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  • General Ontology XVa
    by their corresponding figures in which the Vector Rosette of Actual Growth with its vectors a b c has been added Eupromorphic crystal Homostaura Isopola hexactinota With vector rosette added Eupromorphic crystal Homostaura Isopola tetractinota With vector rosette added Eupromorphic crystal Homostaura Anisopola triactinota With vector rosette added Eupromorphic crystal Heterogyrostaura tetramera With vector rosette added Eupromorphic crystal Tetraphragma radialia With vector rosette added Eupromorphic crystal Tetraphragma radialia With vector rosette added Eupromorphic crystal Tetraphragma interradialia With vector rosette added Eupromorphic crystal Tetraphragma interradialia With vector rosette added As we can see by listing all eupromorphic crystals from the crystal shapes so far discussed Rectangle Square Parallelogram Rhombus Equilateral Triangle and Regular Hexagon we get only a few promorphs out of the many possible Obviously more intrinsic crystal shapes are required to cover all possible promorphs by eupromorphic crystals Or in other words to let crystal shapes fully express all possible promorphs here of course all crystallographically possible promorphs we need much more shapes For example we need the six sided gyroid rectangle to let the intrinsic shape of the crystal express the Heterogyrostaura hexamera and we need the elongated hexagon to express the Autopola Oxystaura hexaphragma And letting express by the intrinsic crystal shape the promorph of D 1 crystals i e crystals possessing as their only symmetry element a mirror line with two symmetrically arranged antimers Zygopleura eudipleura represents an aberrant case because there are no plane straight edged figures with only two corners The straight edged figure with the least number of corners and having D 1 symmetry is the isosceles triangle So this shape must somehow express the mentioned promorph despite the fact that its Vector Rosette of Actual Growth vectors from the center to the corners does not reflect this promorph as in other eupromorphic crystals Nevertheless we must call D 1 crystals with two antimers and having the isosceles triangle as their intrinsic shape eupromorphic crystals just because this shape neatly expresses an instance of the Zygopleura eudipleura which as promorph are indeed geometrically represented by an isosceles triangle The same applies to D 2 crystals with two antimers and having the rectangle or rhombus as their intrinsic shape The rectangle as well as the rhombus then are considered to neatly express an instance of the Autopola Orthostaura diphragma And among the above considered crystal shapes we indeed had two of them rectangle and rhombus that were considered to be already holomorphic crystals but not eupromorphic ones and which must now nevertheless be considered to be eupromorphic crystals It was a rectangular D 2 crystal with two antimers and a rhomb shaped D 2 crystal also with two antimers Eupromorphic crystal Autopola Orthostaura diphragma Eupromorphic crystal Autopola Orthostaura diphragma The same applies to C 2 crystals with two antimers and having the parallelogram as their intrinsic shape Eupromorphic crystal Heterogyrostaura dimera Relations between internal structure and geometric shape in two dimensional crystals Intrinsic symmetry and intrinsic shape of two dimensional crystals are determined by a number of factors Symmetry of translation free residue intrinsic point symmetry of the given crystal Point symmetry often is not expressed geometrically in crystals meromorphic crystals The intrinsic symmetry of the crystal is then represented by a subgroup of the group that represents the symmetry of the crystal shape Atomic aspects of the growing crystal presented to the nutrient environment and the geometry of the lattice together called atomic aspects s l imply a pattern of growth rate vectors and this pattern determines the final crystal shape Pattern of equal atomic aspects s l implies geometric symmetry but often crystallographic asymmetry results nevertheless depending on the symmetry of the translation free residue See example Figure above Diagram of a C 4 crystal i e a crystal with intrinsic symmetry consisting of a 4 fold rotation axis only and no mirror lines The four growth vectors indicated red are equal and represent directions of slowest growth The result is a square crystal So the intrinsic symmetry group C 4 of the crystal is a subgroup of the symmetry group of its shape D 4 Un equal atomic aspects s l generally imply unequal growth which in turn implies geometric asymmetry In the first series of Shapes considered earlier viz Rectangle Square Parallelogram Rhombus Equilateral Triangle and Regular Hexagon it was shown that there is no direct unequivocal universally valid clue for the assessment of the nature and identity of the promorph of the given crystal and not even for the crystal s intrinsic point symmetry provided by the intrinsic s h a p e of such a crystal The following example illustrates this once more Figure above Diagram of a quadratic two dimensional crystal with motif red inserted to indicate the crystal s D 1 symmetry and with four of its growth rate vectors indicated blue Atomic aspects asp indicated Symmetry of Intrinsic Shape Square D 4 Intrinsic Point Symmetry of Crystal D 1 Promorph Pentamphipleura i e bilateral forms with five antimers and not Tetractinota and also not Homogyrostaura tetramera for that matter Even when of the above depicted crystal the atomic aspects asp1 asp2 asp3 asp4 and asp5 and also asp1 asp8 asp7 asp6 and asp5 are d i f f e r e n t while asp2 asp8 asp3 asp7 asp4 asp6 corresponding growth rates could be the same nevertheless in virtue of a certain configuration of determining factors Different atomic aspects generally will entail different growth rates but not necessarily so unless we define atomic aspects according to the resulting growth rates And indeed in the above depicted case with respect to the growth rate vectors V 1 asp1 V 3 asp3 V 4 asp5 V 2 asp7 And the same applies to the corresponding diagonal growth rate vectors equal among themselves different from and higher than the non diagonal growth rates resulting in a case where the intrinsic shape is a square and as such not reflecting the promorph and even also not the intrinsic point symmetry of the crystal However not all shapes are like that In the next series of crystal shapes to be discussed viz the Amphitect gyroid hexagon Amphitect Gyroid Octagon Amphitect Hexagon Bi isosceles Trapezium etc we suppose the internal structure of the crystals having these shapes to be such not because they possess one of these shapes that the crystal s intrinsic symmetry is the same as that of their intrinsic shape holomorphic crystals What we mean is that each crystal having one of these shapes or whatever other shape for that matter comes in two not mutually excluding varieties viz meromorphic and holomorphic and we will in the following only discuss the holomorphic variety And of each such a holomorphic crystal we will discuss 1 the case that it is supposed to be such that its shape not only precisely reflects its intrinsic symmetry but also precisely reflects its promorph eupromorphic crystals and 2 the case that it is such that its shape although precisely reflecting the intrinsic symmetry does not precisely reflect the promorph but is still a holomorphic crystal These new crystal shapes generally consist of more than four faces where some faces cut off corners of say a rectangle and do not grow out of existence when certain conditions are met Their growth rates balance with those of adjacent faces in such a way that they will persist i e do not vanish after prolonged crystal growth Before we will consider these crystal shapes in detail let us first once again show that once a particular empty building block say a particular rectangle and thus a particular lattice is given not just all crystal faces in our 2 dimensional case in fact crystal lines are possible making any arbitrary angle with a lattice line one or another side of the given building block Only certain specific faces are possible either precisely along the sides of the building block s or oblique but then in such a way that the face staggers by a whole number of building blocks while toughing precisely the corners of the building blocks involved The next four Figures illustrate this Figure above Some possible faces in a given rectangular crystal lattice The next Figure shows again the oblique possible faces but now they are drawn as thin lines in order to enhance accuracy Figure above The same possible oblique faces of the previous Figure now drawn as thin lines The next Figure shows three faces that are impossible Figure above Some im possible faces in the given in previous Figure rectangular crystal lattice They do not climb up by a whole number of building blocks and at the same time just toughing the corners of the building blocks involved See also next Figure Figure above The same im possible faces as in the previous Figure Only whole building blocks can be involved in constituting the crystal To the right of each impossible face whole building blocks are indicated that are closest to it One can see that irregular crystal faces i e not straight ones are implied i e not genuine faces Let us now then discuss the above mentioned crystal shapes For each shape a possible Vector Rosette of Actual Growth with vectors a b c is indicated Figure above Two dimensional crystal built up from a stacking of rectangular building blocks resulting in a amphitect gyroid hexagon six fold gyroid amphitect polygon If the motifs not drawn in the building blocks have C 2 symmetry or are such that their translation free residue has this symmetry then the intrinsic point symmetry of the crystal is according to the group C 2 The growth rates of the six faces are supposed to be such that they balance resulting in the persistence of all of them Figure above Resulting crystal shape amphitect gyroid hexagon from the above stacking of rectangular building blocks And with its Vector Rosette of Actual Growth added Figure above Two dimensional crystal built up from a stacking of rectangular building blocks resulting in an amphitect gyroid octagon eight fold gyroid amphitect polygon If the motifs not drawn in the building blocks have C 2 symmetry or are such that their translation free residue has this symmetry then the intrinsic point symmetry of the crystal is according to the group C 2 The growth rates of the eight faces are supposed to be such that they balance resulting in the persistence of all of them Figure above Resulting crystal shape amphitect gyroid octagon from the above stacking of rectangular building blocks And with its Vector Rosette of Actual Growth added Figure above Two dimensional crystal built up from a stacking of rectangular building blocks resulting in an amphitect hexagon elongated hexagon If the motifs not drawn in the building blocks have D 2 symmetry or are such that their translation free residue has this symmetry then the intrinsic point symmetry of the crystal is according to the group D 2 The growth rates of the six faces are supposed to be such that they balance resulting in the persistence of all of them Figure above Resulting crystal shape amphitect hexagon from the above stacking of rectangular building blocks And with its Vector Rosette of Actual Growth added Figure above Two dimensional crystal built up from a stacking of rectangular building blocks resulting in a amphitect octagon If the motifs not drawn in the building blocks have D 2 symmetry or are such that their translation free residue has this symmetry then the intrinsic point symmetry of the crystal is according to the group D 2 The growth rates of the six faces are supposed to be such that they balance resulting in the persistence of all of them Figure above Resulting crystal shape amphitect octagon from the above stacking of rectangular building blocks And with its Vector Rosette of Actual Growth added Figure above Two dimensional crystal built up from a stacking of rectangular building blocks resulting in an bi isosceles trapezium bilateral hexagon If the motifs not drawn in the building blocks have D 1 symmetry or are such that their translation free residue has this symmetry then the intrinsic point symmetry of the crystal is according to the group D 1 The growth rates of the six faces are supposed to be such that they balance resulting in the persistence of all of them Figure above Resulting crystal shape bi isosceles trapezium from the above stacking of rectangular building blocks And with a possible Vector Rosette of Actual Growth added Figure above Two dimensional crystal built up from a stacking of rectangular building blocks resulting in a bilateral octagon If the motifs not drawn in the building blocks have D 1 symmetry or are such that their translation free residue has this symmetry then the intrinsic point symmetry of the crystal is according to the group D 1 The growth rates of the eight faces are supposed to be such that they balance resulting in the persistence of all of them Figure above Resulting crystal shape bilateral octagon from the above stacking of rectangular building blocks And with a possible Vector Rosette of Actual Growth added Figure above Two dimensional crystal built up from a stacking of rectangular building blocks resulting in an isosceles trapezium bilateral tetragon If the motifs not drawn in the building blocks have D 1 symmetry or are such that their translation free residue has this symmetry then the intrinsic point symmetry of the crystal is according to the group D 1 The growth rates of the four faces are supposed to be such that they balance resulting in the persistence of all of them Figure above Resulting crystal shape isosceles trapezium from the above stacking of rectangular building blocks And with its Vector Rosette of Actual Growth added Figure above Two dimensional crystal built up from a stacking of rectangular building blocks resulting in an isosceles triangle bilateral trigon If the motifs not drawn in the building blocks have D 1 symmetry or are such that their translation free residue has this symmetry then the intrinsic point symmetry of the crystal is according to the group D 1 Figure above Resulting crystal shape isosceles triangle from the above stacking of rectangular building blocks And with a possible Vector Rosette of Actual Growth added Figure above Possible Vector Rosette of Actual Growth of the above depicted crystal It indicates the presence of three antimers The next Figure adds the growth vectors red of the three faces of the above isoscelesly triangular crystal For the case of only two antimers the following Vector Rosette of Actual Growth can be suggested Figure above A possible Vector Rosette of Actual Growth for the above depicted isoscelesly triangular crystal indicating the presence of only two antimers The next Figure adds growth vectors red of the faces growth vectors of which in the present case there are supposed to be only two Figure above The isoscelesly triangular crystal of the above Figures It has only two facial growth vectors The third face is just implied by the other two without actually having grown in a direction perpendicular to it See also next Figure Figure above In the present case the isoscelesly triangular crystal D 1 two antimers of the above Figures has grown two faces while a third is automatically implied without actually having grown from the center of the vector rosette All this is a more or less symbolic way to indicate that the crystal has only two antimers Figure above Two dimensional crystal built up from a stacking of rectangular building blocks resulting in a bilateral pentagon If the motifs not drawn in the building blocks have D 1 symmetry or are such that their translation free residue has this symmetry then the intrinsic point symmetry of the crystal is according to the group D 1 The growth rates of the five faces are supposed to be such that they balance resulting in the persistence of all of them Figure above Resulting crystal shape bilateral pentagon from the above stacking of rectangular building blocks And with a possible Vector Rosette of Actual Growth added Another possible Vector Rosette of Actual Growth of this crystal could be the following Figure above Another possible Vector of Actual Growth for the above crystal shape The crystal has a vertical mirror line as its only symmetry element and thus the center of the vector rosette must lie on this mirror line Where on this mirror line depends on the growth that has actually taken place in several directions from the starting point of the crystal i e from the initial microscopically small crystal seed Figure above Two dimensional crystal built up from a stacking of quadratic building blocks resulting in a regular gyroid ditetragon If the motifs not drawn in the building blocks have C 4 symmetry or are such that their translation free residue has this symmetry then the intrinsic point symmetry of the crystal is according to the group C 4 The growth rates of the eight faces are supposed to be such that they balance resulting in the persistence of all of them This shape could also and equivalently be called regular gyroid octagon because regular and gyroid already determine the nature of the octagon namely that it is composed of two different squares expressed by regular and thus that it is a ditetragon that are superimposed upon each other and involving a rotation such that all mirror lines disappear expressed by gyroid But because the term regular is as such more or less vague and because the term gyroid is probably not so familiar we prefer to call this shape regular gyroid ditetragon Figure above Resulting crystal shape regular gyroid ditetragon from the above stacking of quadratic building blocks And with its Vector Rosette of Actual Growth added Figure above Two dimensional crystal built up from a stacking of quadratic building blocks resulting in a

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  • General Ontology XVb
    This concludes our consideration about the relation between the congruency equality similarity and dissimilarity of the antimers in a natural body a crystal or an organism and the promorph The next Figures are going to illustrate that the above enumerated shapes are indeed p o s s i b l e in two dimensional crystals even when growth of them is prolonged indefinitely i e that the faces involved in these shapes will not grow out of existence The latter depends on certain intrinsic conditions involving the relative growth rates and direction of the faces making up these shapes Figure above A case of a two dimensional crystal with relative growth rate vectors solid red lines radiating from a center of faces blue These relative growth rate vectors are perpendicular to the faces of which they indicate their growth Seven of these are equal indicated by the circle but in different directions One of them is short indicating a lower growth rate as compared to that of the other faces The vectors are symmetrically arranged with respect to a mirror line Figure above The pattern of the eight faces as a result of the relative growth rate vectors as indicated in the present and previous Figure Figure above Resulting initial crystal and its shape in virtue of the above given relative growth rate vectors The shape is that of a bilateral octagon as one of the crystal shapes to be discussed Figure above A two dimensional crystal intrinsically shaped as a bilateral octagon The eight arrows are the vectors of relative growth rate Prolonged growth phases of the initially present crystal blue is indicated by newly appearing layers indicated as yellow green yellow green onto the crystal As one can see especially by inspecting the lines a b c d e f g and h the possible faces that are drawn will persist no matter how far growth will proceed We have constructed the crystal such that the growth rates of seven of its faces viz one upper horizontal face and six oblique faces are equal We have done this by drawing equal growth rate vectors red lines with arrow heads The corresponding faces are then perpendicular to these vectors So with respect to the equal growth rates we can indicate several bi triangles i e areas consisting of two triangles indicated by yellow coloring Each of such a bi triangle in fact consists of two right angled triangles And because of the common hypotenusae and the equality of two other sides the equal growth rate vectors the two right angled triangles making up one bi triangle are congruent And from this it follows that the marked i e marked in the Figure acute angles must be equal This equality of angles then holds for every bi triangle associated with equal growth vectors An it is this equality of the mentioned angles that guarantees that the layers of the crystal do not change width when they pass from one direction to the other And this indeed neatly expresses the equal growth rate of the relevant faces of the crystal Figure above Growth layers of equal width in virtue of the equality of angles marked And because the relevant angles in the bottom section green colored bi triangles of the crystal are not equal the layers do change width indicating a different growth rate of the bottom face which is lower than that of the other faces The next Figure indicates the eight antimers of the above crystal which is supposed to be an eupromorphic crystal Figure above The eight antimers green yellow of the crystal of the previous Figure Here we assume that the boundaries of the antimers go through the corners of the crystal which means that these boundaries coincide with the Vector Rosette of Actual Growth of the crystal In the present case we have eight faces with growth rates such that they balance each other despite some difference between them resulting in the prolonged coexistence of these eight faces In this way they point to the existence of a corresponding number of a n t i m e r s in the present case eight The pattern of relative growth rates of the possible faces of a given crystal determines the crystal s intrinsic shape It is therefore important to ruminate for a while about the phenomenon of crystal growth The growth rate of a crystal face depends on several factors Let s enumerate and discuss them briefly The atomic aspect s str presented to the nutrient environment consists of the chemical nature of the constituents that make up the corresponding crystal face In ionic crystals these constituents could entirely and exclusively consist of certain ions with say a positive electrical charge These will attract negative ions that are present in the nutrient environment and the corresponding face is expected to grow fast as compared with another face where equal numbers of positive and negative ions are exposed which is therefore electrically neutral and lacks a net electrical attraction for cations positive ions and anions negative ions and consequently grows more slowly Generally the fastest growing faces have the highest surface energy The surface of a crystal generally contains unsatisfied or distorted chemical bonds and therefore represents a higher energy configuration of atoms ions than the same elements chemically bonded within the center of the crystal Adding additional atoms ions to the surface satisfies the chemical bonds and allows distorted bonds to realign themselves thereby reducing the energy level of the atoms ions on the old surface The new surface then bonds the next layer of atoms ions to reduce its energy level and so forth as the crystal grows The surfaces with the most unsatisfied or distorted chemical bonds that is with the highest surface energy should grow fastest because the greatest energy reduction is achieved here All of this is entirely consistent with the principles of thermodynamics Natural systems tend to change in directions that minimize their energy level thereby releasing entropy to the environment The net entropy of the system increases despite the decrease of entropy of and within the forming crystal Rapid growth of high energy faces reduces their size when certain geometric conditions are met tapering of adjacent slower growing faces The faces with low surface energy grow more slowly and are therefore larger under the mentioned geometric conditions Minimizing the area of high surface energy faces and maximizing the area of low surface energy faces tend to ensure that the total surface energy is minimized See for all this NESSE W D Introduction to Mineralogy 2000 p 80 81 In addition to the atomic aspect s str also the geometry of the lattice which together with the atomic aspect s str is the atomic aspect s l is a growth rate determining factor Faces that cut across the highest number of lattice nodes tend to grow slowest because more material is needed to complete a new layer See next Figure Also the thickness of a single repeating layer which also is implied by the geometry of the given crystal lattice co determines the growth rate perpendicular to that layer and thus perpendicular to the face that is parallel to that layer This factor is not as such mentioned by NESSE It is however mentioned by him as interplanar spacing p 81 Where the density of lattice nodes encountered by a face is lowest the interplanar spacing is smallest But growth is fastest in the case of lowest density of nodes but the layers are thinnest which fact reduces growth in terms of distance from the center of the crystal again The next Figure illustrates in what way the geometry of the crystal lattice co determines the relative growth rates of possible crystal faces Figure above Besides the chemical composition chemical aspect presented to the environment the growth rate of a face is inversely proportional to the number of lattice nodes that are encountered by that face The figure shows a primitive rectangular point net The argument to come also applies to a 3 D point lattice The intersections of the lines making up the rectangular cells meshes are lattice nodes These nodes represent the locations of chemical units Four possible faces are indicated by a b c and d The arrows indicate their respective directions of growth At the same time there is also a corresponding growth in the opposite directions of the arrows Generally the slowest growing faces become the most prominent faces on the crystal The fastest growing faces quickly grow themselves out of existence This Figure which concerns the growth rates of faces only pays attention to one feature of the atomic aspect The atomic aspect determines the growth rate of a face namely the density of nodes encountered by the crystal face in question The other feature refers to the chemical composition of the crystal The slowest growing face on the basis of the feature mentioned is face b because here the density of nodes encountered by that face is highest Next comes face c and then a The fastest growing face is d So the faces b and c will become the most prominent faces The other faces will eventually grow themselves out of existence Because the faces b and c are perpendicular to each other the faster growing face of these two namely c will not grow itself out of existence It keeps on growing and keeps on growing faster than b so the crystal will finally adopt a rectangular shape with its longest dimension oriented vertically Of course a crystal grows faster when there is a greater supply of material And when this greater supply comes from one particular direction the crystal shape will be distorted resulting in an extrinsic shape of the crystal However in the present discussion we only refer to relative growth rates when comparing the faces of a single individual crystal growing in a uniform environment resulting in the crystal s intrinsic shape All the factors mentioned above are simultaneously present They can amplify the result or more or less neutralize each other In the Figure above we have set the growth rates of all faces but one equal They do not therefore grow themselves out of existence when growth is prolonged Also the lower horizontal face will not disappear because its growth rate is lower than that of the other faces The next Figure on the other hand assumes that the two upper oblique faces grow much faster than all the others And because their adjacent faces are tapering these fast growing faces will finally disappear Figure above Two faces of the above depicted octogonal D 1 crystal grow much faster than all the other faces They will finally disappear The blue area is supposed to represent an initially present crystal seed from which growth proceeds The next Figure is the same but without auxiliary lines Figure above Same as previous Figure A two dimensional crystal of which two possible faces grow much faster than all its other faces They will finally disappear resulting in a hexagonal D 1 crystal bilateral hexagon instead of an octogonal D 1 crystal bilateral octagon Remark One should not take this drawing too seriously because I am not entirely sure about its correctness It only serves to illustrate the disappearance of the two fast growing faces In the following we study some other and generally more simple cases of two dimensional crystals and the retention or disappearance of certain faces Figure above Growth of a two dimensional crystal having as its intrinsic and persistent shape that of the isosceles trapezium Oblique layers green yellow change to horizontal layers green yellow which again change to oblique layers If the lines red connecting the points of these directional changes of all the layers involved bisect the top angles of the trapezium indicated by black dots the layers will not change their width as they go from the oblique direction to the horizontal direction and then to the oblique direction again This means that we have depicted e q u a l g r o w t h r a t e s of the two oblique faces and the top horizontal face And this implies that the top horizontal face will not disappear when growth is prolonged In addition to this it is clear from the drawing that the bottom horizontal face to which we have assigned a lower growth rate will also not disappear when growth is prolonged See also next Figure And this means that the crystal s shape as being an isosceles trapezium will persist throughout growth and will then represent the intrinsic shape of the crystal The next Figure depicts the complete right half of the crystal of the previous Figure to show that also the other faces oblique and bottom horizontal face do not vanish during prolonged growth of the crystal Figure above Same crystal as in previous Figure Complete right half revealed All four faces are persistent during further growth of the crystal The next Figures illustrate the case where the growth rate of the upper horizontal crystal face of our initially trapezoid crystal is significantly higher than that of its adjacent faces We have constructed this by not letting the lines red connecting the points of the directional changes of all the involved layers bisect the upper two angles of the trapezium This unequality of the angles is indicated by marks in the Figure The effect of the higher growth rate of the top face with respect to that of its adjacent faces together with the fact that the latter are tapering is that the top face grows out of existence i e when growth is prolonged it will eventually disappear and this process necessarily results in a crystal having as its intrinsic shape not an isosceles trapezium but an isosceles triangle Figure above Growth of a two dimensional initially trapezoid crystal blue though having as its intrinsic and persistent shape that of the isosceles triangle Oblique layers green yellow change to horizontal layers green yellow which again change to oblique layers green yellow The lines red connecting the points of these directional changes of all the layers involved do in the present case not bisect the top angles of the trapezium indicated by black marks The layers will therefore change their width as they go from the oblique direction to the horizontal direction they get wider and then to the oblique direction again they get thinner again This means that we have depicted u n e q u a l g r o w t h r a t e of the top horizontal face as compared to the two adjacent oblique faces In fact the top face grows faster And together with the fact that the two adjacent faces taper this implies that the top horizontal face will eventually disappear when growth is prolonged In addition to this it is clear from the drawing that the bottom horizontal face to which we have assigned a lower growth rate will not disappear when growth is prolonged And all this means that the crystal s shape becomes an isosceles triangle and this new shape will then persist throughout growth and represents the intrinsic shape of the crystal The next Figure depicts the same crystal but now without auxiliary features Figure above Same as previous Figure Two dimensional triangular crystal partially depicted Auxiliary features omitted In the next Figure we follow the accomplished growth further back towards the crystal s center meaning that we consider its growth stages from a still smaller initial crystal blue in the Figure Figure above Same as previous Figure Two dimensional triangular crystal partially depicted Earlier growth stages also indicated Initial crystal indicated by blue coloring The next Figure is the same as the previous one but now with the auxiliary lines omitted Figure above Same as previous Figure Two dimensional triangular crystal partially depicted Growth stadia indicated auxiliary lines omitted The intrinsic shape of the present two domensional crystal is that of an isosceles triangle There are three persistent faces In an eupromorphic crystal this could indicate the presence of t h r e e antimers But because a triangle is the simplest possible figure bounded by straight lines in two domensional space the so called symplex figure of 2 D space a third side in addition to two is always necessary in two dimensional space to obtain a polygon So the number of antimers based on the number of balanced i e persistent crystal faces can be three or two The next Figure depicts the case of three antimers This number i e either two or three is dependent on the morphology of the translation free residue of our present triangular crystal Figure above Three antimers green yellow purple of the present triangular two dimensional crystal partially depicted The case of two antimers is depicted in the next Figure Figure above Two antimers green yellow of the present triangular two dimensional crystal partially depicted Having investigated the bilateral octagon the isosceles trapezium and the isosceles triangle as to their possible and persistent existence as two dimensional crystals we now consider the six fold amphitect polygon in the same respect Figure above A two dimensional crystal having as its intrinsic shape that of the 6 fold amphitect polygon amphitect hexagon It is assumed that the growth rates of all six faces is the same resulting in a succession of layers yellow green yellow etc with the same width geometrically guaranteed by the equality of the angles as indicated The next Figure omits all auxiliary features Figure above Same as previous Figure Two dimensional crystal with the 6 fold amphitect polygon as its intrinsic shape Growth rate of all six faces is set equal If the crystal is an eupromorphic crystal then the six antimers could be conceived as follows Figure above Suggestive configuration

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  • General Ontology XVc
    bi isosceles trapezium bilateral hexagon Figure above A two dimensional crystal with D 1 intrinsic symmetry and with an intrinsic shape according to a bi isosceles trapezium bilateral hexagon The thick blue lines bisect the angles between adjacent faces exactly into two equal angles as indicated by marks in the Figure guaranteeing that the macroscopic growth layers drawn in the next Figure have equal width with respect to different directions expressing in this way the equal growth rates of all the six faces as we want them to be so This equal growth in turn guarantees that the six faces will persist during prolonged growth of the crystal As in the foregoing cases of crystal shapes and supposed equal growth of faces we consider only holomorphic crystals which for the present crystal implies that its intrinsic symmetry is according to the group D 1 A crystal with this shape could in other cases have a C 1 intrinsic symmetry It is then a meromorphic crystal Figure above Macroscopic growth layers increments green yellow of the two dimensional crystal with D 1 intrinsic symmetry and with an intrinsic shape according to a bi isosceles trapezium bilateral hexagon of the above Figure The Figure shows equal growth of the six faces guaranteeing their persistence during prolonged crystal growth And with the auxiliary lines removed Figure above Macroscopic growth layers increments green yellow of the two dimensional crystal with D 1 intrinsic symmetry and with an intrinsic shape according to a bi isosceles trapezium bilateral hexagon of the above Figures Auxiliary lines removed The next two Figures show two possible antimer configurations of the crystal presently under investigation Figure above A configuration of six antimers in the two dimensional crystal with D 1 intrinsic symmetry and with an intrinsic shape according to a bi isosceles trapezium bilateral hexagon of the above Figures With this configuration the crystal is eupromorphic Figure above A configuration of two antimers in the two dimensional D 1 crystal presently under investigation With this configuration the crystal is non eupromorphic The next Figures depict and discuss a two dimensional crystal with intrinsic shape according to a bilateral pentagon Figure above A two dimensional crystal with D 1 intrinsic symmetry and with an intrinsic shape according to a bilateral pentagon The blue lines bisect the angles between adjacent faces exactly into two equal angles as indicated by marks in the Figure guaranteeing that the macroscopic growth layers drawn in the next Figure have equal width with respect to different directions expressing in this way the equal growth rates of all the five faces as we want them to be so This equal growth in turn guarantees that the five faces will persist during prolonged growth of the crystal As in the foregoing cases of crystal shapes and supposed equal growth of faces we consider only holomorphic crystals which for the present crystal implies that its intrinsic symmetry is according to the group D 1 A crystal with this shape could in other cases have a C 1 intrinsic symmetry It is then a meromorphic crystal Figure above Macroscopic growth layers increments green yellow of the two dimensional crystal with D 1 intrinsic symmetry and with an intrinsic shape according to a bilateral pentagon of the above Figure The Figure shows equal growth of the five faces guaranteeing their persistence during prolonged crystal growth And with the auxiliary lines removed Figure above Macroscopic growth layers increments green yellow of the two dimensional crystal with D 1 intrinsic symmetry and with an intrinsic shape according to a bilateral pentagon of the above Figures Auxiliary lines removed The next three figures give some possible antimer configurations in the crystal presently under investigation Figure above Possible arrangement of five antimers in the D 1 bilateral pentagonal crystal of the above Figures With this configuration of five antimers the crystal is eupromorphic Figure above Another possible arrangement of five antimers in the D 1 bilateral pentagonal crystal of the above Figures Also with this configuration of five antimers the crystal is eupromorphic Figure above The case of two antimers in the D 1 bilateral pentagonal crystal of the above Figures The crystal is now non eupromorphic The next Figures depict and discuss a two dimensional crystal having as its intrinsic shape that of a regular gyroid ditrigon Figure above A two dimensional crystal with C 3 intrinsic symmetry and with an intrinsic shape according to a regular gyroid ditrigon The thick blue lines bisect the angles between adjacent faces exactly into two equal angles as indicated by marks in the Figure guaranteeing that the macroscopic growth layers drawn in the next Figure have equal width with respect to different directions expressing in this way the equal growth rates of all the six faces as we want them to be so This equal growth in turn guarantees that the six faces will persist during prolonged growth of the crystal As in the foregoing cases of crystal shapes and supposed equal growth of faces we consider only holomorphic crystals which for the present crystal implies that its intrinsic symmetry is according to the group C 3 Figure above Macroscopic growth layers increments green yellow of the two dimensional crystal with C 3 intrinsic symmetry and with an intrinsic shape according to a regular gyroid ditrigon of the above Figure The Figure shows equal growth of the six faces guaranteeing their persistence during prolonged crystal growth And with the auxiliary lines removed Figure above Macroscopic growth layers increments green yellow of the two dimensional crystal with C 3 intrinsic symmetry and with an intrinsic shape according to a regular gyroid ditrigon of the above Figures Auxiliary lines removed The next two Figures give two possible positions of the three antimers of the above C 3 regular gyroid ditrigonal crystal Figure above Possible position of the three equal antimers green yellow blue of the C 3 regular gyroid ditrigonal two dimensional crystal presently under investigation The promorph of this

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  • General Ontology XVI
    i g u r a t i o n s of the above discussed amphitect gyroid hexagonal crystal with intrinsic symmetry according to the group C 2 The number and arrangement of a n t i m e r s in a crystal depend on the geometry of its translation free residue This residue is the motif as it remains after having everything telescoped in i e after elimination of all translational elements in the crystal In the above Figures that would mean that the translation free residue is the following And here we clearly have to do with two antimers However in the present document concerning C 2 crystals we will interpret motifs as they are drawn in the ensuing Figures not as motifs just like that but as only r e p r e s e n t i n g the existence of a C 2 motif w i t h a n y p o s s i b l e n u m b e r o f a n t i m e r s The precise relation of antimers and the internal structure of the crystal will be studied in later Parts The next Figure depicts the case of our amphitect gyroid hexagonal C 2 crystal having six antimers Figure above A two dimensional amphitect gyroid hexagonal C 2 crystal with six similar antimers green yellow blue The inserted C 2 motifs have as they are drawn here only two antimers represented by two motif units related to each other by a half turn but are in the present case meant to represent C 2 motifs with six antimers The Vector Rosette of Actual Growth is here supposed to coincide with the boundary lines of the antimers letting the crystal be eupromorphic But also when the vectors of the mentioned Rosette coincide with the median lines of the antimers or correspond in some other way with the antimers the crystal is eupromorphic As was already discussed earlier the lines connecting the crystal s center with its corners together with the colors green yellow and blue are supposed to indicate the a n t i m e r s of the crystal But in fact the only slight and as it seems indirect macroscopic indications of the presence of antimers and thus the presence of a definite promorph in a crystal is the constancy and definiteness of its intrinsic shape There we generally see plane faces 3 D or straight lines 2 D and always encounter definite and constant angles between them Every single crystal grown in a uniform medium is a definite polyhedron 3 D or polygon 2 D and thus allows for the existence of genuine and definite symmetry and promorph Internally however nothing macroscopically morphological is seen that could indicate the presence of antimers See next Figures Figure above The two dimensional amphitect gyroid hexagonal C 2 crystal of the above Figures which in the present case is supposed to consist of six similar antimers green yellow blue The line b is supposed to separate two antimers blue green but many more lines parallel to it as for instance the lines a and c are totally equivalent to it except for the fact that line b ends up in a corner while the others do not So line b only differs from the many other lines parallel to it by an external feature of the crystal See also next Figures Figure above Part of the interior of the crystal of the previous Figure containing parts of the lines a b and c Figure above Same as previous Figure Color difference removed Figure above Same as previous Figure One can clearly see that the lines a b and c are wholly equivalent The only straightforward evidence of antimers and with them of a definite promorph in crystals is the symmetry and morphology of the microscopic translation free residue of the crystal that can be obtained by eliminating all translational features of the crystal If we do so while considering only intrinsic symmetry we in fact go from the crystal s Space Group 3 D or Plane Group 2 D describing its total symmetry including translations to its Point Group describing all the non translational symmetries of the crystal A non translational symmetry is a superimposition operation of some object where at least one point remains where it was before the operation was applied to that object The translation free residue so obtained is a certain configuration of atoms or sometimes just one single atom and as such not only has a certain definite point symmetry representing a crystal class but also has a definite promorph in so far as antimers can be unequivocally distinguished in this residue So a promorph of a crystal turns out to be only present microscopically and if it were not for some macroscopic indications of it as mentioned above we would never have devoted so much attention to promorphs of crystals and not have indicated them macroscopically in our drawings as we have indeed done In organisms on the other hand matters are different with respect to the expression of their promorph While intrinsic symmetry and promorph are in most cases more or less concealed in virtue of functional demands of the organism or because of evolutionary contingencies they are macroscopically expressed as we can clearly see in many Echinoderms Coelenterates Radiolarians and many other animals and also in many subindividuals of plants flowers fruits leaves and pollen grains So roughly we can say that both organisms and crystals possess a definite promorph but in organisms this promorph is expressed macroscopically while in crystals it is expressed microscopically So expressing macroscopically the antimers of a crystal by means of radiating lines 2 D or planes 3 D together with distinguishing colors as we have done in our drawings is partially only a symbolic way to express the promorph of the crystal Only in eupromorphic crystals it is

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  • General Ontology XVII
    residue is the motif as it remains after having everything telescoped in i e after elimination of all translational elements in the crystal In the above Figures that would mean that the translation free residue is the following And here we clearly have to do with two antimers However in the present document concerning C 2 crystals we will interpret motifs as they are drawn in the ensuing Figures not as motifs just like that but as only r e p r e s e n t i n g the existence of a C 2 motif w i t h a n y p o s s i b l e n u m b e r o f a n t i m e r s The precise relation of antimers and the internal structure of the crystal will be studied in later Parts Eight antimers Figure above The case of e i g h t similar antimers green yellow blue in a two dimensional C 2 amphitect gyroid octogonal crystal The inserted C 2 motifs have as they are drawn here only two antimers represented by two motif units related to each other by a half turn but are in the present case meant to represent C 2 motifs with eight antimers The Vector Rosette of Actual Growth is supposed to coincide with the boundary lines of the antimers letting the crystal be eupromorphic But also when the vectors of the mentioned Rosette coincide with the median lines of the antimers or correspond in some other way with the antimers the crystal is eupromorphic The next Figure gives a macroscopic view of this crystal by removing the lattice connection lines and the motifs Figure above Macroscopic view of the above two dimensional C 2 amphitect gyroid octogonal crystal with eight similar antimers green yellow blue The promorph of the eupromorphic crystal of the previous Figures is belonging the Heterogyrostaura octomera This promorph is depicted in the next Figure Figure above Possible representation of the promorph of the above discussed two dimensional C 2 amphitect gyroid octogonal crystal with eight similar antimers It is an eight fold amphitect gyroid polygon and as such the two dimensional analogue of an eight fold amphitect gyroid pyramid which represents the promorph of corresponding three dimensional crystals or other objects To represent this promorph geometrically we could have chosen a somewhat simpler figure in fact we can use the drawing of the crystal itself without motifs and building blocks and thus its macroscopic view as given above We have nevertheless decided for the present figure because it emphasizes very clearly the gyroid nature of the polygon Six antimers Figure above A two dimensional C 2 amphitect gyroid octogonal crystal The case of six similar antimers green yellow blue The crystal is non eupromorphic because its intrinsic shape suggests eight antimers while in fact there are six The next Figure gives a macroscopic view of this crystal by removing the lattice connection lines

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  • General Ontology XVIII
    of the amphitect hexagonal D 2 two dimensional crystal of the previous Figure The promorph of the above amphitect hexagonal eupromorphic crystal is as two dimensional analogue belonging to the Autopola Oxystaura hexaphragma This promorph is depicted in the next Figure Figure above The promorph of the amphitect hexagonal crystal with six antimers It is a 6 fold amphitect polygon amphitect hexagon and as such the two dimensional analogue of the 6 fold amphitect pyramid which represents the promorph of corresponding three dimensional crystals or other objects Radial R and interradial IR directions seen from the center of the polygon are indicated Four antimers radial configuration Figure above A two dimensional amphitect hexagonal crystal with intrinsic D 2 symmetry Its D 2 motifs black have four antimers in radial configuration Microscopic view Figure above The amphitect hexagonal D 2 two dimensional crystal of the previous Figure The case of f o u r similar antimers green yellow Note the correspondence between the morphology of the microscopic motif as translation free residue and the arrangement of the macroscopic antimers of the crystal In this way the promorph and in particular the number of antimers is based on the morphology of the translation free residue of the crystal This residue is explicitly given in the form of a D 2 motif black inside each rectangular building block It is or represents an atomic configuration such that four antimers can be distinguished in it The four antimers of this microscopic atomic configuration are radially arranged perpendicular directional cross axes passing through antimers which will reflect itself in the promorph of the crystal The crystal is non eupromorphic because its intrinsic shape suggests six antimers while in fact there are only four of them The next Figure gives the macroscopic view of the crystal of the previous Figure obtained by removing lattice lines and motifs Figure above Macroscopic view of the amphitect hexagonal D 2 two dimensional crystal with four radially arranged antimers of the previous Figure The promorph of the above amphitect hexagonal non eupromorphic crystal is as two dimensional analogue belonging to the Autopola Orthostaura Tetraphragma radialia This promorph is depicted in the next Figure Figure above The promorph of the amphitect hexagonal crystal with four antimers It is a 4 fold amphitect polygon rhombus and as such the two dimensional analogue of the rhombic pyramid which represents the promorph of corresponding three dimensional crystals or other objects Note the difference in shape between this promorph rhombus amphitect tetragon and that of the crystal amphitect hexagon of which it is the promorph The above promorph Autopola Orthostaura Tetraphragma radialia can be expressed in a more general way than was done in the previous Figure to express the possible non congruity and thus non equality of the antimers only two by two equal In the previous Figure which depicted a representation of this promorph the antimers were drawn equal The more general representation is depicted in the next Figure Figure above Slightly different

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  • General Ontology XIX
    D 2 two dimensional crystal Its D 2 motifs black have four radially arranged antimers Microscopic view Figure above The amphitect octagonal D 2 two dimensional crystal of the previous Figure The case of f o u r similar antimers green yellow Note the correspondence between the morphology of the microscopic motif as translation free residue and the arrangement of the macroscopic antimers of the crystal In this way the promorph and in particular the number of antimers is based on the morphology of the translation free residue of the crystal This residue is explicitly given in the form of a D 2 motif black inside each rectangular building block It is or represents an atomic configuration such that four antimers can be distinguished in it The four antimers of this microscopic atomic configuration are radially arranged perpendicular directional cross axes passing through antimers which will reflect itself in the promorph of the crystal The crystal is non eupromorphic because its intrinsic shape suggests eight antimers while in fact there are only four of them By removing lattice lines and motifs we obtain a macroscopic view of the crystal Figure above Macroscopic view of the amphitect octagonal D 2 two dimensional crystal of the previous Figure with four radially arranged antimers green yellow The promorph of the above amphitect octagonal non eupromorphic crystal is as two dimensional analogue belonging to the Autopola Orthostaura Tetraphragma radialia This promorph is depicted in the next Figure Figure above The promorph of the amphitect octagonal crystal with four antimers It is a 4 fold amphitect polygon rhombus and as such the two dimensional analogue of the rhombic pyramid which represents the promorph of corresponding three dimensional crystals or other objects Note the difference in shape between this promorph rhombus amphitect tetragon and that of the crystal amphitect octagon of which it is the promorph As noted earlier previous document the above promorph Autopola Orthostaura Tetraphragma radialia can be expressed in a more general way than was done in the previous Figure to express the possible non congruity and thus non equality of the antimers only two by two equal In the previous Figure which depicted a representation of this promorph the antimers were drawn equal The more general representation is depicted in the next Figure Figure above Slightly different representation of the promorph Autopola Orthostaura Tetraphragma radialia of the amphitect octagonal crystal with four antimers in radial configuration It also is a 4 fold amphitect polygon rhombus but now expresses the unequality of the antimers as they are in the above amphitect octagonal crystal Four antimers interradial configuration Figure above An amphitect octagonal D 2 two dimensional crystal Its D 2 motifs black have four interradially arranged antimers Microscopic view Figure above The amphitect octagonal D 2 two dimensional crystal of the previous Figure The interradial case of f o u r congruent two by two equal antimers green yellow Note the correspondence between the morphology of the microscopic motif as translation free residue

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  • General Ontology XX
    next Figure is the same as the previous Figure but now with the lattice lines and motifs omitted and in this way presenting a macroscopic view of the crystal Figure above Macroscopic view of the bilateral crystal under investigation The non congruity of the six antimers is clearly visible The division of the crystal into areas representing the six antimers i e the allocation of the the boundaries of these antimers as presented in the above drawings has been partly arbitrary As always the only macroscopic indication of the crystal s antimers is its intrinsic shape And as we know this is only a weak and more or less indirect indication In crystals their promorph is microscopically defined See for this Basic Forms of Crystals in Second Part of Website The next Figure shows an alternative allocation of the macroscopic boundaries of the six antimers of our crystal Figure above Alternative division into antimers of the above discussed bilateral hexagonal D 1 crystal with six antimers The promorph of the above bilateral hexagonal eupromorphic crystal with six antimers is as two dimensional analogue belonging to the Allopola Amphipleura hexamphipleura Allopola hexamphipleura This promorph is depicted in the next Figure Figure above The promorph two images of the bilateral hexagonal crystal with six antimers It is half a 12 fold amphitect polygon and as such the two dimensional analogue of half a 12 fold amphitect pyramid which represents the promorph of corresponding three dimensional crystals or other objects Note the slight difference in shape between this promorph half a 12 fold amphitect polygon and that of the crystal bilateral hexagon of which it is the promorph Radial R and interradial IR directions are indicated Short Intermezzo on Crystals Quasicrystals and Organisms Just above we spoke about some arbitrariness in how a promorph of a crystal is macroscopically expressed This arbitrariness reveals some degree of indeterminateness with respect to promorphs in crystals When the promorph of a crystal is only defined microscopically i e on an atomic or molecular scale aren t we then in fact defining promorphs only of chemical radicals or of other atomic groupings and not of crystals It is indeed the definite intrinsic shape which is a macroscopic feature of crystals that allows us to assign a promorph to them too In crystals two dimensional imaginary crystals as well as three dimensional real crystals we can describe their overall structure by indicating their intrinsic symmetry and their microscopically defined promorph In the case of eupromorphic and by implication holomorphic crystals their intrinsic shape is then implied with respect to a t y p e for instance a bilateral hexagonal shape a quadratic shape a regularly hexagonal shape a regularly gyroid hexagonal shape an amphitect octagonal shape an amphitect gyroid hexagonal shape a rectangular shape etc And even in non eupromorphic crystals and in meromorphic crystals their intrinsic shape is definite and precisely describable In organisms on the other hand we also can describe their overall structure by their intrinsic symmetry and their macroscopically defined promorph But the intrinsic symmetry of organisms is often more or less hidden and distorted by secondary processes and so is their promorph We have to abstract this symmetry and promorph from a more or less asymmetric organic body Moreover the majority of organisms most Vertebrates Arthropods and many others has D 1 intrinsic symmetry with two antimers implying their promorph to belong to the Allopola Zygopleura eudipleura meaning that Symmetry Theory and Promorphology do not come a long way to differentiate between the many organismic species and especially letting their intrinsic shape virtually undetermined Almost all organismic shapes are the result of evolutionary process and adaptation All the different shapes that we can see among eudipleural i e bilateral s str animals having thus D 1 intrinsic symmetry are in fact biological shapes generated by the animal species response to demands from the environment and from their specific ways of life Their generation is not comparable to crystallization like processes Think for instance about the shape of a housefly of a scorpion or a mouse and try to compare these shapes with those of Radiolarians or with the crystal shapes of minerals These biological shapes demand a different approach different not only from the approach to crystal shapes but also from that to say flowers or Radiolarians In the latter there is more freedom for generating and maintaining different symmetries while most actively and fast moving animals must restrict to a D 1 symmetry In addition to that intense differentiation of the body then prohibits the repetition of body parts promorphologically resulting in bodies with only two antimers a right and left side of the body and thus restricting their promorph to the Allopola Zygopleura eudipleura The majority of other promorphs can be found in many lower animals and in flowers But in most of these cases the intrinsic symmetry and promorph is not directly evident It must be dug up as it were While in crystals the intrinsic symmetry and promorph are definitely defined and unequivocally assessible there are also a few cases where the intrinsic symmetry and with it their promorph is as in most organisms not present in a definite way and not unequivocally assessible namely in so called quasi crystals which can be encountered in certain aluminum manganese alloys This is because their internal structure although definitely not totally random as in glasses is not precisely periodic too but only approximately so A microscopic definition of intrinsic symmetry and consequently of promorph is then not very well possible Organisms are wholly aperiodic but their intrinsic symmetry and promorph is defined macroscopically The next four Figures illustrate this vagely defined symmetry for imaginary two dimensional quasicrystals Figure above A rough model of the internal structure in terms of building blocks of a two dimensional quasicrystal After BALL Ph 1994 Figure above In several directions one can roughly discriminate periodic layers in the above rough model of the internal structure of

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