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- General Ontology XXI

is explicitly given in the form of a D 1 motif black inside each rectangular building block It is or represents an atomic configuration such that six antimers can be distinguished in it The crystal is non eupromorphic because its intrinsic shape suggests eight antimers while there are only six present The 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 just depicted bilateral octagonal D 1 two dimensional crystal with six antimers 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 partition into antimers green yellow blue of the bilateral octagonal D 1 two dimensional crystal inder investigation Compare with the earlier version above The promorph of the above bilateral octagonal non eupromorphic crystal is as two dimensional analogue belonging to the Allopola Amphipleura hexamphipleura This promorph is depicted in the next Figure Figure above The promorph two images of the bilateral octagonal 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 Eight antimers Figure above A two dimensional bilateral octagonal crystal with intrinsic D 1 symmetry Its D 1 motifs black have eight antimers Microscopic view Figure above The bilateral octagonal D 1 two dimensional crystal of the previous Figure The case of e i g h t similar antimers green yellow Note the correspondence between the morphology of the microscopic motif as translation free residue of the crystal 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 1 motif black inside each rectangular building block It is or represents an atomic configuration such that eight antimers can be distinguished in it The crystal is eupromorphic because its intrinsic shape suggests eight antimers which are indeed present The 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 eight

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Open archived version from archive - General Ontology XXII

Radial R and interradial IR directions are indicated Six antimers Figure above A two dimensional bilateral tetragonal crystal with intrinsic D 1 symmetry Its D 1 motifs black have six antimers Microscopic view Figure above The bilateral tetragonal D 1 two dimensional crystal of the previous Figure The case of s i x similar antimers green yellow Note the correspondence between the morphology of the microscopic motif as translation free residue of the crystal 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 1 motif black inside each rectangular building block It is or represents an atomic configuration such that six antimers can be distinguished in it The crystal is non eupromorphic because its intrinsic shape suggests four antimers while in fact six are present These six antimers are indicated by means of an alternative set of colors green yellow blue in the next Figure Figure above Same as previous Figure The six antimers are indicated by the colors green yellow and blue The 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 green yellow blue is clearly visible The promorph of the above bilateral tetragonal non 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 Four antimers radial configuration Figure above A bilateral tetragonal two dimensional D 1 crystal Its D 1 motifs black have four radially arranged antimers Microscopic view Figure above The bilateral tetragonal D 1 two dimensional crystal of the previous Figure The case of f o u r similar antimers green yellow in radial configuration Note the correspondence between the morphology of the microscopic motif as translation free residue of the crystal 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 1 motif black inside each rectangular building block It is or represents

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Open archived version from archive - General Ontology XXIII

each rectangular building block It is or represents an atomic configuration such that three similar antimers can be distinguished in it The crystal is eupromorphic because its intrinsic shape suggests three antimers which are indeed present The next Figure gives an alternative distribution of the three macroscopical antimers Figure above The bilateral trigonal D 1 two dimensional crystal under investigation Alternative distribution of the three macroscopic antimers green yellow blue Removing the lattice connection lines and the motifs results in a macroscopic view of the crystal Figure above Macroscopic view of the bilateral trigonal D 1 two dimensional crystal of the previous Figure with three symmetrically arranged antimers green yellow blue The promorph of the bilateral crystal is as two dimensional analogue belonging to the Allopola Amphipleura triamphipleura Allopola triamphipleura and is depicted in the next Figure Figure above The promorph of the bilateral trigonal D 1 crystal with three similar antimers previous Figures It is half a six fold amphitect polygon and as such the two dimensional analogue of half a six fold amphitect pyramid which is the promorph of corresponding three dimensional crystals or other objects Six antimers Figure above A two dimensional bilateral trigonal crystal with intrinsic D 1 symmetry Its D 1 motifs black have six antimers Microscopic view Figure above The bilateral trigonal D 1 two dimensional crystal of the previous Figure The case of s i x similar antimers green yellow blue Note the correspondence between the morphology of the microscopic motif as translation free residue of the crystal 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 1 motif black inside each rectangular building block It is or represents an atomic configuration such that six antimers can be distinguished in it The crystal is non eupromorphic because its intrinsic shape suggests three antimers while in fact six are present The 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 green yellow blue is clearly visible The promorph of the above bilateral trigonal non 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 trigonal 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 difference in shape between this promorph half a 12 fold amphitect polygon and that of the crystal bilateral trigon of

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Open archived version from archive - General Ontology XXIV

Possible a n t i m e r c o n f i g u r a t i o n s for holomorphic bilateral pentagonal two dimensional crystals Two antimers Figure above A bilateral pentagonal D 1 two dimensional crystal The case of t w o congruent symmetric 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 1 motif black inside each rectangular building block It is or represents an atomic configuration such that two antimers related to each other by a reflection can be distinguished in it The crystal is non eupromorphic because its intrinsic shape suggests five antimers while in fact there are only two Removing the lattice connection lines and the motifs results in a macroscopic view of the crystal Figure above Macroscopic view of the bilateral pentagonal D 1 two dimensional crystal of the previous figure with two congruent symmetric antimers green yellow The promorph of the above bilateral non eupromorphic crystal is as two dimensional analogue belonging to the Allopola Zygopleura eudipleura This promorph is depicted in the next Figure Figure above The promorph of the bilateral pentagonal D 1 crystal with two antimers It is an isosceles triangle half a rhombus and as such the two dimensional analogue of the isosceles pyramid half a rhombic pyramid which represents the promorph of corresponding three dimensional crystals or other objects Note the difference in shape between this promorph bilateral trigon and that of the crystal bilateral pentagon of which it is the promorph Radial R and interradial IR directions are indicated Five antimers Figure above A bilateral pentagonal two dimensional D 1 crystal Its D 1 motifs black have five symmetrically arranged antimers Microscopic view Figure above The bilateral pentagonal D 1 two dimensional crystal of the previous Figure The case of f i v e similar antimers green yellow blue Note the correspondence between the morphology of the microscopic motif as translation free residue of the crystal 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 1 motif black inside each rectangular building block It is or represents an atomic configuration such that five similar antimers can be distinguished in it The crystal is eupromorphic because its intrinsic shape suggests five antimers which are indeed present Removing the lattice connection lines and the motifs results in a macroscopic view of the crystal Figure above Macroscopic view of the bilateral pentagonal D 1 two dimensional crystal of the previous Figure with five symmetrically arranged

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Open archived version from archive - General Ontology XXV

and interradial IR directions are indicated If we omit the four buckled lines that only serve to emphasize the gyroid nature ending up in the corners we have this same promorph where its four equal antimers are indicated by the colors green and yellow Figure above The promorph of the regular gyroid octogonal C 4 crystal with four antimers Buckled auxiliary lines omitted The three dimensional counterpart of this promorph is depicted in the next Figure Figure above Regular gyroid 4 fold pyramid slightly oblique top view representing all three dimensional Homogyrostaura tetramera i e all regular 4 fold gyroid promorphs of three dimensional objects A holomorphic two dimensional crystal having the shape of a regular gyroid ditetragon depicted in the Figures above has an intrinsic symmetry according to the group C 4 because this is the symmetry of its intrinsic shape This intrinsic symmetry C 4 allows for only 4n antimers to be present When n 1 as in the above case the four antimers are equal because they can be transformed into each other by a mechanical operation in the present case by a 4 fold rotation axis If n is different from 1 the antimers are not equal but similar As long as the intrinsic symmetry of our crystal is according to the group C 4 there must be one or another four fold repetition of some area around the 4 fold axis These four areas must be equal Each of them can be an antimer as in the above case or represent two or more non congruent and thus similar antimers All this means that a number of antimers different from 4n cannot occur because then there is no four fold repetition which means that the symmetry has changed The latter has become different from C 4 which in turn means that the crystal is no longer holomorphic it has become meromorphic The next Figures give an example of such a meromorphic crystal namely a crystal having as its intrinsic shape the regular gyroid ditetragon as also in the above Figures but where the C 4 motifs have been repaced by C 2 motifs resulting in the transformation of C 4 symmetry into C 2 symmetry making the crystal meromorph In our example the C 2 motif has two antimers while in other cases a C 2 motif can have 4 6 8 etc antimers We insert this example because it is instructive with respect to the difference between holomorphic and meromorphic crystals After it we continue our study of holomorphic crystals Meromorphic Two antimers Figure above A regular gyroid octogonal C 2 two dimensional meromorphic crystal Its C 2 motifs black have two equal antimers Figure above The regular gyroid octogonal C 2 two dimensional meromorphic crystal of the previous Figure The case of t w o equal antimers green yellow related to each other by a 2 fold rotation axis 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 C 2 motif black inside each quadratic building block It is or represents an atomic configuration such that two antimers related to each other by a half turn can be distinguished in it The crystal is meromorphic because its intrinsic symmetry C 2 is different from namely lower than the symmetry of its intrinsic shape The crystal is non eupromorphic because its intrinsic shape suggests eight antimers while in fact there are only two Removing the lattice connection lines and the motifs results in a macroscopic view of the meromorphic crystal Figure above Macroscopic view of the meromorphic regular gyroid octogonal C 2 two dimensional crystal of the previous figure with two equal antimers green yellow The promorph of the above gyroid non eupromorphic crystal belongs as two dimensional analogue to the Heterogyrostaura dimera This promorph is depicted in the next Figure Figure above The promorph of the meromorphic non eupromophic regular gyroid octogonal C 2 crystal with two antimers It is a 2 fold and therefore amphitect gyroid polygon i e an amphitect polygon meant to express the presence of two antimers and as such the two dimensional analogue of the 2 fold gyroid pyramid which represents the promorph of corresponding three dimensional crystals or other objects Note the difference in shape between this promorph 2 fold gyroid polygon and that of the crystal regular gyroid ditetragon of which it is the promorph As has been explained above holomorphic two dimensional crystals having as their intrinsic shape that of a regular gyroid ditetragon and thus implying their intrinsic symmetry to be according to the group C 4 can have only 4n antimers In fact this holds for all C 4 two dimensional crystals Above i e in the beginning of the document we discussed the case for n 1 and now we will consider the case for n 2 i e the case for eight antimers These antimers cannot be equal because that would imply the presence of an 8 fold rotation axis while our crystal has by definition only a 4 fold rotation axis and in addition to this an 8 fold rotation axis cannot occur in crystals whether they be two dimensional or three dimensional Figure above Construction of a two dimensional crystal microscopic view having as its intrinsic shape that of a regular gyroid ditetragon Its C 4 motifs black possess eight similar antimers A pair of two similar antimers is repeated four times around the center of the motif effecting that center to be a 4 fold rotation axis of the motif Figure above Result of the above construction of a two dimensional crystal microscopic view having as its intrinsic shape that of a regular gyroid ditetragon and possessing an

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Open archived version from archive - General Ontology XXVI

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 4 motif black inside each quadratic building block It is or represents an atomic configuration such that eight similar and four by four equal and in themselves mirror symmetric antimers forming pairs that relate to each other by a 4 fold rotation axis can be distinguished in it The crystal is eupromorphic because its intrinsic shape suggests eight similar and in themselves mirror symmetric antimers which are indeed present Removing the lattice connection lines and the motifs in the above Figure results in a macroscopic view of the crystal Figure above Macroscopic view of the regular ditetragonal D 4 two dimensional crystal of the previous figure with eight similar and in themselves mirror symmetric antimers green yellow The promorph of the above eupromorphic crystal is as two dimensional analogue belonging to the Dihomostaura Isopola tetractinota This promorph is depicted in the next Figure Figure above The promorph of the regular ditetragonal D 4 crystal with eight similar antimers It is a regular ditetragon and as such the two dimensional analogue of the ditetragonal pyramid which represents the promorph of corresponding three dimensional crystals or other objects Note the identical shapes of the promorph regular ditetragon and the crystal regular ditetragon Radial R and interradial IR directions are indicated It is important to realize that not only ditetragonal but also q u a d r a t i c D 4 crystals i e D 4 crystals having as their intrinsic s h a p e not a ditetragon but that of a square have this same promorph if they possess eight similar antimers configured such that they display alternation See next Figure and some more explanation further down Figure above A regular tetragonal i e quadratic D 4 two dimensional crystal The case of e i g h t similar and in themselves mirror symmetric 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 4 motif black inside each quadratic building block It is or represents an atomic configuration such that eight similar and four by four equal and in themselves mirror symmetric antimers forming pairs that relate to each other by a 4 fold rotation axis can be distinguished in it The crystal is like the other crystals discussed here holomorphic The symmetry of its shape is the same as the intrinsic symmetry of the crystal both D 4 It is non eupromorphic because its intrinsic shape suggests four equal and in themselves mirror symmetric antimers while eight similar antimers are actually present Removing the lattice connection lines and the motifs in the above Figure results in a macroscopic view of the crystal Figure above Macroscopic view of the regular tetragonal i e quadratic D 4 two dimensional crystal of the previous figure with eight similar and in themselves mirror symmetric antimers green yellow The promorph of this crystal like the ditetragonal crystal discussed earlier belongs to the Dihomostaura Isopola tetractinota and was depicted above Note with respect to the present case the difference in shape obtaining between that of the promorph ditetragon and that of the present crystal square of which it is the promorph Let s return to the d i t e t r a g o n a l D 4 crystal that we were discussing earlier The eight antimers of the translation free residue are each for themselves mirror symmetric with respect to a mirror line passing through the residue s center See Figure above This feature must be expressed in the macroscopically conceived antimers as has been correctly depicted in the Figure further above and in the Figure preceding that Therefore the next Figure depicts as macroscopic view an incorrect delineation of the eight macroscopically conceived antimers Figure above Macroscopic view of the regular ditetragonal D 4 two dimensional crystal again under investigation with eight supposedly similar antimers green yellow The delineation of the eight antimers is incorrect because as depicted they are not each for themselves mirror symmetric with respect to a mirror line passing through the crystal s center The length of the line segment AB is to that of BC as 4 square root 2 to 5 implying that DB is not equal to BE Moreover the antimers as depicted here green yellow are congruent not just similar which they shouldn t be according to the morphology of the translation free residue motif See Figure above In fact the antimers as depicted here are not really antimers Only when we consider pairs consisting of two antimers the latter related to each other by a reflection we have genuine antimers And then we get the regular ditetragonal crystal with four antimers that was discussed earlier And the promorph of that crystal is consequently belonging to the Homostaura Isopola tetractinota which promorph was depicted above Above we have established that the promorph of our two dimensional ditetragonal D 4 crystal with eight similar antimers See Figure above belongs to the Dihomostaura Isopola tetractinota Let us explain this promorph and its name Homostaura are regular polygons The cross axes stauri of such a polygon are within each type equal therefore these polygons are named homostaura When referring to the promorph of three dimensional objects they are regular pyramids The base of such a pyramid is a regular polygon The geometric figure as promorph representing the Dihomostaura Isopola tetractinota is a d i t

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Open archived version from archive - General Ontology XXVII

small crystal is built up by a huge number of such building blocks And because of this huge number the crystal s outline in our case the regular gyroid ditrigonal outline is much more clearly expressed So to accommodate for this we will stick to the drawings made earlier in this document where also half rhombi such that these halves are equilateral triangles are admitted to participate in the crystal s formation resulting in its shape being more clearly expressed Moreover as one can see the three by three equivalence of the corresponding faces of the crystal is particularly evident in these earlier Figures The next Figure shows a macroscopic view of our regular gyroid ditrigonal two dimensional crystal Figure above Macroscopic view of the two dimensional C 3 crystal with intrinsic shape according a regular gyroid ditrigon obtained by the removal of all lattice connection lines i e including the lines that divide the rhombi into equilateral triangles lines which are also point lattice connection lines The next Figure shows the pattern of symmetry elements of our crystal Figure above Pattern of symmetry elements of the two dimensional C 3 crystal with intrinsic shape according a regular gyroid ditrigon It consists of one 3 fold rotation axis only passing through its center See also next Figure Figure above Clarification of the 3 fold rotation axis of the crystal under investigation Two crystallographic Forms are needed to construct our two dimensional C 3 regular gyroid ditrigonal crystal Figure above Two crystallographic Forms are needed to construct the faces and with them the outline of our C 3 regular gyroid ditrigonal crystal An initially given face red implies two more faces in virtue of the 3 fold rotation axis The result is a closed Form consisting of three faces equilateral triangle A second initially given face dark blue not parallel to one of the faces of the first Form also implies two other faces in virtue of that same 3 fold rotation axis also resulting in a closed Form equilateral triangle The two Forms combine to give our regular gyroid ditrigonal crystal Our crystal is supposed to have an intrinsic shape according to a regular gyroid ditrigon The symmetry of this shape is according to the group C 3 And because we only consider holomorphic crystals the intrinsic symmetry of our crystal must also be according to that group This must now be expressed by appropriate motifs These motifs as we will draw them represent the translation free residue of the crystal The symmetry of this residue is in all cases of holomorphic regular gyroid ditrigonal crystals according to the group C 3 while further details of its morphology determine the number of its antimers And this is then at the same time the number of antimers of the crystal The intrinsic C 3 symmetry allows for either three equal antimers to be present or a multiple of three antimers which are then similar three by three equal Possible a n

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Open archived version from archive - General Ontology XXVIII

configurations of holomorphic two domensional crystals with an intrinsic shape of a regular gyroid dihexagon A two dimensional crystal with intrinsic C 6 symmetry can have either six equal antimers or a multiple of six antimers in which case they are similar This number of antimers depends on the morphology of the translation free residue which we can call the motif Six antimers Figure above Construction of a C 6 two dimensional crystal with an intrinsic shape according to a regular gyroid dihexagon Two C 6 motifs black with six equal antimers are inserted in each rhombic building block of the hexagonal net Figure above Same as previous Figure The remaining half rhombic building blocks which are equilateral triangles that are not yet filled in are highlighted yellow Figure above Completion of the construction of the C 6 two dimensional crystal with an intrinsic shape according to a regular gyroid dihexagon The left over equilateral triangles are now also provided with a C 6 motif black So now every equilateral triangle of the hexagonal net is provided with a C 6 motif The intrinsic symmetry of this crystal is now explicitly C 6 and its total symmetry is according to the plane group P6 And restoring the lines dark blue that indicate the faces of the crystal Figure above The C 6 two dimensional crystal with an intrinsic shape according to a regular gyroid dihexagon Its C 6 motifs have six equal antimers Figure above Same as previous Figure One can see that the motifs and their distribution are completely compatible with a 6 fold rotation axis of the crystal and that they forbid the presence of any mirror line So indeed this crystal has C 6 symmetry Figure above The regular gyroid dihexagonal C 6 two dimensional crystal of the previous Figures The case of s i x equal and in themselves asymmetric antimers green yellow blue 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 C 6 motif black inside each equilateral triangle of the hexagonal net It is or represents an atomic configuration such that six equal and in themselves asymmetric antimers relating to each other by a 6 fold rotation axis can be distinguished in it The crystal is non eupromorphic because its intrinsic shape suggests 12 similar and in themselves asymmetric antimers while in fact there are six equal antimers present Removing the lattice connection lines and the motifs in the above Figure results in a macroscopic view of the crystal Figure above Macroscopic view of the regular gyroid dihexagonal C 6 two dimensional crystal of the previous Figure The promorph of our regular gyroid dihexagonal crystal with intrinsic C 6 symmetry belongs as a two dimensional analogue to the Homogyrostaura hexamera and is depicted in the next Figure Figure above The promorph of the regular gyroid dihexagonal C 6 crystal with six equal but asymmetric antimers It is a regular gyroid six fold polygon with six antimers and as such the two dimensional analogue of the regular gyroid six fold pyramid which represents the promorph of corresponding three dimensional crystals or other objects Note the similar shapes of the promorph regular gyroid six fold polygon and the crystal regular gyroid dihexagon In fact the above drawing of the crystal could be used to represent this promorph In the present drawing however the gyroid nature is more conspicuously displayed Radial R and interradial IR directions are indicated Twelve antimers The next case to be considered is the presence of 12 similar antimers that are six by six equal And this is based on the morphology of the C 6 translation free residue motif which itself is supposed to consist of 12 similar antimers six by six equal Such a motif could look like this Figure above A possible motif as translation free residue of a C 6 two dimensional crystal The motif has 12 similar antimers six by six equal and its symmetry is according to the group C 6 In the next Figure we suppose such motifs C 6 symmetry 12 antimers six by six equal to be present in the crystal but depict the latter directly as it is macroscopically seen Because the motifs cannot properly be drawn in the triangles of the hexagonal net we have used above Figure above Macroscopic view of a regular gyroid dihexagonal C 6 two dimensional crystal The case of t w e l v e similar and six by six equal antimers green yellow There is a correspondence between the morphology of the microscopic motif as translation free residue not shown 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 The crystal is eupromorphic because its intrinsic shape suggests 12 similar and six by six equal antimers which are indeed present The promorph of our regular gyroid dihexagonal crystal with intrinsic C 6 symmetry belongs as a two dimensional analogue to the Homogyrostaura dihexamera and is depicted in the next Figure Figure above The promorph of the regular gyroid dihexagonal C 6 crystal with 12 similar six by six equal antimers It is a regular gyroid six fold polygon with twelve antimers and as such the two dimensional analogue of the regular gyroid six fold pyramid with 12 antimers which represents the promorph of corresponding three dimensional crystals or other objects Note the similar shapes of the promorph regular gyroid six fold polygon and the crystal regular gyroid dihexagon In fact the above drawing of the crystal could be used to represent this promorph In the present drawing however the

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