below the unit face can be denoted by a b c The gray shaded face of the larger bipyramid cuts off twice as much from the Z axis than the unit face does but the same amount from the X axis and from the Y axis as the unit face does So we can denote this face as a b 2c If the larger bipyramid were steeper then this face could be denoted by say a b 3c If the steepness of the face increases more and more it will eventually become parallel to the Z axis which means that its cut off point lies infinitely far away from the origin of the set of axes So the coefficient in front of c becomes infinitely large and will be denoted by the sign Thus if indeed that face assumed a vertical position it could then belong to a prism instead of a bipyramid then still relating this face to the unit face it could be describes as a b c If we imagine a face that cuts off a piece of the Z axis twice as long as the unit face does but which is horizontally oriented then it is parallel to both the X axis and the Y axis so the symbol for this face would read a b 2c And if it were parallel to both the X axis and the Z axis then its symbol would read a b c If the cut off piece concerns a part of the axis which is negative then the symbol will be placed in front of the symbol corresponding to that particular axis For example a b 2c denotes a face which goes through the lower tip of the larger bipyramid of Figure 2 but is parallel to both the X axis and the Y axis When we consider a certain face of a crystal belonging to a certain Crystal Class in isolation then we can subject this one face to all the symmetry operations rotatation reflection etc of that Crystal Class In fact this is equivalent to the demand that precisely those symmetries must finally all be present in the finished structure having the crystallographic axes as its scaffold Generally we then will obtain more faces in virtue of the workings of those symmetries For instance the presence i e the operation of a mirror plane as belonging to the set of symmetry elements of that particular Crystal Class will normally duplicate that face except when that mirror plane is perpendicular to that face If that mirror plane is parallel to that face then we will obtain a pair of parallel faces The structure resulting from subjecting a particular face to all the symmetry elements i e symmetry operations of the relevant Crystal Class is called a basic Form Such a Form could be something that completely encloses a volume of space within its faces like a cube or an octahedron or it could be open like a prism without bottom and top faces or just a pair of faces It even could be one face only namely in the case where the Crystal Class does not have any symmetry at all like the Triclinic Pedial Crystal Class Of course the open Forms cannot exist as real crystals but should be combined with other Forms So while the familiar meaning of the term Form refers to the outward appearance of something its meaning in Crystallography is somewhat different and more restricted the outward appearance is indicated by the term habit A Form consists of a group of crystal faces all of which have the same relation to the elements of symmetry and display the same chemical and physical properties because all are underlain by like atoms in the same geometrical arrangement A Form will be denoted by one of its faces namely the one with positive coefficients like a b c To denote the Form we place this symbol between brackets a b c The shapes of Crystals belonging to a certain Crystal Class can be those closed Forms and also any combination of whatever Forms belonging to that Crystal Class An example of the derivation of Forms It is now possible to derive the crystal Forms and their combinations and thus all crystal shapes of ideally grown crystals We shall work out some examples with respect to the Isometric System The Forms and their combinations of Crystals belonging to the Isometric System can be described by means of a set of three equivalent crystallographic axes perpendicular to each other Figure 5 Basic Form regular octahedron of the Isometric Crystal System and the set of crystallographic Isometric axes In this Figure we have drawn the basic Form of the Isometric Crystal System the regular octahedron One face is shaded This face cuts off equal pieces of the three crystallographic axes and can accordingly be denoted by a a a in which the three a s express the fact that the three axes are equivalent We will now derive some Forms belonging to the most symmetrical Crystal Class of the Isometric System the Cubic Hexakisoctahedral Crystal Class Recall that the symmetry content of crystals of this Class is Three 4 fold rotation axes coincident with the three crystallographic axes Four 3 fold rotation axes in diagonal directions Six 2 fold rotation axes bisecting the angles between the crystallographic axes Three primary mirror planes symmetry planes each one parallel to two crystallographic axes Six secondary mirror planes symmetry planes bisecting the angle between two primary mirror planes while perpendicular to a third Center of symmetry Let us depict some symmetry elements of this Class Figures 6 7 and 8 Figure 6 Some symmetry elements of the Cubic Hexakisoctahedral Class Depicted All three 4 fold rotation axes 4 and two out of six 2 fold rotation axes 2 Figure 7 Some symmetry elements of the Cubic Hexakisoctahedral Class Depicted two out of four 3 fold rotatation axes 3 Figure 8 Some symmetry elements of the Cubic Hexakisoctahedral Class Depicted one out of nine mirror plane and center of symmetry represented by the dot in the middle of the image In explaining the faces and Forms we concentrate on the notational method described above This is the method of Chr WEISS There are two other equivalent methods one developed by C NAUMANN and another by W H MILLER They will be explained below Now if we consider the shaded face which is the unit face for the Isometric Crystal System in Figure 5 in isolation i e if we think of having only this one face then we can generate the complete Form the regular octahedron of Figure 5 when we subject this face to all the symmetry elements symmetry operations belonging to our Crystal Class here the Cubic Hexakisoctahedral Crystal Class resulting in the finished form having all the symmetries of that Class A vertical 4 fold rotation axis then generates starting from the one face mentioned the upper part of the regular octahedron to subject our face to this four fold rotational symmetry will result in the generation of the three others The horizontal mirror plane coinciding with the plane in which the X and Y axes lie will generate from the top half we now have the bottom half of the octahedron So we now have our complete octahedron All the remaining symmetries are by now already implied i e when we subject our regular octahedron to those remaining symmetry operations no new faces are generated anymore The generated regular octahedron possesses all the symmetries of the Cubic Hexakisoctahedral Class The Weiss symbol for the regular octahedron the basic Form of the Cubic Hexakisoctahedral Crystal Class is a a a the Naumann symbol is O and the Miller symbol is 111 Recall that these symbols are equivalent it s just a difference in the method of description of faces and Forms Figure 9 Regular octahedron A next basic Form of this Class is the cube To accomplish its derivation we let the shaded face of Figure 5 become more and more steep which means that the length of the cut off piece of the Z axis becomes bigger and bigger In the limit this piece becomes infinitely long The face has now aquired a vertical position Next we turn this vertical face such that it becomes parallel to the X and Z axes while remaning vertical In this way we obtain the face of a cube oriented correctly namely its right vertical face and this face can be denoted by a a a If we now subject this vertical face to the symmetry elements of our Class then a cube will be generated The vertical 4 fold rotation axis will result in the four vertical faces of the cube and a secondary mirror plane demanded to be present in the finished structure stretching between diametrically opposite horizontal edges of the until now generated structure generates the top and bottom faces of the cube The figure we thus have generated not only is a cube but also one with a convenient orientation The crystallographic axes still coincide as in the case of the above generated octahedron the basic Form with 4 fold axes The Weiss symbol for the cube regular hexahedron is a a a the Naumann symbol is O and the Miller symbol is 100 Figure 10 Cube or regular hexahedron A next basic Form of this Class to be derived is the rhombic dodecahedron To accomplish this we let the shaded face of Figure 5 become vertical This face can then accordingly be denoted by a a a When we now subject this face to the symmetry elements of our Crystal Class we obtain a rhombic dodecahedron Figure 9 Another basic Form of the Cubic Hexakisoctahedral Crystal Class the rhombic dodecahedron The Weiss symbol for this Form is accordingly a a a the Naumann symbol is O and the Miller symbol is 101 Another Form of this Class is the ikositetrahedron We can derive this Form as follows Starting again with the shaded face of the regular octahedron of Figure 5 we will let it turn horizontally such that the length of the cut off piece of the Y axis i e the length of the piece cut off from the Y axis by that face has been increased without having this increase carried to an extreme meaning that the length of the cut off piece does not become infinite Many such positions of the face are possible for example a case in which the cut off piece of the Y axis has become twice as long as the one of the X axis Here we will consider the general case and say that the cut off piece of the Y axis has become m times as long as the one of the X axis The symbol of the face would then read a ma a Next we are going to further subject the face a ma a to a turn such that the length of the cut off piece of the Z axis increases in precisely the same way as the cut off piece of the Y axis did before The resulting face can then be described with the symbol a ma ma If we now subject this face to all the symmetries of our Crystal Class a polyhedron will be generated that is called an ikositetrahedron Figure 10 Yet another basic Form of the Cubic Hexakisoctahedral Crystal Class the ikositetrahedron The Weiss symbol for this Form is accordingly a ma ma the Naumann symbol is mOm and the Miller symbol is hkk In figure 10 we see this Form for the case of m 2 which means that its symbols are respectively a 2a 2a 2O2 and 211 In this way we can derive the rest of the Forms of this Crystal Class We can easily see that for the Isometric Crystal System having three equivalent crystallographic axes there are only seven types of basic faces possible and from these we can derive the seven basic Forms of the Crystal System s most symmetrical Class the Cubic Hexakisoctahedral Class by subjecting a basic face to the symmetries of that Class We shall list the seven types of faces together with the Forms that can be generated from them a a a Octahedron a a ma Pyramid octahedron a a a Rhombic dodecahedron a ma ma Ikositetrahedron a a a Cube a ma na 48 hedron i e a polyhedron with 48 faces a ma a Pyramid cube Crystals of this Class not only can have the shapes of these seven basic Forms but also of combinations of them For example a combination of an octahedron and a cube We will then see an octahedron but with its six corners cut off by the six cube faces We will expound and illustrate all this in more detail in the next Essay on The Isometric Crystal System The derivation of the shapes of the remaining Classes of the Isometric System will be dealt with later For The Crystal Classes of the other Crystal Systems other sets of crystallographic axes are used in which not all axes are equivalent To express this fact different symbols for their respective cut off pieces are used implying that we can have symbols like a a mc or a b mc The method just described extensively of indicating faces and Forms is as has been said developed by Chr WEISS and we can call it the method of Weiss The above derived Forms and their combinations all belong to the Cubic Hexakisoctahedral Crystal Class of the Isometric Crystal System this Class served as an example of denotation and derivation of faces and Forms It is the most symmetrical Class of this system which means that the symmetry of regular grown crystals is the same as the empty building block a cube by which every crystal of this System can be thought to be constructed by stacking those building blocks in an equivalent way along the three directions indicated by the crystallographic axes of the Isometric System What about the derivation of the shapes of crystals belonging to the less symmetrical Classes of this System Well that is in principle very simple We again start from the seven basic faces and subject each one of them to all the symmetries of the particular lower symmetrical Crystal Class Because these remaining Crystal Classes of the Isometric System are all less symmetric than the Cubic Hexakisoctahedral Class and because we can order these Classes according to a decreasing symmetry we can generate the shapes of crystals of these less symmetrical Classes by starting not with the basic faces but with the basic Forms We then eliminate more and more symmetries of those Forms and will in so doing obtain Forms belonging to those lesser symmetrical Classes together with their combinations This approach of deriving conceptually the Forms of the lesser symmetric Crystal Classes of each Crystal System uses the concepts of holohedric hemihedric hemimorphic tetartohedric and ogdohedric We shall expound this approach shortly But let us first expound a second method to denote basic Forms developed by C NAUMANN While the above expounded method the method of Weiss can denote faces as well as Forms the method now to be discussed is only geared to symbolize Forms But it is a very convenient one So let us explain it In each of the six Crystal Systems that Form is chosen as to be the basic Form of that System of which each one of its faces cuts off finite pieces of each crystallographic axis like in the Weissian method and this basic Form is now denoted by the Naumannian symbol P pyramid with the exception of the basic Form of the Isometric System which is denoted by the symbol O octahedron The derivation coefficients used in the first method the m n and in symbols like a ma a or like a b mc will when they relate to the Z axis the vertical crystallographic axis be placed in front of the symbol P or O and when they relate to the horizontal axes they will be placed after the symbol P or O So the derived pyramid bipyramid of Figure 4 i e the larger bipyramid denoted by the Weissian method as a b 2c will then be represented by the Naumannian symbol 2P and a Form a b c will be denoted as P etc We already have used Naumannian symbols namely in the Figures 9 and 10 In drawings we will as we did in Figure 9 use the conventional symbol for infinity a horizontal eight instead of the symbol Further details of this very convenient method will be given in the ensuing Essays devoted to the different Crystal Systems and their Classes A third method presently widely used of symbolizing faces and Forms uses indices the so called Miller indices Here each crystal face will be symbolized as follows We take the reciprocal values of the Weissian coefficients for that face Next these reciprocals are reduced to whole numbers by multiplying them with an appropriate number which of course does not change their ratio Finally the letters a b and c and the s are omitted and when a minus sign is needed it will be placed above the relevant number In the Essays of this website however we will use instead because of typographical problems an asterisk placed after the relevant number to symbolize the fact that a negative part of a crystallographic axis is involved The number relating to the X axis and its corresponding cut off piece a is the first number of the Millerian symbol the number relating to the Y axis and its corresponding cut off piece b is the second number of that symbol while the number relating to the Z axis and its corresponding cut off piece c is the third number of this symbol Because the reciprocal value of infinity is zero the number 0 will appear where in the Weissian symbol there was a symbol for infinity The Miller indices representing a face will be enclosed between circular brackets So the face a b c becomes 111 and the Miller indices of the face a b c are 111 and a b c becomes 110 The Miller indices of the face a b 2c can be calculated as follows the Weissian coefficients are 1 1 2 The reciprocal values of these are 1 1 1 1 1 2 We can bring these values to whole numbers by multiplying them by 2 resulting in the Millerian symbol 221 When the weissian coefficients are fractions then we must first reduce them to whole numbers For instance 3a 3 2b c becomes 6a 3b 2c The reciprocals of these are then 1 6 1 3 1 2 Multiplication by 6 gives 6 6 6 3 6 2 and thus the Millerian symbol 123 For the general Weissian symbol ma nb oc one writes hkl as the general Millerian symbol The whole Form is indicated in this method by placing the face symbol between braces Thus the smaller bipyramid a b c of Figure 4 can be denoted by 111 as well as by the Naumannian symbol P The derivation of the Forms of the lower Symmetry Classes of each Crystal System We will now expound some general concepts that are needed to conceptually derive all lower Symmetry Classes Crystal Classes of each Crystal System from its highest symmetrical Class each Crystal System consists of several Crystal Classes Symmetry Classes one of which has the highest symmetry i e it has the same symmetry as its empty building block Holohedric hemihedric hemimorphic tetartohedric and ogdohedric It is possible to conceptually derive certain Forms from others by letting a part of the faces disappear in a systematic way This can be accomplished by letting certain mirror planes disappear resulting for each mirror plane in the supression of all the faces lying at one side of that plane while letting the other faces extend in a corresponding way i e letting them extend till they meet In this way we obtain new Forms in which the relevant mirror planes are absent and which possess only a part of the faces of the original Forms The orientation of these left over faces is however the same as the corresponding ones in the original Form Such new Forms are called merohedrons while the original Forms are called holohedrons and one distinguishes between hemihedrons Tetartohedrons and ogdohedrons in so far as the new Forms possess respectively half a quarter or one eight of the faces of the original holohedric Forms Naturally two hemihedrons originate from a holohedron four tetartohedrons and eight ogdohedrons and one calls such Forms correlated Forms i e the two hemihedrons originating from a holohedron are correlated Forms so also the four tetartohedrons and also the eight ogdohedrons Most correlated Forms are congruent and are only distinguished by their orientation with respect to the relevant crystallographic axes But some relate to each other as being left and right handed implying that they cannot be transformed into each other by any mechanical operation like for instance a rotation They are called enantiomorphous Forms There exist merohedrons which are not distinguished from their corresponding holohedrons by their outer shape i e morphologically and that is always the case when the disappearing mirror planes were perpendicular to the faces of the holohedron The fact that they possess a lower symmetry than the corresponding holohedrons and thus are merohedrons in a crystallographical sense is visible in their physical properties and also by the fact that they can combine with other merohedrons Where several types of mirror planes are involved also several types of merohedrons can result when either one set of mirror planes is suppressed or another set One signifies that type of hemihedron as being hemimorphous hemimorphic that originates when the one for the crystal unique singular mirror plane is suppressed for example the equatorial symmetry plane of a tall or fat bipyramid The effect of all this is that the axis that was perpendicular to that mirror plane has now become polar implying that the crystal is differently developed at the two sides of that axis in a geometrical as well as in a physical sense Remark It should be noted that the concepts of holohedron hemihedron etc and the derivation of the one from the other and also the grouping of several Crystal Classes into Crystal Systems is essentially just an aid to obtain a clear and convenient overview of the forms and shapes of crystals In themselves the Crystal Classes Symmetry Classes are independent of each other and wholly equivalent In the next Essays we will apply this method of derivation of the lower Symmetry Classes from the highest one the holohedric one of the Crystal System concerned How the above concept of basic Form has its foundation in the crystal s internal structure One might wonder why we can so easily speak of faces that together build up a Form consisting of the same type of face i e consisting of equivalent faces Surely Crystals are the product of some Lego set of faces that can be attached to each other But no crystals are not hollow structures but filled solids without container walls In order to explain why the above constructions are nevertheless conceptually possible and founded in real state of affairs let us consider a certain fictitious two dimensional crystal consisting of two different sorts of constituents say two different species of ions electrically charged atoms or groups thereof with opposite electrical charge Let us represent those ions by two kinds of disks gray ones and red ones Because like charges repel each other and opposite charges attract each other each negatively charged ion wants to collect as many positive ions as its nearest neighbors as possible and each positively charged ion wants to collect as many negative ions as its nearest neighbors as possibe And every ion avoids a position next to any ion having a charge of the same sign The following crystalline structure will be the result of this in this figure we depict only the internal structure and not its external boundaries so we depict just an arbitrary fragment of the crystal Figure 11 Internal structure of a two dimensional ionic crystal consisting of two different sorts of ions of opposite electrical charge This structure allows several kinds of flat faces to be developed Each type of face presents a different atomic aspect to the growing environment the nutrient environment In the next figure we show that the structure under consideration allows three different types of faces A B and C corresponding to three atomic aspects to the environment The blue lines indicate these possible faces and we must interpret the aspects as true aspects in so far as they are facing outwardly i e towards the environment Figure 12 Examples of the three different kinds of faces allowed by the structure according to the three different atomic aspects pictured below Figure 13 Three different atomic aspects of the structure of the figures 11 and 12 When we look to aspect A we see that six faces are possible i e six faces differing in orientatation but showing the same aspect to the growing environment The nutrient material cannot distinguish among these six When conditions in the environment are the same around the entire crystal all six faces will grow at the same rate And if they are the only faces appearing then the ideal crystal will have the shape of a regular hexagon as shown in the next figure and will not reveal the lower symmetry of the structure which has the symmetry of an equilateral triangle not of a regular hexagon Figure 14 Regular hexagon bounded by A faces only This hexagon does not however possess the full symmetry of a regular hexagon One set of mirror lines is suppressed and the rotation axis is three fold not six fold To show this clearly let us look to a two dimensional structure which has the full symmetry of the hexagon Figure 15 A structure showing the full hexagonal two dimensional symmetry This structure

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  • Derivation of the Crystal Classes I
    center of the sphere onto the faces representing their orientations These perpendiculars are then extended till they intersect pierce the sphere These piercing points are the so called face poles To obtain stereographic projections of the symmetry elements we extend all the symmetry axes of the crystal till they pierce the sphere and thus resulting in piercing points on the sphere Further we extend the mirror planes of the crystal if present in it till they intersect with the sphere Such a mirror plane is then represented on the sphere as a circle In the case of an oblique mirror plane we only extend the part that lies above the equatorial plane projection plane resulting in half a circle on the sphere Now all these piercing points and when the case may be circle halves lying on the sphere must be projected onto the equatorial plane projection plane of the sphere This is done by connecting these points when lying on the northern half of the sphere with the south pole of that sphere The intersections of those connecting lines with the equatorial plane the projection plane are now the stereographic projections of those points lying initially on the sphere and at the same time they are the projections of the corresponding features of the crystal Those piercing points that lie on the southern half of the sphere are connected with its north pole Their intersections with the projection plane again are the projections of those points lying initially on the sphere and at the same time they are the projections of the corresponding features of the crystal Piercing Points on the equator of the sphere can be connected either with the south or the north pole which does not make a difference Their projections will end up om the perimeter of the equatorial plane In this way faces and axes are represented on the projection plane as points a horizontal mirror plane as a circle coinciding with the perimeter of the projection plane vertical mirror planes as straight lines An inclined mirror plane then is represented on the projection plane as an arc i e the hemicircle on the sphere becomes an arc on the projection plane The above Figure depicts the projection plane and the relevant projections onto it The fact that the circular outline of the projection plane is dashed means that a horizontal mirror plane is absent in the crystal When such a mirror plane were present the outline would be drawn as a solid circular outline Two small solid ellipses opposite to each other signify a horizontal 2 fold rotatation axis The sign in the center of the projection plane signifies the vertical rotational axis which in our example is a 4 fold roto inversion axis The solid straight lines are the projections of vertical mirror planes when a mirror plane is horizontal it will be projected as the just mentioned solid circular outline of the projection plane when a mirror plane is inclined its projection will have the shape of an arc going from one side of the projection plane to the other The face poles of crystal faces i e the piercing points of the perpendiculars to those faces will end up as points on the projection plane In our ensuing stereograms they will be depicted in red When such a point was a projection from above thus representing an upper face of the crystal it will be depicted by a red dot small solid red circle When such a point was a projection from below thus representing a lower face of the crystal it will be depicted as a small red open circle When there are two points on the sphere one on the northern hemisphere and the other on the southern hemisphere whose projections onto the projection plane coincide they will be depicted as a small red circle centered with a red dot When a face pole lies on the equator of the sphere then representing a vertical face its projection will end up on the perimeter of the projection plane and will be given in the stereograms as small solid red circles The same goes for piercing points of symmetry axes when those piercing points lie on the equator of the sphere Their respective symbols come to lie on the perimeter of the projection plane When an axis is inclined one of its piercing points will come to lie on the northern hemisphere and the other on the southern hemisphere In this case only the northern piercing point will be projected onto the projection plane resulting in a point lying inward from the perimeter of the projection plane We shall now prove that the value of n expressing the foldness of a rotation axis or of a roto inversion axis will be limited to 1 2 3 4 and 6 in crystals because of their three dimensional periodic structure The three dimensional periodicity of a crystal is given by its translational lattice A translational lattice is a lattice of points such that every vector a vector is a line of a certain length and having a certain direction from a lattice point to another such point determines a parallel shift translation which maps the crystal s atomic piling pattern onto itself when this pattern is conceived as being extended indefinitely Let a crystal possess a n fold axis then this axis will also be present in the translational lattice of that crystal Perpendicular to a symmetry axis in a translational lattice is always a lattice plane of that lattice In this lattice plane there is a shortest lattice vector say p In virtue of the n fold axis there will be equal lattice distances in directions that make angles of 360 n 0 2 x 360 n 0 n 1 x 360 n 0 with p In figure 4 AB is the shortest lattice vector p p AC forms an angle with it of 360 n 0 and p AD an angle of n 1 360 n 0 360 n 0 Figure 4 Some lattice points of a lattice plane separated by a minimum distance p i e the length of the shortest translation vector p The lattice has n fold symmetry Because AD represents a lattice vector and thus a translation E will be a lattice point if C is one and indeed it is one i e from lattice point C we must find another lattice point at a distance equal to the length of AD and in the direction of AD because the structure is repeated over distances equal to the length of p in the directions dictated by the n fold axis E lies on the line through AB Because AB is the shortest translation all lattice points on the line through AB are separated by a distance equal to the length of p So E being a lattice point should lie at a distance from A that is a whole number times the length of p this length can be denoted by p Then the following relation should hold AE 2pcos 360 n Np in which N is a whole number Let us derive this relation See Figure 4 Recall that the cosinus cos of an angle of a triangle having one other angle equal to 90 0 is equal to the ratio of the length of the opposite side and the length of the oblique side In figure 4 we have two angles at A 360 n and 360 n We can see that cos 360 n AF AC AF p This is equivalent to pcos 360 n AF And now we can say AE 2AF 2pcos 360 n Np in which N is a whole number This is equivalent to the equation cos 360 n N 2 Now we can solve this equation for n We know that the cosinus of any angle varies only between 1 and 1 Further we know that N is a whole positive number So we can proceed as follows If N 2 then cos 360 n is equal to 1 and this is the case when 360 n 180 0 and then n 180 360 so n 2 In the same way we can proceed for the other whole values of N such that the cosinus lies between 1 and 1 inclusive Then we obtain the following solutions for n N cos 360 n 360 n n 2 1 180 0 2 1 1 2 120 0 3 0 0 90 0 4 1 1 2 60 0 6 2 1 0 0 360 0 1 infinity When n 1 then a figure having a 1 fold axis will be mapped onto itself every 360 1 0 360 0 When n infinity then a figure having an infinity fold axis will be mapped onto itself every 360 infinity 0 0 0 Because a rotation of 360 0 an one of 0 0 is a same rotation a 1 fold axis is identical to an infinity fold axis so we can call them by the same name say a 1 fold axis So the table above has as its result that for crystals the value of n can only be 1 2 3 4 or 6 These five values are also the only possible values for n with respect to roto inversion axes There a rotation is immediately followed by an inversion in a point on that axis Let me explain An n fold rotation axis generates a real image of the initial motif every 360 n 0 But an n fold roto inversion axis does not always generate a real image every 360 n 0 It generates either a real image every 360 n 0 or an intermediate image every 360 n 0 See the next Figures Figure 4a Stages 1 2 3 4 5 6 in the generation of a Form i e a set of equivalent crystal faces from an initial crystal face by a 3 fold roto inversion axis depicted in stereographic projections stereograms Small red solid circles represent projections from above so they represent upper crystal faces small red open circles represent projections from below so they represent lower crystal faces 1 Stereogram of the 3 fold roto inversion axis and an initial motif an upper crystal face 2 Illustration of the generation of a second crystal face by the action of the 3 fold roto inversion axis The initial motif is rotated 120 0 clockwise generating an intermediate image green dot This intermediate image is then inverted through a point on the axis resulting in the second crystal face a lower face 3 In the same way a third crystal face can be generated from the second 4 Generation of the fourth face 5 Generation of the fifth face 6 Generation of the sixth face This sixth face will generate the first face i e the initial face so after this no new faces are generated anymore In the end stage of the generation just given we see six crystal faces They clearly show a 3 fold symmetry Indeed we see an identical image appearing every 120 0 So in this case a roto inversion axis here a 3 fold roto inversion axis successively replaces all the intermediary images wherever they initially are generated by real images Indeed a 3 fold roto inversion axis is equivalent to a 3 fold rotation axis and a center of symmetry Said in other words a figure possessing a 3 fold roto inversion axis which combines the operation of rotation with the operation of inversion in a point on that axis is equivalent to it possessing a 3 fold rotation axis and possessing a center of symmetry Other roto inversion axes are however not like that They do not finally replace their intermediate images with real images so for instance a 4 fold roto inversion axis Figure 4b Stages 1 2 3 4 in the generation of a Form i e a set of equivalent crystal faces from an initial crystal face by a 4 fold roto inversion axis depicted in stereographic projections stereograms Small red solid circles represent projections from above so they represent upper crystal faces small red open circles represent projections from below so they represent lower crystal faces 1 Stereogram of a vertical 4 fold roto inversion axis and an initial motif an upper crystal face 2 Illustration of the generation of a second crystal face by the action of the 4 fold roto inversion axis The initial motif is rotated 90 0 clockwise generating an intermediate image green dot This intermediate image is then inverted through a point on the axis resulting in the second crystal face a lower face 3 In the same way a third crystal face can be generated from the second 4 Generation of the fourth face This fourth face then generates the initial face so no new faces are generated anymore when the process is continued The intermediate image in 2 of the above Figure stands for an upper intermediate crystal face and as can be seen is not replaced by a real upper face Also in the case of a 6 fold roto inversion axis the intermediate images are not replaced by real images Figure 4c Stages 1 2 3 4 5 6 in the generation of a Form i e a set of equivalent crystal faces from an initial crystal face by a 6 fold roto inversion axis depicted in stereographic projections stereograms Small red solid circles represent projections from above so they represent upper crystal faces small red open circles represent projections from below so they represent lower crystal faces A small red open circle centered with a red dot represents two coinciding projections one from above and one from below 1 Stereogram of the 6 fold roto inversion axis and an initial motif an upper crystal face 2 Illustration of the generation of a second crystal face by the action of the 6 fold roto inversion axis The initial motif is rotated 60 0 clockwise generating an intermediate image green dot This intermediate image is then inverted through a point on the axis resulting in the second crystal face a lower face 3 In the same way a third crystal face can be generated from the second 4 Generation of the fourth face 5 Generation of the fifth face 6 Generation of the sixth face This sixth face will generate a face that has already been generated before so after this no new faces are generated anymore In the above Figure we see that there is no replacement of all the intermediate images The final face configuration has a 3 fold symmetry One can see that this face configuration could equally well be generated from the initial face by subjecting this face first to an ordinary 3 fold rotation axis and then subject the result to the action of a horizontal mirror plane indicated by the solid outline of the projection plane So a 6 fold roto inversion axis is equivalent to a 3 fold rotation axis plus a mirror plane perpendicular to it Thus 6 3 m Above Figure 4 we proved that only 1 2 3 4 and 6 fold rotation axes can exist in crystals This was based on the fact that the lattice points are being repeated in directions dictated by the rotational symmetry of the axis When we proceed to prove that the same values for n are also the only possible for roto inversion axes then we must allow for the real as well as the intermediate images And then the proof will go in exactly the same way as for the ordinary rotation axes so we will find the same only possible values for n with respect to the roto inversion axes This concludes our proof of the above given limited possibilities with respect to symmetry axes that can occur in crystals The symmetry of crystals can accordingly only be represented by those axes namely 1 2 3 4 and 6 fold rotation axes and 1 2 3 4 and 6 fold roto inversion axes either occurring alone or combined i e the symmetry of a crystal is either determined by such an axis or by a combination of them a mirror plane m is equivalent to a 2 fold roto inversion axis 2 and a center of symmetry i is equivalent to a 1 fold roto inversion axis 1 It turns out that only 32 combinations including symmetries determined by just one such axis are possible They represent the 32 Crystal Classes Derivation of the 32 Crystal Classes We will now derive the 32 Crystal Classes symmetry classes albeit not rigorously by means of stereographic projections We ll begin with a group of 27 Classes 17 of them will be derived directly or indirectly from the remaining 10 which are represented by the above mentioned ten symmetry axes These 27 Classes will be ordered in a table having the following outline In this outline matrix their corresponding entries are given in blue A 1 B 1 C 1 D 1 E 1 F 1 G 1 A 2 B 2 C 2 D 2 E 2 F 2 G 2 A 3 B 3 C 3 D 3 E 3 F 3 G 3 A 4 B 4 C 4 D 4 E 4 F 4 G 4 A 5 B 5 C 5 D 5 E 5 F 5 G 5 The above table only serves as a matrix of the possible entries of the ensuing real table of the stereograms of the first to be discussed 27 Crystal Classes later to be supplemented with those of the remaining five Classes Each stereogram in this real table represents a Class It shows the symmetry elements of that Class and the projection of the face poles of the most general Form i e it

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  • Derivation of the Crystal Classes II
    is a mirror plane parallel to the n fold axes of the Classes of column A We will get three new Classes Addition of a mirror plane parallel to the n fold axis of the Class represented by the entry A1 yields B2 See Figure 12 Figure 12 A1 B2 Derivation of B2 from A1 1 Stereogram of the symmetry elements of the Class 1 and of the face poles of the most general Form 2 Addition of a mirror plane generates a second motif 3 Reorientation of the stereogram The resulting motif pattern is consistent with 2 m symmetry Addition of a mirror plane parallel to the n fold axis of A2 yields E2 See Figure 13 Figure 13 A2 E2 Derivation of E2 from A2 1 Stereogram of the symmetry elements of the Class 2 and of the face poles of the most general Form 2 Addition of a mirror plane parallel to the n fold axis generates two more motifs 3 The resulting motif pattern is consistent with mm2 symmetry Addition of a mirror plane parallel to the n fold axis of A3 yields F3 See Figure 14 The motif pattern of 3 of this Figure is drawn rotated with respect to that of 2 and also a bit rearranged but this is immaterial because the position of the motifs with respect to the crystallographic axes remains a general one So the stereograms involved are equivalent in so far as they express the symmetry Figure 14 A3 F3 Derivation of F3 from A3 1 Stereogram of the symmetry elements of the Class 3 and of the face poles of the most general Form 2 Addition of a mirror plane parallel to the n fold axis generates three more motifs 3 The resulting motif pattern implies two more vertical mirror planes and is consistent with 3m symmetry Addition of a mirror plane parallel to the n fold axis of A4 yields F4 See Figure 15 Figure 15 A4 F4 Derivation of F4 from A4 1 Stereogram of the symmetry elements of the Class 4 and of the face poles of the most general Form 2 Addition of a mirror plane parallel to the n fold axis generates four more motifs 3 The resulting motif pattern implies three more vertical mirror planes and is consistent with 4mm symmetry Addition of a mirror plane parallel to the n fold axis of A5 yields F5 See Figure 16 The slight rearrangement of motifs in 3 of this Figure is immaterial because their position is still general so the stereograms involved are equivalent in so far as they express the symmetry Figure 16 A5 F5 Derivation of F5 from A5 1 Stereogram of the symmetry elements of the Class 6 and of the face poles of the most general Form 2 Addition of a mirror plane parallel to the n fold axis generates six more motifs 3 The resulting motif pattern implies five more vertical mirror planes and is consistent with 6mm symmetry By adding a mirror plane parallel to the n fold axis of the Classes of column A we have derived three new Classes namely F3 F4 and F5 i e we now have derived all the Classes of column F Let us indicate this progress in the matrix of our table of the first 27 Crystal Classes A 1 B 1 C 1 D 1 E 1 F 1 G 1 A 2 B 2 C 2 D 2 E 2 F 2 G 2 A 3 B 3 C 3 D 3 E 3 F 3 G 3 A 4 B 4 C 4 D 4 E 4 F 4 G 4 A 5 B 5 C 5 D 5 E 5 F 5 G 5 Addition of a mirror plane parallel to the n fold axis of column B does not yield any new Classes Addition of a mirror plane parallel to the n fold axis of the Class represented by B1 yields C2 The Class 1 B1 does not have intrinsic directionality imposed by an intrinsic axis The addition of a mirror plane induces a 2 fold rotation axis which is now the n fold rotation axis of the set of motifs According to the convention of our table the orientation will be such that this 2 fold rotation axis is perpendicular to the projection plane and thus to the plane of the drawing of the stereogram See Figure 17 Figure 17 B1 C2 Derivation of C2 from B1 1 Stereogram of the symmetry elements of the Class 1 and of the face poles of the most general Form 2 Addition of a mirror plane parallel to the n fold axis generates two more motifs 3 This addition of the mirror plane and the resulting motif pattern imply a 2 fold rotation axis perpendicular to the added mirror plane and the motif pattern is now consistent with 2 m symmetry The resulting stereogram is equivalent to the stereogram of 4 which is obtained by reorienting it 4 Stereogram after reorientation of that of 3 5 Stereogram with rearranged motifs causing no change of symmetry Addition of a mirror plane parallel to the n fold axis of B2 yields E2 See Figure 18 Figure 18 B2 E2 Derivation of E2 from B2 1 Stereogram of the symmetry elements of the Class 2 and of the face poles of the most general Form 2 Addition of a mirror plane parallel to the n fold axis generates two more motifs 3 Reorientation of the stereographic projection such that the resulting equivalent stereogram is as 2 when viewed in the direction of the bold blue arrow In this reorientation we see again the two mirror planes perpendicular to each other 4 The resulting motif pattern implies a 2 fold rotation axis as indicated Addition of a mirror plane parallel to the n fold axis of B3 yields E3 See Figure 19 The stereogram of 4 of this figure is drawn rotated with respect to that of 3 but this is immaterial insofar as symmetry is concerned Figure 19 B3 E3 Derivation of E3 from B3 1 Stereogram of the symmetry elements of the Class 3 and of the face poles of the most general Form 2 Addition of a mirror plane parallel to the n fold axis generates six more motifs 3 The resulting motif pattern implies two more vertical mirror planes 4 Also three 2 fold rotation axes are implied The motif pattern is consistent with 3 2 m symmetry Addition of a mirror plane parallel to the n fold axis of B4 yields E4 See Figure 20 The stereogram of 4 is drawn rotated with respect to that of 3 but this is immaterial in so far as symmetry is concerned Figure 20 B4 E4 Derivation of E4 from B4 1 Stereogram of the symmetry elements of the Class 4 and of the face poles of the most general Form 2 Addition of a mirror plane parallel to the n fold axis generates four more motifs 3 The resulting motif pattern implies one more vertical mirror plane 4 Also two 2 fold rotation axes are implied The motif pattern is consistent with 4 2m symmetry Addition of a mirror plane parallel to the n fold axis of B5 yields E5 See Figure 21 The stereogram of 4 of this figure is drawn rotated with respect to that of 3 but this is immaterial insofar as symmetry is concerned Figure 21 B5 E5 Derivation of E5 from B5 1 Stereogram of the symmetry elements of the Class 6 and of the face poles of the most general Form 2 Addition of a mirror plane parallel to the n fold axis generates six more motifs 3 The resulting motif pattern implies two more vertical mirror planes 4 Also three 2 fold rotation axes are implied The motif pattern is consistent with 6 m2 symmetry So by adding a mirror plane parallel to the n fold axes of the Classes of column B the following Classes given by their entries in our table are generated C2 E2 E3 E4 and E5 C2 we had aready generated before and also all the members of column E were already generated earlier So no new classes are generated The next symmetry element to be added is a mirror plane perpendicular to the n fold rotation axis When we add such a mirror plane to the Classes of columns A and B no new combinations of symmetry elements will be generated Addition of a mirror plane perpendicular to the n fold axis of A1 yields B2 The direction of the n fold axis is not fixed because it is a 1 fold axis that can have any direction See Figure 22 Figure 22 A1 B2 Addition of a mirror plane perpendicular to the n fold axis of A2 yields C2 See Figure 23 Figure 23 A2 C2 Addition of a mirror plane perpendicular to the n fold axis of A3 yields B5 See Figure 24 Figure 24 A3 B5 Addition of a mirror plane perpendicular to the n fold axis of A4 yields C4 See Figure 25 Figure 25 A4 C4 Addition of a mirror plane perpendicular to the n fold axis of A5 yields C5 See Figure 26 Figure 26 A5 C5 Summing up addition of a mirror plane perpendicular to the n fold axes of the Classes of column A yielded the following Classes as represented by their entries in our table B2 C2 B5 C4 and C5 All these five classes were already derived earlier so nothing new is generated Addition of a mirror plane perpendicular to the n fold axis of B1 yields C2 B1 only has a center of symmetry So the direction of the n fold axis which is here a 1 axis is not fixed Axial direction is finally given by the implied 2 fold rotation axis that comes perpendicular to the added mirror plane See Figure 27 Figure 27 B1 C2 Addition of a mirror plane perpendicular to the initial n fold axis implies a 2 fold rotation axis that imposes a definite axial direction by being the new n fold axis Addition of a mirror plane perpendicular to the n fold axis of B2 yields B2 The Class represented by B2 namely 2 m has a 2 axis which is perpendicular to the plane of the drawing of our stereogram Such a 2 axis then already implies a mirror plane perpendicular to it So nothing changes See Figure 28 Figure 28 B2 B2 Addition of a mirror plane perpendicular to the n fold axis does not change the Class because that mirror plane was already present Addition of a mirror plane perpendicular to the n fold axis of B3 yields C5 See Figure 29 Figure 29 B3 C5 Addition of a mirror plane perpendicular to the n fold axis of B4 yields C4 See Figure 30 Figure 30 B4 C4 Addition of a mirror plane perpendicular to the n fold axis of B5 yields B5 See Figure 31 Figure 31 B5 B5 Addition of a mirror plane perpendicular to the n fold axis does not change the Class because that mirror plane was already present Summing up addition of a mirror plane perpendicular to the n fold axes of the Classes of column B yielded the following Classes as represented by their entries in our table C2 B2 C5 C4 and B5 All these five classes were already derived earlier so nothing new is generated Finally we must see what happens when symmetry elements are added to columns C D E and F If we combine for example column C with a 2 fold rotation axis perpendicular to the n fold axis then three new classes will be generated constituting column G Adding a 2 fold rotation axis perpendicular to the n fold axis of C2 the first Class in column C yields G2 Two vertical mirror planes and one more 2 fold rotation axis are implied by the resulting motif pattern The slight displacement of the motifs as drawn in the stereograms and the rotation of the drawing are immaterial as far as symmetry is concerned See Figure 32 Figure 32 C2 G2 Derivation of G2 from C2 1 Stereogram of the symmetry elements of the Class 2 m and of the face poles of the most general Form 2 Addition of a 2 fold rotation axis perpendicular to the n fold axis generates four more motifs 3 The resulting motif pattern implies two vertical mirror planes and one more 2 fold rotation axis The resulting stereogram is equivalent to that of 4 4 Rotated stereogram as it is given in the table of the first 27 Classes to be derived Adding a 2 fold rotation axis perpendicular to the n fold axis of C4 the second Class in column C yields G4 See Figure 33 Figure 33 C4 G4 Addition of a 2 fold rotation axis perpendicular to the n fold axis generates eight new motifs and implies four vertical mirror planes and three more 2 fold rotation axes Adding a 2 fold rotation axis perpendicular to the n fold axis of C5 yields G5 See Figure 34 The motifs are drawn a little reshuffled but their positions remain general and so still reflect the effect of the generated new symmetry Figure 34 C5 G5 Addition of a 2 fold rotation axis perpendicular to the n fold axis generates 12 new motifs and implies six vertical mirror planes and five more 2 fold rotation axes Summarizing By adding a 2 fold rotation axis perpendicular to the n fold axis of the Classes of column C yielded the following new Classes G2 G4 and G5 i e all the Classes of column G Let us indicate this result in the matrix of entries of our table of the first 27 Crystal Classes A 1 B 1 C 1 D 1 E 1 F 1 G 1 A 2 B 2 C 2 D 2 E 2 F 2 G 2 A 3 B 3 C 3 D 3 E 3 F 3 G 3 A 4 B 4 C 4 D 4 E 4 F 4 G 4 A 5 B 5 C 5 D 5 E 5 F 5 G 5 We see that we now have derived all the Crystal Classes listed in our table Of course this table was already presupposed from the beginning so we must check if or if not the table is in fact bigger i e we must check whether the addition of the above mentioned symmetry elements to Classes having at most one more than 2 fold axis an addition such that this one more than 2 fold axis will not be multiplied will now have generated all the Classes with at most one more than 2 fold axis So we must still add those symmetry elements to the Classes of the columns C D E F and G For column C we have done so already with respect to the addition of a 2 fold rotation axis perpendicular to the n fold axis The Classes of column C all have already a center of symmetry so the addition of that symmetry element cannot have any effect Let s explain this See Figures 34a and b As an example we take the first Class of column C i e Class C2 and show that it already has a center of symmetry Figure 34a 1 Stereogram of the Class C2 2 Invocation of real motifs represented by comma s at the locations of the projections of the face poles In the next Figure continuing the explanation that the Classes of column C already possess a center of symmetry we depict the configuration of Figure 34a such that our direction of view is perpendicular to that of Figure 34a Figure 34b Four motifs related to each other by a 2 fold rotation and a reflection In this drawing we should imagine that a comma also differs in the direction of viewing indicated by the coloring If a comma with its black side toward the beholder is rotated by 180 0 about the axis indicated by the small blue solid ellipses This axis thus lies in the plane of the drawing then it will show its red side So there is no mirror plane perpendicular to the projection plane and containing that 2 fold axis We can clearly see that motif b is the inverted image of motif a and vice versa and that motif b is the inverted image of motif a and vice versa because every detail of the comma including the color is interchanged And that is precisely what happens when the motifs are related to each other by the operation of inversion with the intersection of the 2 fold axis with the projection plane as inversion point Then the whole motif configuration indeed possesses a center of symmetry About a center of symmetry see also Figure 3 in Part One and Figure 38b in the present Part The same reasoning applies to the other two Classes of column C Addition of a mirror plane perpendicular to the n fold axis of the Classes of column C also cannot have any effect because these Classes already have such a mirror plane Addition of a mirror plane parallel to the n fold axis of the Classes of column C will yield the same Classes as did the addition of a 2 fold rotation axis perpendicular to the n fold axis Addition of a mirror plane parallel to the n fold axis of C2 yields G2 See Figure 35 The displacements of motifs as drawn in this Figure are immaterial insofar symmetry is concerned Figure 35 C2 G2 Derivation of G2 from C2 1 Stereogram of the symmetry elements of the Class 2 m and of the face poles

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  • Derivation of the Crystal Classes III
    added 2 fold axis making an angle of 30 0 with axis aa so also in this case a 6 fold rotational symmetry results for the n fold axis the axis perpendicular to the plane of the drawings The next Figure shows that when a 2 fold rotation axis perpendicular to the n fold axis making an angle of 1 2 45 0 twenty two and a half degrees with the axis aa as denoted in Figure 11a is added an 8 fold symmetry results an impossibility in crystals Figure 11b 1 Stereogram of the symmetry elements and the face poles of the most general Form of the Class 222 represented by the entry D2 of our table of the first 27 Crystal Classes to be derived 2 Addition of a 2 fold rotation axis perpendicular to the n fold axis and making an angle of 1 2 45 0 with the horizontal 2 fold axis axis aa in Figure 11a already present The added axis is duplicated by the latter 2 fold axis In compliance with these new 2 fold axes new motifs are generated 3 The new axes in their turn duplicate the original axes The implied axis generates another axis red by duplicating the original one The resulting axis then duplicates the implied axis The resulting axis blue is duplicated by the existing axis dd as denoted in Figure 11a resulting in yet another axis The axis dd is duplicated by that new axis resulting in yet another axis which is again duplicated resulting in an eighth axis So now we have a total of eight horizontal 2 fold rotation axes The resulting pattern of horizontal 2 fold rotation axes and the new motif pattern are consistent with 8 fold rotational symmetry which is not possible in crystals because of their periodic internal order The above Figure 11b shows that if a 2 fold rotation axis is added such that it makes a 1 2 45 0 angle with one or the other horizontal 2 fold axis already present as in Class D2 then in multiplying the already existing axes this results in a rotational symmetry higher than 6 which as has been proved cannot occur in crystals Here we end up with an eight fold rotational symmetry When we want to add a 2 fold rotation axis perpendicular to the n fold axis it should make an angle of either 45 0 or 30 0 or 60 0 with an already existing 2 fold rotation axis So we will first add a 2 fold rotation axis to the Class represented by D2 making a 45 0 angle with the other axes Then another 2 fold axis is automatically implied in compliance with the original horizontal 2 fold rotation axes The motif configuration consistent with these axes has then become such that the n fold axis has become a 4 fold rotation axis In this way the Class represented by D4 is derived from D2 See Figure 11c Figure 11c D2 D4 Addition of a 2 fold rotation axis perpendicular to the n fold axis and lying exactly midway between the other two horizontal 2 fold axes generates four new motifs and implies one more 2 fold rotation axis because the newly added axis is duplicated by an existing 2 fold axis The new motif configuration becomes such that the n fold axis which was originally a 2 fold rotation axis perpendicular to the plane of the drawing becomes a 4 fold rotation axis The new motif pattern will be mapped onto itself with every rotation by 90 0 about the n fold axis perpendicular to the drawing When we add a 2 fold rotation axis perpendicular to the n fold axis and making an angle of either 30 0 or 60 0 with the existing horizontal 2 fold axes then we will end up with the Class represented by D5 See Figure 11d Figure 11d D2 D5 Addition of a 2 fold rotation axis perpendicular to the n fold axis and making an angle of 30 0 or 60 0 with an existing 2 fold rotation axis generates eight new motifs and implies four more horizontal 2 fold rotation axes while the n fold axis which was originally a 2 fold axis perpendicular to the plane of the drawing has become a 6 fold rotation axis See also 8 of Figure 11a Addition of a 2 fold rotation axis perpendicular to the n fold axis of D3 yields D5 See Figure 12 Figure 12 D3 D5 1 Stereogram of the symmetry elements and of the face poles of the most general Form of Class 32 represented by the entry D3 of our table 2 Addition of a 2 fold rotation axis perpendicular to the n fold axis and bisecting the angle between two already existing 2 fold axes It accordingly makes an angle of 30 0 with those axes but at the same time it makes an angle of 90 0 with another already existing horizontal 2 fold rotation axis After the new motif pattern is generated by these axes other 2 fold rotation axes are then implied The initially added axis makes an angle of 60 0 with them So we in one stroke have tried out all the permitted angles that the added axis can make with existing axes In the present case an angle of 45 0 implies an angle of 15 0 leading to a forbidden symmetry 3 The added 2 fold rotation axis is consistent with the extended pattern of motifs as indicated 4 The generated pattern is consistent with 622 symmetry and is represented by the entry D5 of our table Addition of a 2 fold rotation axis perpendicular to the n fold axis of D4 yields D4 See Figure 13 Figure 13 D4 D4 1 Stereogram of the symmetry elements and of the face poles of the most general Form of Class 422 represented by the entry D4 of our table 2 Addition of a 2 fold rotation axis perpendicular to the n fold axis and making angles of 30 0 and 60 0 with existing 2 fold axes implies still more 2 fold axes such that an 8 fold symmetry will be the result See also Figure 11b which is impossible in crystals So in this case the only angles such an added axis can make with existing axes are 45 0 and 90 0 But such axes are already present Consequently the symmetry content will not be affected by the addition of a 2 fold rotation axis perpendicular to the n fold axis Addition of a 2 fold rotation axis perpendicular to the n fold axis of D5 yields D5 See Figure 14 Figure 14 Stereogram of the symmetry elements and of the face poles of the most general Form of the Class 622 represented by the entry D5 of our table When we would add a 2 fold rotation axis making an angle of 45 0 with one of the existing horizontal 2 fold axes then as can be seen in the right image of the Figure an angle between 2 fold axes of 15 0 is implied which is not permissible in crystals Angles of 30 0 60 0 and 90 0 between existing horizontal 2 fold axes are already present in Class D5 so an addition of a 2 fold rotation axis perpendicular to the n fold axis of D5 does not affect its symmetry content Summarizing Addition of a 2 fold rotation axis perpendicular to the n fold axes of the Classes of column D yielded D4 and D5 so nothing new is generated Now we will add a 2 fold rotation axis perpendicular to the n fold axis the vertical axis of the Classes of column E Addition of a 2 fold rotation axis perpendicular to the n fold axis to the symmetry content of the Class represented by the entry E2 the first Class of column E and making angles of 45 0 with existing mirror planes yields E4 See Figure 15 Figure 15 E2 E4 1 Stereogram of the symmetry elements and of the face poles of the most general Form of the Class mm2 represented by the entry E2 2 Addition of a 2 fold rotation axis red making angles of 45 0 with existing mirror planes The existing mirror planes duplicate this added axis The motifs i e faces represented by the projections of face poles are multiplied by the new 2 fold axes resulting in the motif pattern as indicated 3 The newly generated motif pattern turns out to be equivalent to the pattern as it is depicted in the stereogram of the Class E4 so we have generated Class E4 from E2 If we add such a 2 fold rotation axis to the symmetry of E2 such that it makes angles of 30 0 and 60 0 with the existing mirror planes while it is still perpendicular to the n fold axis then we will obtain the Class G5 See Figure 16 Figure 16 Derivation of G5 from E2 1 Stereogram of the symmetry elements and of the face poles of the most general Form of the Class mm2 represented by the entry E2 Two vertical mirror planes perpendicular to each other are present The vertical n fold axis is here a 2 fold rotation axis 2 Addition of a 2 fold rotation axis indicated by 2 perpendicular to the n fold axis making an angle of 30 0 with an existing mirror plane indicated by m and an angle of 60 0 with the other existing mirror plane 3 The first mentioned mirror plane is duplicated by the added 2 fold axis while also two more motifs are generated here two lower motifs 4 The second mentioned mirror plane the one that is horizontally drawn in the Figure reflects that new mirror plane and the new motifs resulting in yet another mirror plane and two more lower motifs 5 The added 2 fold axis is duplicated by the first new mirror plane resulting in a new 2 fold axis lying in an existing mirror plane and as such indicated by m2 So m2 means mirror plane 2 fold axis See arrow But because the motifs lying above and below m2 are now not only related by a reflection but also by a rotation of 180 0 these motifs should have upper counter motifs because the action of the 2 fold rotation axis generates upper motifs from lower ones Also the motifs drawn in the upper part of the stereogram should have their lower counter motifs in order to comply with the added 2 fold axis 6 The mirror plane indicated by a green arrow generates another m2 indicated by a blue arrow 7 The mirror plane indicated by the blue arrow duplicates the motifs lying on the right of it such that we now find such motifs also at the six o clock position in the stereogram One of the originally existing mirror planes namely the one vertically drawn in the stereograms produces yet another m2 indicated by the red arrow which means that the added 2 fold rotation axis now lies in a mirror plane This m2 duplicates the motifs as indicated The resulting motifs are again duplicated by reflection in an already originally existing mirror plane horizontally drawn in the images The final result is a motif pattern consisting of six pairs of upper motifs and their corresponding lower motifs 8 This motif configuration demands that now all the m s must be m2 s To understand why this must be see the Remark inframe section Implication of Mirror Planes below Figure 1 in Part Four There we see that after every 30 0 we must find an m2 The symmetry configuration so obtained is fully equivalent with that depicted in 9 which is the symmetry configuration of Class G5 9 Stereogram of the resulting Class 6 m 2 m 2 m represented by the entry G5 of our table of the stereograms of the first 27 Crystal Classes to be derived Addition of a 2 fold rotation axis perpendicular to the n fold axis of E2 and making angles of 0 0 and 90 0 with the existing mirror planes yields G2 See Figure 17 Figure 17 Derivation of G2 from E2 1 Stereogram of the symmetry elements and of the face poles of the most general Form of the Class mm2 represented by the entry E2 Two vertical mirror planes and one vertical 2 fold rotation axis are present 2 Addition of a 2 fold rotation axis perpendicular to the vertical n fold rotation axis and making angles of 0 0 and 90 0 with existing mirror planes 3 The upper motifs must obtain their lower counter motifs in order to comply with the added 2 fold axis 4 The resulting motif pattern then implies a horizontal mirror plane and one more 2 fold rotation axis now making up a 2 m 2 m 2 m symmetry That a horizontal mirror plane is indeed implied as stated in 4 of Figure 17 is explained in the Remark inframe in Part Four below Figure 1 From the lower part of Figure 1a of that same Remark we can deduce that also a 2 fold rotation axis is implied Addition of a 2 fold rotation axis perpendicular to the n fold axis of E3 and making an angle of 45 0 with an existing 2 fold axis causes a proliferation of 2 fold axes beyond the the number permissible for crystals See Figure 18 Figure 18 1 Stereogram of the symmetry elements and of the face poles of the most general Form of the Class 3 2 m represented by the entry E3 2 Addition of a 2 fold rotation axis perpendicular to the vertical n fold axis and making angles of 45 0 and 15 0 with existing 2 fold axes 3 The added 2 fold axis is duplicated by an existing 2 fold axis The resulting 2 fold axis is then duplicated by an existing mirror plane The resulting axis is in turn duplicated by an existing 2 fold axis etc finally resulting in a forbidden symmetry Addition of a 2 fold rotation axis perpendicular to the vertical n fold axis and making an angle of 30 0 with existing 2 fold axes to the symmetry content of E3 yields G5 See Figure 19 Figure 19 Derivation of G5 from E3 1 Stereogram of the symmetry elements and of the face poles of the most general Form of the Class 3 2 m represented by the entry E3 2 Addition of a 2 fold rotation axis perpendicular to the vertical n fold axis and making an angle of 30 0 with existing 2 fold axes 3 The added 2 fold axis is multiplied by existing 2 fold axes such that now each vertical mirror plane contains a 2 fold axis perpendicular to the n fold axis 4 In order to comply with all these horizontal 2 fold rotation axes each existing motif must have its counter motif An upper one must have its lower a lower must have its upper counter motif The resulting motif configuration now demands that all the horizontal 2 fold axes must be contained in vertical mirror planes See Figure 1e of the Remark inframe in Part Four below Figure1 there For the implication of a horizontal mirror plane see that same Remark Figure 1a All this results in a symmetry configuration consistent with that of G5 i e with 6 m 2 m 2 m symmetry Our case of the added 2 fold axis making an angle of 30 0 with existing 2 fold axes is equivalent to the case where it makes an angle of 90 0 as can be seen in the stereogram of E3 2 in the above Figure We do not have to check the case of the added 2 fold rotation axis making an angle of 60 0 with existing 2 fold axes because those existing axes themselves make already an angle of 60 0 with each other Addition of a 2 fold rotation axis perpendicular to the n fold axis of E4 and making angles of 30 0 and 60 0 with existing 2 fold axes yields a 12 fold rotational symmetry forbidden i e impossible for crystals See Figure 20 Figure 20 Generation of 12 fold rotational symmetry out of E4 1 Stereogram of the symmetry elements and of the face poles of the most general Form of the Class 4 2m represented by the entry E4 2 Addition of a 2 fold rotation axis perpendicular to the vertical n fold axis and making angles of 30 0 and 60 0 with existing 2 fold rotation axes 3 This new axis is then duplicated by one of the existing mirror planes and the resulting axis is in turn duplicated by one of the existing 2 fold axes The result is then duplicated by one of the existing mirror planes 4 By the action of the horizontal 2 fold axes new mirror planes are generated as indicated The mirror planes are indicated by the symbol m 5 Here and already in 4 we can see that we have 12 sets of symmetry elements each set consisting of half a vertical mirror plane m and half a horizontal 2 fold axis The latter is indicated by the symbol 2 So we have generated a 12 fold rotational symmetry indicated by 12 sectors each consisting of a white and a red half sector Such a symmetry is as we know impossible in crystals Consequently we cannot add a 2 fold rotation axis perpendicular to the n fold axis making angles of 30 0 and 60 0 with existing 2 fold axes We do not need to check the effect of an addition

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  • Derivation of the Crystal Classes IV
    the existing mirror planes generates three more vertical mirror planes 4 All these mirror planes generate eight more motifs 5 The resulting motif configuration demands that the vertical n fold axis which originally was a 2 fold rotation axis becomes a 6 fold rotation axis The symmetry configuration is now that of F5 Addition of a mirror plane parallel to the n fold axis of E2 and making angles of 45 0 with existing mirror planes yields F4 See Figure 9 Figure 9 Derivation of F4 from E2 1 Stereogram of the symmetry elements and of the face poles of the most general Form of the Class mm2 represented by the entry E2 2 Addition of a mirror plane parallel to the n fold axis of E2 and making angles of 45 0 with existing mirror planes 3 A second mirror plane is generated 4 The mirror planes generate four more motifs 5 The resulting motif pattern now implies a 4 fold vertical rotation axis The symmetry configuration is that of F4 A mirror plane 90 0 apart from another mirror plane is already present in the symmetry configurarion of E2 Addition of a mirror plane parallel to the n fold axis of E3 can only be such that it contains one of the existing horizontal 2 fold axes In all other cases it either coincides with an already existing mirror plane and then of course nothing is added at all or an angle smaller then 30 0 will be involved leading to a forbidden symmetry Addition of a mirror plane parallel to the n fold axis of E3 and coinciding with one of the horizontal 2 fold axes yields G5 See Figure 10 Figure 10 Derivation of G5 from E3 1 Stereogram of the symmetry elements and of the face poles of the most general Form of the Class 3 2 m represented by the entry E3 2 Addition of a mirror plane parallel to the n fold axis and coinciding with one of the horizontal 2 fold rotation axes 3 The added mirror plane generates 12 new motifs as indicated 4 The resulting motif pattern demands a horizontal mirror plane and three more horizontal 2 fold rotation axes The vertical mirror planes now present generate two more vertical mirror planes The vertical n fold axis which was originally a 3 fold roto inversion axis becomes a 6 fold rotation axis The symmetry configuration is that of G5 Addition of a mirror plane parallel to the n fold axis of E4 can only be such that it coincides with an existing horizontal 2 fold axis otherwise it would either coincide with an already existing mirror plane or involve an angle of 15 0 implied when an angle of 30 0 or 60 0 is taken leading to a forbidden symmetry Addition of a mirror plane parallel to the n fold axis of E4 and coinciding with an existing horizontal 2 fold rotation axis yields G4 See Figure 11 Figure 11 Derivation of G4 from E4 1 Stereogram of the symmetry elements and of the face poles of the most general Form of the Class 4 2m represented by the entry E4 2 Addition of a mirror plane parallel to the n fold axis of E4 and coinciding with an existing horizontal 2 fold rotation axis 3 The added mirror plane generates eight new motifs 4 The resulting motif pattern implies a horizontal mirror plane and two more horizontal 2 fold rotation axes The originally existing mirror planes duplicate the added mirror plane The motif pattern also demands that the vertical n fold axis which originally was a 4 fold roto inversion axis becomes a 4 fold rotation axis The symmetry configuration is that of G4 The fact that the motifs of 4 are drawn rotated with respect to those of 3 is immaterial insofar symmetry is concerned Addition of a mirror plane parallel to the n fold axis of E5 can only be such that it bisects the angle between existing mirror planes otherwise it either coincides with an already existing mirror plane or involves an angle of 15 0 which will be implied when an angle of 45 0 is taken leading to a forbidden symmetry Addition of a mirror plane parallel to the n fold axis of E5 and bisecting the angle between existing mirror planes yields G5 See Figure 12 Figure 12 Derivation of G5 from E5 1 Stereogram of the symmetry elements and of the face poles of the most general Form of the Class 6 m2 represented by the entry E5 2 Addition of a mirror plane parallel to the n fold axis of E5 and bisecting the angle between existing mirror planes 3 The added mirror plane generates 12 new motifs as indicated 4 The resulting motif pattern implies three more horizontal 2 fold axes It also demands that the vertical n fold axis which was originally a 6 fold roto inversion axis becomes a 6 fold rotation axis The mirror planes now present generate two more vertical mirror planes The symmetry configuration is that of G5 Summing up Addition of a mirror plane parallel to the n fold axis of the Classes of column E generated the following Classes F5 F4 G5 and G4 So nothing new is actually generated Next we will investigate the addition of a mirror plane parallel to the n fold axis of the Classes of column F Addition of a mirror plane parallel to the n fold axis of the Class represented by the entry F3 the first Class of column F can only be such that it bisects the angle between existing mirror planes otherwise it will either coincide with an already existing mirror plane or involve an angle of 15 0 implied by taking an angle of 45 0 leading to a forbiddden symmetry Addition of a mirror plane parallel to the n fold axis of F3 and bisecting the angle between existing mirror planes

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