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  • General Ontology VIII
    is equivalent to c we insert a motif black itself having D 2 symmetry as such representing the crystal s point symmetry but clearly having only two antimers See next Figure Figure above Same as previous Figure but now with a D 2 motif inserted representing the crystal s point symmetry and indicating the two antimers The next Figure gives the promorph and its name of the above square D 2 crystal with two antimers D 2 symmetry and six antimers A two dimensional intrinsically square crystal with intrinsic D 2 symmetry can in some cases possess six antimers See next Figure Figure above Two dimensional square crystal with intrinsic D 2 symmetry and six antimers green yellow indicated by the numerals 1 2 3 4 5 and 6 The next Figure is the same as the previous one but with the symmetry elements one 2 fold rotation axis small solid dark blue ellipse and two mirror lines red inserted In the next Figure the Vector Rosette of Actual Growth with its four vectors a b c d is indicated dark blue lines Figure above Two dimensional square crystal with intrinsic D 2 symmetry and six antimers The Vector Rosette of Actual growth is indicated blue lines The next Figure finally gives the promorph and its name of our crystal Figure above Promorph of the two dimensional square crystal considered above D 2 six antimers It is a six fold amphitect polygon and as such the two dimensional analogue of a six fold amphitect pyramid In the foregoing we considered d i a g o n a l mirror lines to be present in a square crystal with intrinsic point symmetry D 2 They represented the two only mirror lines of any D 2 object But of course the two mirror lines of D 2 symmetry could alternatively be represented by the two non diagonal mirror lines of the square crystal Where then the diagonals would not be mirror lines And because a minimum of crystallographic Forms namely one are needed to construct a D 2 crystal if its two mirror lines coincide with the diagonals of the square we started considering this case And now it is time to consider the other case as well In a much earlier case Part V and VI where we considered rectangular crystals we did not have this choice because a rectangle cannot have diagonal mirror lines For the square we do and so we will now turn to the case of intrinsically square crystals also having intrinsic D 2 symmetry but having its two mirror lines not coinciding with the square s diagonals but with the two lines connecting the centers of opposite sides The next Figure indicates the pattern of symmetry elements of such a crystal Figure above Symmetry elements of a two dimensional intrinsically square crystal with intrinsic D 2 point symmetry One 2 fold rotation axis and two mirror lines m intersecting at the piercing point of the rotation axis The two mirror lines do not coincide with the diagonals of the square Two crystallographic Forms are needed to mentally construct such a crystal An initial crystal face parallel to one of the mirror lines and perpendicular to the other implies a second face in virtue of either the 2 fold axis or the parallel mirror line No further faces can be developed so these two faces two parallel faces together constitute a crystallographic Form And because it is an open Form it cannot represent a crystal In order to do so it must combine with another Form resulting in a closed figure So a second initial face is needed one perpendicular to the first one And this second initial face implies another face parallel to it in virtue of the 2 fold rotation axis or the mirror line parallel to it and resulting in a second Form Combining these two Forms yields four faces forming a square and representing in this way the whole two dimensional crystal See next Figure Figure above The two Forms blue red of a two dimensional square crystal with intrinsic D 2 symmetry and non diagonal mirror lines With such a crystal i e with a square crystal having D 2 symmetry we consider again the case of four antimers Figure above The four antimers green yellow of a square D 2 crystal with non diagonal mirror lines Symmetry elements indicated All figures with D 2 symmetry have two and no more than two mirror lines perpendicular to each other And because they are perpendicular they must correspond with the directional axes of the corresponding promorphs of such D 2 figures In the crystal of the present Figure we can see that these directional axes are radial because they coincide with the median lines of the antimers geen yellow and so the corresponding promorph must also be radial A second possible configuration of the four antimers is the interradial configuration Figure above The four antimers green yellow of a square D 2 crystal with non diagonal mirror lines Symmetry elements indicated The antimers correspond to the corner areas of the square and their configuration is interradial because the mirror lines necessarily representing to the promorphological directional axes are interradial They run between antimers Figure above Same as previous Figure The four antimers green yellow of a square D 2 crystal with non diagonal mirror lines Symmetry elements not drawn Radial R and interradial IR directions indicated Configuration of the four antimers interradial The next Figure indicates the Vector Rosette of Actual Growth of our square crystal Figure above Same as previous Figure The four antimers green yellow of a square D 2 crystal with non diagonal mirror lines Vector Rosette of Actual Growth with its four vectors a b c d indicated dark blue lines The above three Figures are however only partially adequate because they suggest that each antimer is in itself mirror symmetric and that they relate to each other by a 4 fold rotation axis All this cannot be because it is not compatible with the pattern of symmetry elements in these three Figures In order to express the true nature of the four antimers we insert a motif Figure above Same as previous Figure Motif black inserted in order to highlight the D 2 intrinsic symmetry of the crystal i e the intrinsic point symmetry of the crystal is not that of its quadratic shape which is D 4 Compare with the quadratic crystal with intrinsic D 4 symmetry considered above Compare also with the quadratic D 2 crystal with its two mirror lines coinciding with the diagonals of the square also considered above The configuration of the four antimers is interradial The next Figure shows the second configuration mentioned earlier viz the radial configuration of the four antimers elucidated by an inserted motif black Figure above A two dimensional intrinsically square crystal with intrinsic point symmetry according to the group D 2 and having four antimers green yellow Radial configuration of antimers indicated by an inserted motif black which also expresses the D 2 intrinsic symmetry of the crystal To see in what way the motif shown by a comparable motif also having D 2 symmetry is actually only microscopically present and then periodically repeated in the quadratic crystal see the relevant Figure in Part VII Sometimes such a motif is smeared out as it were because of glide reflections of which only the reflection component is macroscopically visible because the translations are very small When in addition to the simple translations also the translational component of such glide reflections is eliminated turning them into ordinary reflections which is what we do when looking to the crystal macroscopically we are in the present case left with a translation free residue with D 2 symmetry See for this latter case the relevant Figure in Part VII The promorph of our quadratic D 2 crystal with non diagonal mirror lines and four antimers is the same viz Autopola Orthostaura Tetraphragma radialia or interradialia as that of the quadratic D 2 crystal with diagonal mirror lines i e with its only two mirror lines being diagonal and four antimers as considered above D 2 symmetry with two antimers A quadratic D 2 crystal with non diagonal mirror lines and with only two antimers is depicted in the follwing two Figures Figure above The case of having only two antimers green yellow of a quadratic D 2 crystal with its two mirror lines non diagonally positioned And with the Vector Rosette of Actual Growth added Figure above The case of having only two antimers green yellow of a quadratic D 2 crystal with its two mirror lines non diagonally positioned Vector Rosette of Actual Growth added All vectors a b c d are equivalent The promorph of an intrinsically quadratic two dimensional crystal with intrinsic D 2 symmetry and two antimers and having its mirror lines non diagonally positioned i e the crystal just discussed is the same as that of precisely such a crystal but having its mirror lines diagonally positioned This promorph was depicted above D 2 symmetry and six antimers We now consider the case of a quadratic D 2 crystal with its two mirror lines still being non diagonal but having six antimers Figure above The six antimers green yellow of a quadratic D 2 crystal with its two mirror lines non diagonally positioned The antimers are also indicated by numerals The next Figure adds the Vector Rosette of Actual Growth Figure above The six antimers green yellow of a quadratic D 2 crystal with its two mirror lines non diagonally positioned The antimers are also indicated by numerals The Vector Rosette of Actual Growth with its four vectors a b c d is indicated blue lines All four vectors are equivalent Compare with the quadratic D 2 crystal also with six antimers but with its mirror lines in a diagonal position considered above The promorph of such a crystal i e of an intrinsically quadratic D 2 crystal with its mirror lines non diagonally positioned and with six antimers is the same as that of an intrinsically quadratic crystal also with intrinsic D 2 symmetry and also with six antimers but with its mirror line diagonally positioned This promorph was depicted above D 1 symmetry We now consider crystals again possessing the square as their intrinsic shape but now with intrinsic symmetry according to the point group D 1 Again we have two possibilities The position of the mirror line could either be diagonal or non diagonal with respect to the quadratic crystal We begin with the first case mirror line diagonal The next Figure depicts the symmetry elements of such a two dimensional D 1 crystal Figure above The only symmetry element with respect to point symmetry a mirror line m of a square crystal with intrinsic D 1 point symmetry To generate the whole crystal conceptually two crystallographic Forms are needed An initial crystal face at 45 0 to the mirror line implies a second face in virtue of this mirror line The so obtained face pair is a Form an open Form A face perpendicular perpendicular to the first one also implies a second face in virtue of the mirror line The so obtained face pair is another Form also an open Form The two Forms together constitute a square and thus represent the whole crystal See next Figure A crystal with intrinsic D 1 symmetry in principle admits of any number 1 of antimers as long as they are symmetrically arranged with respect to a mirror line We give some examples D 1 symmetry and two antimers Figure above Two dimensional and intrinsically square crystal with point symmetry D 1 and two antimers green yellow A motif black is added to indicate that there is only one mirror line present The next Figure is the same as the previous one but now provided with the crystal s Vector Rosette of Actual Growth with its four vectors a b c d The inserted motif black demonstrates the non equivalence of a and c of b and d of c and d of b and c and finally also of a and b and it demonstrates the equivalence of a and d Further it expresses the intrinsic D 1 point symmetry of the crystal despite its intrinsic square shape To see in what way the motif shown by a comparable motif also having D 1 symmetry is actually only microscopically present and then periodically repeated in the square crystal see the relevant Figure in Part VII Sometimes such a motif is smeared out as it were because of glide reflections of which only the reflection component is macroscopically visible because the translations are very small When in addition to the simple translations also the translational component of such glide reflections is eliminated turning them into ordinary reflections which is what we do when looking to the crystal macroscopically we are in the present case left with a translation free residue with D 1 symmetry See for this latter case the relevant Figure in Part VII The next Figure gives and names the promorph of our intrinsically quadratic D 1 two dimensional crystal with its mirror line as an element of its point group coinciding with a diagonal of the square and with two antimers Promorph of a quadratic D 1 crystal with two antimers D 1 symmetry and four antimers We will now consider the case of a quadratic D 1 two dimensional crystal still with its mirror line diagonal but having four antimers Figure above Two dimensional quadratic D 1 crystal with four antimers The inserted motif black clearly shows which lines are going to represent the directional axes of the promorph One of them is the mirror line coincident with the NE diagonal see Figure above the other is perpendicular to it and is not a mirror line And we see that these lines run between antimers so the configuration of the antimers is interradial Figure above Same as previous Figure Vector Rosette of Actual growth with its four vectors a b c d added The four antimers allow for two types of configuration One type we have just depicted viz the interradial configuration It promorphologically corresponds to the Allopola Zygopleura Tetrapleura interradialia while a second type promorphologically corresponds to the Allopola Zygopleura Tetrapleura radialia See for this second type of configuration of the four antimers the next two Figures Figure above Two dimensional quadratic D 1 crystal with four antimers Alternative configuration viz radial of the four antimers Also here the inserted motif expresses the intrinsic symmetry and shows where the lines that are going to represent the directional axes of the promorph are to be found NE diagonal mirror line and the line perpendicular to it and they are radial Figure above Same as previous Figure Vector Rosette of Actual growth with its four vectors a b c d added As has been noted earlier the motif black in the above figures does not as such occur in a crystal It here only serves to indicate the intrinsic D 1 point symmetry of the crystal as caused by comparable microscopic motifs which as translation free residues of the relevant plane group represent the crystal s point symmetry The next Figures give and name the promorphs respectively possessed by the two types radial interradial of quadratic D 1 crystal with diagonal mirror line One can see that whether the mirror line is diagonal or non diagonal does not make a difference in promorph Figure above Promorph of the quadratic D 1 crystal with four antimers It is an isosceles trapezium and as such a two dimensional analogue of the trapezoid pyramid i e it is its base The four antimers green yellow and the radial and interradial directions are indicated The configuration of the antimers is interradial Figure above Promorph of the quadratic D 1 crystal with four antimers Alternative configuration viz radial of the four antimers This promorph is a bi isosceles triangle and as such the two dimensional analogue of the bi isosceles pyramid i e it is its base The four antimers green yellow and the radial and interradial directions are indicated As we did already in Part VI concerning rectangular crystals the next three Figures elaborate a little more on the directional axes of D 1 promorphs Allopola with four antimers Allopola Zygopleura eutetrapleura in fact their two dimensional analogues The two possible promorphs were given just above Figure above Cross axes two pairs blue red of the isosceles trapezium the basic form of the Eutetrapleura interradialia The perpendicular cross axes red are the directional axes while the other axes blue are just ordinary cross axes The latter are bent but this is neither typical nor necessary as the next Figure shows Figure above Cross axes of the isosceles trapezium the basic form of the Eutetrapleura interradialia Two sets of straight cross axes red blue The perpendicular ones red are the directional axes of the isosceles trapezium The horizontal directional axis does not connect the centers of opposite sides but this is to be expected because of the absence in the promorph of a horizontal mirror line Also the basic form of the Allpola Zygopleura Eutetrapleura r a d i a l i a can be depicted by a polygon viz a bi isosceles triangle such that it has straight cross axes Figure above Cross axes of a bi isosceles triangle the basic form of the Eutetrapleura radialia All cross axes are straight The perpendicular ones red are the directional axes of the bi isosceles triangle D 1 symmetry and six antimers A two dimensional quadratic crystal i e a crystal of which the intrinsic shape is a square with intrinsic D 1 point symmetry and having its mirror line in the diagonal position can depending on the

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  • General Ontology IX
    of a parallelogram which as such has C 2 symmetry wheras the crystal itself has a point symmetry according to the group C 1 because its parallelogrammatic meshes are provided with an asymmetric motif destroying the crystal s symmetry i e lowering its symmetry from C 2 to C 1 In the next document we will investigate the promorphs of these parallelogrammatic two dimensional crystals while in the present document we will consider intrinsically rhombus shaped crystals as a prelude to the study of their promorphs Now we shall deal with the Rhombus So let us consider a fully developed two dimensional crystal having a rhombus as its intrinsic shape a rhombus is a parallelogram with all sides of equal length Any rhombus shaped crystal can be conceived as being built up by a periodic stacking of microscopic rhombus shaped units as the next Figure illustrates The true point symmetry of such an intrinsically rhombic crystal is either according to the Dihedral Group D 2 crystallographically denoted by 2mm or to the Dihedral Group D 1 crystallographically denoted by m or to the Cyclic Group C 2 crystallographically denoted by 2 or finally to the Asymmetric Group C 1 all depending on the crystal s internal structure That a rhombic crystal i e a two dimensional crystal having as its intrinsic shape the rhombus can possess either of these symmetries can be explained succinctly as follows and will be further evident in the sequel A Rhombus as such has the following symmetries i e it will be superposed upon itself by the following transformations which are then for that reason symmetry transformations 0 0 or 360 0 rotation about any axis 180 0 rotation half turn about the axis through its center Reflection in a line connecting two opposite corners Reflection in a line connecting two opposite corners and perpendicular to the one just mentioned c The rhombus shaped crystal originates by development of crystal faces These faces will appear in accordance with the available symmmetry i e according to the set of symmetry transformations available in each case And this set is dependent on the symmetry of the motif An initially given face will imply copies of itself according to this set of available symmetry transformations The configuration of faces so obtained then constitutes a Form in the crystallographic sense So it is clear that When no symmetry is available a of the above list four Forms are needed to produce a rhombus When only a half turn is available b of the above list two Forms are needed to produce a rhombus When only one reflection is available c or d of the above list two Forms are needed to produce a rhombus When two reflections are available c and d of the above list only one Form is needed to produce a rhombus The next table shows which plane groups can support rhombus shaped two dimensional crystals It further indicates the corresponding point groups with the crystallographic

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  • General Ontology X
    faces Consequently any such initially given crystal face is a Form So to construct a parallelogrammatic two dimensional crystal four Forms are needed blue red green purple The next Figure two images gives the possible promorphs and their names of this parallelogrammatic crystal The promorph is either left image half an isosceles triangle and equivalently a quarter of a rhombus and is as such the two dimensional analogue of a quarter of a rhombic pyramid or right image an irregular triangle indicating two unequal antimers yellow green and is as such the two dimensional analogue of an irregular pyramid or equivalently a 1 fold pyramid This concludes our investigation concerning the relation of intrinsic parallelogrammatic 2 dimensional crystal shape to intrinsic point symmetry and to promorph Next we consider the Rhombus The Vector Rosette of Actual Growth of a rhombus shaped crystal i e of any two dimensional crystal having as its intrinsic shape when fully developed the Rhombus which itself has D 2 symmetry and possessing whatever possible intrinsic point symmetry is as follows blue lines This Vector Rosette viz a vector rosette of a rhombic two dimensional crystal in fact consists of four vectors a b c d each originating in the center of the crystal and ending up at a corner As such it encodes the crystal s shape The following plane groups support a rhombus as the intrinsic shape of a two dimensional crystal P1 P2 Pm Cm Pg P2mm C2mm P2mg and P2gg See also the table in the previous document indicating these plane groups the implied point groups with the crystallographic notation between brackets and the number of crystallographic Forms needed to construct the rhombus shaped crystal in each case of point symmetry In the table we can see that a rhomb shaped crystal can have either an intrinsic symmetry equal to that of a rhombus or a lower symmetry down to no symmetry at all We will now consider several possible cases of the given example of a rhombus shaped two dimensional crystal as depicted just above D 2 symmetry This is the case of a rhombus shaped two dimensional crystal having as its intrinsic point symmetry the full symmetry of a Rhombus i e D 2 symmetry The pattern of symmetry elements of rhomb shaped D 2 crystals is depicted in the next Figure Figure above Symmetry elements of an intrinsically rhomb shape two dimensional crystal with intrinsic D 2 symmetry Two mirror lines m 1 m 2 perpendicular to each other and connecting opposite corners One two fold rotation axis at the intersection point of the mirror lines If we take an initially given crystal face to be such that it is not perpendicular to one of the mirror lines only one crystallographic Form is needed to construct the whole rhomb shaped D 2 crystal See next Figure Figure above One Form blue is needed to conceptually construct a rhomb shaped D 2 crystal An initially given crystal face not parallel to one of the mirror lines implies three more faces together making up a closed Form having the shape of a rhombus The initial face is reflected in one of the mirror lines resulting in two faces This set of two faces is then reflected in the other mirror line resulting in a rhombus Crystals having intrinsic D 2 symmetry can have several different promorphs depending on the geometry of the chemical motif Let s consider the several possibilities D 2 symmetry and four antimers The geometry of the chemical motif of a 2 dimensional intrinsically rhomb shaped D 2 crystal could be such that four antimers can be distinguished See next Figure Figure above A two dimensional rhomb shaped D 2 crystal with four antimers green yellow Radial R and interradial IR directions indicated And with the Vector Rosette of Actual Growth added Figure above A two dimensional rhomb shaped D 2 crystal with four antimers green yellow Vector Rosette of Actual Growth added blue lines The vectors of the rosette are two by two equivalent a with d and b with c Depending on the motif translation free residue the four antimers can be configured differently as the next Figures illustrate Figure above Alternative configuration of four antimers green yellow in a two dimensional rhomb shaped D 2 crystal And with the Vector Rosette of Actual Growth added Figure above The two dimensional rhomb shaped D 2 crystal with four antimers green yellow of the previous Figure Vector Rosette of Actual Growth added blue lines The two types of configuration of the four antimers as shown above imply two slightly different promorphs The first configuration corresponds to the following promorph Figure above Promorph of a two dimensional rhomb shaped D 2 crystal with four antimers arranged according to the first configuration It is a rhombus and as such the two dimensional analogue of the rhombic pyramid The directional axes see next Figure i e those cross axes that are perpendicular to each other run through the antimers and are therefore radial directional axes The next Figure indicates the directional axes strong red lines of the just given promorph The second configuration corresponds to the following promorph Figure above Promorph of a two dimensional rhomb shaped D 2 crystal with four antimers arranged according to the second configuration It is a rectangle and as such the two dimensional analogue of the rectangular pyramid The directional axes red lines i e those cross axes that are perpendicular to each other run between the antimers and are therefore interradial directional axes D 2 symmetry with two antimers The motif of a two dimensional intrinsically rhomb shaped D 2 crystal could be such that only two antimers can be distinguished See next Figures Figure above A two dimensional rhomb shaped crystal with intrinsic D 2 symmetry and two antimers green yellow And with the Vector Rosette of Actual growth added Figure above A two dimensional rhomb shaped crystal with intrinsic D 2 symmetry and two antimers green yellow Vector Rosette of Actual growth added The vectors are two by two equivalent a with d and b with c The next Figure gives the promorph of our rhomb shaped D 2 crystal with two antimers It is a rhombus and as such the two dimensional analogue of a rhombic pyramid D 2 symmetry and six antimers The geometry of the motif of a rhomb shape two dimensional crystal with intrinsic D 2 symmetry can be such that six antimers can be distinguished See next Figures Figure above A two dimensional rhomb shaped crystal with intrinsic D 2 symmetry and six antimers green yellow blue And with the Vector Rosette of Actual growth added Figure above A two dimensional rhomb shaped crystal with intrinsic D 2 symmetry and six antimers green yellow blue Vector Rosette of Actual growth added The vectors are two by two equivalent a with d and b with c The next Figure gives the promorph of our rhomb shaped D 2 crystal with six antimers It is a six fold amphitect polygon and as such the two dimensional analogue of a six fold amphitect pyramid D 2 symmetry and eight antimers The geometry of the motif of a rhomb shape two dimensional crystal with intrinsic D 2 symmetry can be such that eight antimers can be distinguished See next Figures Figure above A two dimensional rhomb shaped crystal with intrinsic D 2 symmetry and eight antimers green yellow And with the Vector Rosette of Actual growth added Figure above A two dimensional rhomb shaped crystal with intrinsic D 2 symmetry and eight antimers green yellow Vector Rosette of Actual growth added The vectors are two by two equivalent a with d and b with c The next Figure gives the promorph of our rhomb shaped D 2 crystal with eight antimers It is an eight fold amphitect polygon and as such the two dimensional analogue of an eight fold amphitect pyramid D 1 symmetry Intrinsically rhomb shaped two dimensional crystals can dependent on the symmetry of the translation free residue which represents the chemical motif have a lower symmetry than that of a rhombus Here we consider such crystals with D 1 symmetry The next Figure gives the pattern of symmetry elements of such a crystal a pattern which consists of just one mirror line Figure above A mirror line m as the only symmetry element of a two dimensional rhomb shaped crystal with intrinsic D 1 symmetry Two crystallographic Forms are needed to conceptually construct a rhomb shape crystal with intrinsic D 1 symmetry An initially given crystal face not parallel and also not perpendicular to the mirror line implies one more face The result is an open Form consisting of two faces A second initially given face parallel to the first implies a second face resulting in a second Form The two Forms together constitute the rhombus and thus the rhomb shaped crystal See next Figure Figure above Two crystallographic Forms red dark blue are needed to construct a rhomb shaped D 1 two dimensional crystal D 1 crystals allow for different numbers of antimers to be present We give some examples D 1 symmetry and four antimers As is by now clear these four antimers can be arranged in two ways which we have called the interradial and radial configurations The next Figures illustrate first the one then the other Figure above A two dimensional intrinsically rhomb shaped D 1 crystal with four antimers green yellow indicated by numerals Interradial configuration of antimers And with the Vector Rosette of Actual Growth added Figure above A two dimensional intrinsically rhomb shaped D 1 crystal with four antimers green yellow indicated by numerals Interradial configuration of antimers Vector Rosette of Actual Growth added The vector b is equivalent with the vector c We have just depicted the interradial configuration One should not however think that this is directly based on the image of the crystal with its antimers See next Figure Figure above One would be tempted to interpret the purple axes which are perpendicular to each other as directional axes which would then imply that one of them runs between antimers while the other runs through two antimers so that we cannot decide whether the directional axes taken together as a pair are radial or interradial The directional axes of which we assess whether together they are radial or interradial and which assessment in turn determines whether the promorph is radial or interradial must be the directional axes of the promorph The purple lines in the present Figure are the directional axes of the rhombus which here is the intrinsic shape of the crystal The geometric figure representing the promorph of this crystal i e the D 1 crystal with four antimers arranged as in the above illustration two at one side of the mirror line and two at the other is not a rhombus but an isosceles trapezium See the promorph below And in such a trapezium both directional axes run between antimers They are therefore definitely interradial The next Figures illustrate the radial configuration of antimers Figure above A two dimensional intrinsically rhomb shaped D 1 crystal with four antimers green yellow indicated by numerals Radial configuration of antimers And with the Vector Rosette of Actual Growth added Figure above A two dimensional intrinsically rhomb shaped D 1 crystal with four antimers green yellow Radial configuration of antimers Vector Rosette of Actual Growth added The vector b is equivalent with the vector c The promorph of our two dimensional rhomb shaped D 1 crystal with four antimers arranged according to the interradial configuration two at one side of the mirror line two at the other is depicted in the next Figure It is an isosceles trapezium and as such the two dimensional analogue of a trapezoid pyramid and thus as promorph in turn the two dimensional analogue of an instance of the Allopola Zygopleura Eutetrapleura interradialia Promorph of the D 1 crystal as depicted above The promorph of our two dimensional rhomb shaped D 1 crystal with four antimers arranged according the radial configuration one at each side of the mirror line and two on the mirror line is depicted in the next Figure It is a bi isosceles triangle and as such the two dimensional analogue of the bi isosceles pyramid and thus as promorph in turn the two dimensional analogue of an instance of the Allopola Zygopleura Eutetrapleura radialia Promorph of the D 1 crystal as depicted above D 1 symmetry and two antimers The geometry of the motif as translation free residue of a rhomb shaped D 1 two dimensional crystal can be such that only two antimers can be distinguished See next Figures Figure above A two dimensional intrinsically rhomb shaped D 1 crystal with two antimers green yellow A motif black is inserted in order to express the D 1 symmetry of the crystal i e to express the fact that there is only one mirror line present And with the Vector Rosette of Actual Growth added Figure above A two dimensional intrinsically rhomb shaped D 1 crystal with two antimers green yellow Vector Rosette of Actual Growth added The vector b is equivalent with the vector c The promorph of our two dimensional intrinsically rhomb shaped D 1 crystal with two antimers is depicted in the next Figure It is an isosceles triangle and as such the two dimensional analogue of half a rhombic pyramid and as promorph in turn the two dimensional analogue of an instance of the Allopola Zygopleura eudipleura Promorph of the above D 1 crystal with two antimers D 1 symmetry and three antimers An intrinsically rhomb shaped crystal can contain a motif as translation free residue with D 1 symmetry but with a geometry that allows for three antimers to be distinguished See next Figures Figure above A two dimensional intrinsically rhomb shaped D 1 crystal with three antimers green yellow and indicated by numerals And with the Vector Rosette of Actual Growth added Figure above A two dimensional intrinsically rhomb shaped D 1 crystal with three antimers green yellow Vector Rosette of Actual Growth added The vector b is equivalent with the vector c The promorph of our two dimensional intrinsically rhomb shaped D 1 crystal with three antimers is depicted in the next Figure It is half a six fold amphitect polygon and as such the two dimensional analogue of half a six fold amphitect pyramid and as promorph in turn the two dimensional analogue of an instance of the Allopola Amphipleura triamphipleura Promorph of the above D 1 crystal with three antimers D 1 symmetry and five antimers An intrinsically rhomb shaped crystal can contain a motif as translation free residue with D 1 symmetry but with a geometry that allows for five antimers to be distinguished See next Figures Figure above A two dimensional intrinsically rhomb shaped D 1 crystal with five antimers green yellow and indicated by numerals And with the Vector Rosette of Actual Growth added Figure above A two dimensional intrinsically rhomb shaped D 1 crystal with five antimers green yellow Vector Rosette of Actual Growth added The vector b is equivalent with the vector c The promorph of our two dimensional intrinsically rhomb shaped D 1 crystal with five antimers is depicted in the next Figure It is half a ten fold amphitect polygon and as such the two dimensional analogue of half a ten fold amphitect pyramid and as promorph in turn the two dimensional analogue of an instance of the Allopola Amphipleura pentamphipleura Promorph of the above D 1 crystal with five antimers D 1 symmetry with six antimers A two dimensional rhomb shaped crystal with intrinsic D 1 symmetry can have a motif as translation free residue of which the geometry is such as to allow for six antimers to be distinguished See next Figures Figure above A two dimensional intrinsically rhomb shaped D 1 crystal with six antimers green yellow and indicated by numerals And with the Vector Rosette of Actual Growth added Figure above A two dimensional intrinsically rhomb shaped D 1 crystal with six antimers green yellow Vector Rosette of Actual Growth added The vector b is equivalent with the vector c The promorph of our two dimensional intrinsically rhomb shaped D 1 crystal with six antimers is depicted in the next Figure It is half a twelve fold amphitect polygon and as such the two dimensional analogue of half a twelve fold amphitect pyramid and as promorph in turn the two dimensional analogue of an instance of the Allopola Amphipleura hexamphipleura Promorph of the above D 1 crystal with six antimers C 2 symmetry An intrinsically rhomb shaped two dimensional crystal can have an intrinsic symmetry according to the Cyclic Group C 2 The only symmetry element of such a crystal with respect to point symmetry is a 2 fold rotation axis at the center of the rhombus and of course perpendicular to the plane of the rhombus See next Figure Figure above The only symmetry element of a two dimensional intrinsically rhomb shaped crystal with intrinsic C 2 symmetry is a two fold rotation axis green Two crystallographic Forms are needed to conceptually construct an intrinsically rhomb shaped C 2 crystal See next Figure Figure above Two crystallographic Forms are needed to construct a rhomb shaped C 2 two dimensional crystal An initially given crystal face red implies in virtue of the 2 fold rotation axis one more face parallel to it resulting in an open Form consisting of two faces red A second initially given face blue not parallel to the first implies a second face in virtue of the same rotation axis also resulting in an open Form consisting of two faces blue The two Forms together red blue constitute a rhombus and therefore constitute the whole rhomb shaped crystal C 2 symmetry and two antimers The

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  • General Ontology XI
    equilateral triangle The center of the latter is highlighted As one can see there is a 3 fold rotation axis of the periodic P31m pattern that does coincide with the 3 fold rotation axis of the mentioned equilateral triangle but there are no three mirror lines of the periodic pattern such that their point of intersection coincides with the location of this axis The symmetry of the triangular crystal i e its intrinsic symmetry is consequently according to the Cyclic Group C 3 instead of the Dihedral Group D 3 which is the point group implied by the plane group P31m The next two Figures depict our regularly triangular crystal without the lattice connection lines which do not belong to the internal pattern as such One can see that the intrinsic symmetry of this crystal is not D 3 as should be expected because it is the point symmetry implied by the present plane group i e the group P31m but C 3 i e its only symmetry element is a 3 fold rotation axis Figure above Two dimensional crystal with the intrinsic shape of an equilateral triangle as it was discussed above drawn without lattice connection lines Figure above Same as previous Figure The 3 fold rotation axis as the only symmetry element of the triangular crystal is indicated The blue lines are not mirror lines of the crystal So the plane group P31m does not support crystals with the Equilateral Triangle as their intrinsic shape One could wonder whether this plane group can support crystals at all But of course it does Crystals with intrinsic hexagonal shape i e having the shape of regular hexagon itself possessing a symmetry according to the group D 6 can be supported by the plane group P31m in which case such crystals would have intrinsic D 3 symmetry See next Figures Figure above Two dimensional crystal with intrinsic regularly hexagonal shape supported by the plane group P31m The connection lines of the hexagonal point lattice not drawn The next Figure shows its intrinsic symmetry Figure above Two dimensional crystal with intrinsic regularly hexagonal shape supported by the plane group P31m The connection lines of the hexagonal point lattice not drawn While the hexagon as being the shape of the crystal itself has D 6 symmetry the intrinsic symmetry of the crystal is lower namely according to the group D 3 three mirror lines and one 3 fold rotation axis while the hexagon just as hexagon has six mirror lines and one 6 fold rotation axis Plane Group P3 Figure above Two dimensional regularly triangular crystal with point symmetry C 3 and supported by a hexagonal point lattice indicated by connection lines The next two Figures show the symmetry C 3 of the above crystal Figure above The two dimensional regularly triangular crystal of the previous Figure with point symmetry C 3 and supported by a hexagonal point lattice indicated by connection lines Empty spaces right side removed resulting in a smaller version of the crystal but wholly equivalent Figure above Same as previous Figure Lattice connection lines removed they do not belong to the pattern and the 3 fold rotation axis as the only symmetry element of the crystal indicated The intrinsic symmetry of the triangular crystal is clearly according to the group C 3 which is implied by the supporting plane group P3 The next Figure shows that the periodic stacking of rhombi is equivalent to a periodic stacking of corresponding rectangles as was discussed above Plane Groups Pm and Pg A regularly triangular two dimensional crystal can also be built by the periodic stacking of special rectangles viz rectangles where the diagonals involve angles of 60 0 which means that a rectangular lattice can if satifying the mentioned condition support regularly triangular crystal shape See next Figures Figure above Rectangular building blocks of special dimensions namely such that their diagonals involve angles of 60 0 can be periodically stacked in such a way that possible crystal faces form an equilateral triangle The next Figures provide these rectangular building blocks with appropriate motifs resulting in the plane groups Pm point group D 1 and Pg point group D 1 both based on a primitive rectangular lattice The latter is thus a special primitive rectangular lattice in virtue of the angles mentioned and as such it in fact turns into a hexagonal lattice Plane Group Pm Figure above Two dimensional regularly triangular crystal with point symmetry D 1 and supported by a primitive rectangular point lattice indicated by connection lines with meshes of special dimensions such that their diagonals involve angles of 60 0 As has been said this point lattice is in fact a hexagonal point lattice For convenient overview of shape we give yet a smaller version of the same crystal type Figure above Two dimensional regularly triangular crystal with point symmetry D 1 and supported by a primitive rectangular point lattice indicated by connection lines with meshes of special dimensions such that their diagonals involve angles of 60 0 Smaller version of triangle The next Figure shows the intrinsic symmetry D 1 of the above triangular crystal larger version Figure above The larger version of the two dimensional regularly triangular crystal considered above with point symmetry D 1 Its only symmetry element a mirror line m is indicated It is consistent with the crystal s intrinsic shape because this mirror line is also present in the equilateral triangle representing this shape It is also consistent with the crystal s internal structure In the rectangular point lattice of the above crystals we can inscribe a rhombic net by drawing connection lines coinciding with the diagonals of the rectangles But this rhombic net as drawn cannot be a lattice that underlies the mode of repetition of the motifs of the crystal because it turns out that we then have two types of filled in rhombi instead of just one type meaning that we do not then have one and the same

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  • General Ontology XII
    trigonal 3 fold gyroid pyramid representing the Homogyrostaura trimera as a category of three dimensional promorphs D 1 symmetry The symmetry of the motif as translation free residue could be such that the intrinsic symmetry of the triangular two dimensional crystal is according to the Dihedral Group D 1 The next Figure shows the set of symmetry elements of such a crystal consisting of a mirror line only Figure above The only symmetry element of a two dimensional triangular D 1 crystal is a mirror line m Two crystallographic Forms are needed to conceptually construct a triangular D 1 crystal See next Figure Figure above Two crystallographic Forms are needed to construct a triangular D 1 two dimensional crystal One initially given crystal face blue not perpendicular or parallel to the mirror line implies one other face blue in virtue of that mirror line resulting in one open Form blue consisting of two faces Another initially given face red perpendicular to the mirror line does not imply other faces so this one face constitutes an open Form red The two Forms together make up the whole triangular crystal A crystal with intrinsic D 1 symmetry can in principle have any number 1 of antimers depending on the geometry of the motif as translation free residue We will discuss several examples D 1 symmetry and two antimers The motif as translation free residue of a two dimensional D 1 crystal can be such that two antimers can be distinguished See next two Figures Figure above A two dimensional intrinsically triangular crystal with intrinsic D 1 symmetry and two antimers green yellow Radial R and interradial IR directions indicated And the Vector Rosette of Actual Growth added Figure above A two dimensional intrinsically triangular crystal with intrinsic D 1 symmetry and two antimers green yellow Radial R and interradial IR directions indicated Vector Rosette of Actual Growth added The vectors b and c are equivalent The next Figure depicts and names the promorph of our two dimensional triangular D 1 crystal It is an isosceles triangle and as such the two dimensional analogue of half a rhombic pyramid As promorph it is the two dimensional analogue of an instance of the three dimensional Allopola Zygopleura eudipleura Figure above Promorph of the above discussed two dimensional triangular D 1 crystal The two antimers are indicated green yellow D 1 symmetry with four antimers The motif as translation free residue of a two dimensional D 1 crystal can be such that four antimers can be distinguished As we already know from earlier discussions four antimers allow for two different spatial configurations viz the interradial and the radial configuration Interradial configuration of the four antimers This configuration consists in the fact that two antimers are at one side of the mirror line while the two others are at the other side of the mirror line See next Figures Figure above A two dimensional intrinsically triangular crystal with intrinsic D 1 symmetry and four antimers green yellow and indicated by numerals Interradial configuration Radial R and interradial IR directions indicated And the Vector Rosette of Actual Growth added Figure above A two dimensional intrinsically triangular crystal with intrinsic D 1 symmetry and four antimers green yellow Interradial configuration Radial R and interradial IR directions indicated Vector Rosette of Actual Growth added The vectors b and c are equivalent The symmetry of our crystal only demands that the configuration and shapes of the four antimers as a whole comply with the one mirror line of the crystal And this means that the common point of the antimers can be positioned anywhere on the mirror line which is vertical in the present case dependent on the geometry of the motif as translation free residue The next Figure illustrates this freedom of positioning of the common point Figure above A two dimensional intrinsically triangular crystal with intrinsic D 1 symmetry and four antimers green yellow and indicated by numerals Vector Rosette of Actual Growth added Alternative position of common point The next Figure depicts and names the promorph of our two dimensional triangular D 1 crystal with an interradial configuration of its four antimers It is an isosceles trapezium and as such the two dimensional analogue of a trapezoid pyramid As promorph it is the two dimensional analogue of an instance of the three dimensional Allopola Zygopleura Eutetrapleura interradialia Figure above Promorph of the above discussed two dimensional triangular D 1 crystal The four antimers are indicated green yellow Radial configuration of the four antimers This configuration consists in the fact that two antimers lie on the mirror line while the third antimer lies at one side of it and the fourth at the other See next Figures Figure above A two dimensional intrinsically triangular crystal with intrinsic D 1 symmetry and four antimers green yellow Radial configuration Radial R and interradial IR directions indicated And the Vector Rosette of Actual Growth added Figure above A two dimensional intrinsically triangular crystal with intrinsic D 1 symmetry and four antimers green yellow Radial configuration Radial R and interradial IR directions indicated Vector Rosette of Actual Growth added The vectors b and c are equivalent The next Figure depicts and names the promorph of our two dimensional triangular D 1 crystal with radial configuration of its four antimers It is a bi isosceles triangle and as such the two dimensional analogue of a bi isosceles pyramid As promorph it is the two dimensional analogue of an instance of the three dimensional Allopola Zygopleura Eutetrapleura radialia Figure above Promorph of the above discussed two dimensional triangular D 1 crystal The four antimers are indicated green yellow D 1 symmetry and three antimers The motif as translation free residue of a two dimensional D 1 crystal can be such that three antimers can be distinguished See next Figures Figure above A two dimensional intrinsically triangular crystal with intrinsic D 1 symmetry and three antimers green yellow and indicated by numerals And the Vector

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  • General Ontology XIII
    e implied by the plane group are also symmetries of the Regular Hexagon This does not necessarily hold the other way around A crystal can have an intrinsic shape that is a regular hexagon but nevertheless some or even all symmetries of this hexagon can be absent in the crystal resulting in the fact that the intrinsic symmetry of the crystal is lower than that of its intrinsic shape Figure above The symmetries indicated by symmetry elements of the Regular Hexagon Six mirror lines and a six fold rotation axis at their point of intersection Plane Group P6mm Figure above A pattern according to the plane group P6mm Each rhombic lattice mesh contains two D 3 motifs s str that are rotated 60 0 with respect to each other The pattern must be imagined to be extended indefinitely over the plane On the periodic pattern of the above Figure a hexagonal crystal can be based as follows Figure above A two dimensional hexagonal crystal with point symmetry D 6 based on a hexagonal lattice i e on a periodic stacking of 60 120 0 rhombi At the jagged sides the motif pattern is extended a little bit by adding some motif units yellow at appropriate locations in order to make evident the crystal s point symmetry Figure above Same as previous Figure Lattice connection lines which do not belong to the pattern erased The next Figure shows the D 6 symmetry of the hexagonal crystal of the previous Figures Figure above Symmetry elements of the hexagonal crystal of the previous Figures Six mirror lines and one 6 fold rotation axis at their point of intersection And this pattern of symmetry elements is the fingerprint of D 6 symmetry In the present case the intrinsic point symmetry of the crystal is the same as that of its intrinsic shape regular hexagon Plane Group P6 Figure above A pattern representing the plane group P6 It has its three fold motifs s str at the mid points of the equilateral triangles that make up the rhombic unit meshes of the hexagonal point lattice A lattice mesh is indicated green Hexagonal crystals can be supported by this P6 pattern as the next Figure shows Figure above A two dimensional hexagonal crystal supported by a P6 pattern It can be conceived as consisting of a periodic stacking of 60 120 0 rhombic building blocks The intrinsic point symmetry of the crystal is according to the group C 6 This is shown in the next two Figures where the first one depicts to begin with the pattern proper i e without the lattice connection lines Figure above The above two dimensional C 6 crystal with hexagonal shape Lattice connection lines omitted they do not belong to the pattern Figure above A 6 fold rotation axis as the only symmetry element of the above two dimensional C 6 crystal with hexagonal shape Its point symmetry is therefore C 6 As also all the crystals below the present crystal thus has a lower intrinsic symmetry than that of its intrinsic shape which is a regular hexagon and has D 6 symmetry Plane Group P3m1 Figure above A two dimensional hexagonal crystal supported by a P3m1 pattern The point symmetry of this crystal is D 3 The crystal results from a periodic stacking of 60 120 0 rhombic building blocks according to a hexagonal net Figure above Same as previous Figure Lattice connection lines removed Figure above Symmetry elements three mirror lines and one 3 fold rotation axis at their point of intersection of the above depicted two dimensional hexagonal crystal based on a P3m1 pattern The point symmetry of the crystal is accordingly D 3 Plane Group P31m Figure above A two dimensional hexagonal crystal supported by a P31m pattern The point symmetry of this crystal is D 3 The crystal results from a periodic stacking of 60 120 0 rhombic building blocks according to a hexagonal net Figure above Same as previous Figure Lattice connection lines removed Figure above Symmetry elements three mirror lines and one 3 fold rotation axis at their point of intersection of the above depicted two dimensional hexagonal crystal based on a P31m pattern The point symmetry of the crystal is accordingly D 3 Plane Group P3 Figure above A two dimensional hexagonal crystal supported by a P3 pattern The point symmetry of this crystal is C 3 The crystal results from a periodic stacking of 60 120 0 rhombic building blocks according to a hexagonal net Figure above Same as previous Figure Lattice connection lines removed Figure above Symmetry elements one 3 fold rotation axis only of the above depicted two dimensional hexagonal crystal based on a P3 pattern The point symmetry of the crystal is accordingly C 3 Figure above Slightly smaller version of the above crystal hexagonal C 3 for convenient overview of its hexagonal shape Plane Group P2mm Figure above Two dimensional hexagonal crystal i e a crystal having as its intrinsic shape a regular hexagon consisting of a stacking of special rectangles viz rectangles such that their diagonals involve angles of 60 0 The lattice plus D 2 motifs is a periodic pattern with a symmetry according to the plane group P2mm The crystal therefore despite its regularly hexagonal intrinsic shape has D 2 point symmetry and not D 6 symmetry That the crystal shape so obtained is indeed a regular hexagon is partially proven in the next Figure There we found three corners of the six sided polygon to be 120 0 The other three corners are then also 120 0 by reason of the reflectional symmetry of the pattern Figure above The crystal s six sided shape obtained by periodic stacking of special rectangles having diagonals involving 60 0 angles has corners of 120 0 The next Figure shows that also the six sides are equal in length Figure above Because of the fact that the relevant angles are all 60 0 the polygon with corners

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  • General Ontology XIV
    alternately coincide with the median lines of the antimers and the lines separating adjacent antimers The three antimers green yellow blue are indicated by numerals The next Figure depicts and names the promorph of our intrinsically hexagon shaped two dimensional crystal with intrinsic D 3 symmetry It is an equilateral triangle regular trigon and as such the two dimensional analogue of the regular three fold pyramid As promorph it is the two dimensional analogue of an instance of the three dimensional Homostaura Anisopola triactinota Figure above Promorph of the above discussed two dimensional hexagon shaped D 3 crystal The three antimers are indicated green yellow blue The next Figure gives the three dimensional analogue of our just established two dimensional promorph Figure above Slightly oblique top view of a regular pyramid with three antimers indicated by colors as the stereometric basic form representing the Homostaura Anisopola triactinota C 3 symmetry Two dimensional crystals with intrinsic C 3 symmetry do not have the full symmetry of the regular hexagon so in this case the intrinsic shape of the crystal a regular hexagon does not express the crystal s intrinsic symmetry It suggests a higher symmetry than the crystal actually possesses The next Figure indicates the symmetry elements of an intrinsic C 3 crystal Figure above Symmetry elements with respect to point symmetry of a two dimensional crystal with intrinsic C 3 symmetry only a 3 fold rotation axis at the center of the crystal Two crystallographic Forms are needed to conceptually construct a hexagonal crystal with intrinsic C 3 symmetry Figure above Two crystallographic Forms are needed to conceptually construct a hexagon shaped intrinsic C 3 two dimensional crystal An initially given crystal face red implies two others in virtue of the 3 fold rotation axis resulting in one open Form consisting of three faces A second initially given crystal face blue also implies two other faces also in virtue of the 3 fold rotation axis resulting in a second open Form also consisting of three faces The two Forms together constitute the whole hexagon shaped crystal C 3 symmetry and three antimers All intrinsic C 3 crystals have three antimers See next Figures Figure above The three asymmetric antimers yellow green blue of an intrinsically hexagon shaped two dimensional crystal with intrinsic C 3 symmetry Radial R and interradial IR directions indicated And with the Vector Rosette of Actual Growth added Figure above Two dimensional hexagonal C 3 crystal with Vector Rosette of Actual Growth added Each of the three vector pairs a b d f and e c of the six vectors a b c d e f coincides with an antimer i e lies within an antimer The next Figure depicts and names the promorph of our intrinsically hexagon shaped two dimensional crystal with intrinsic C 3 symmetry It is a regular gyroid trigon regular gyroid triangle and as such the two dimensional analogue of the regular gyroid three fold pyramid As promorph it is the two dimensional analogue of an instance of the three dimensional Homogyrostaura trimera Figure above Promorph of the above discussed two dimensional hexagonal C 3 crystal The three antimers are indicated green yellow blue The next Figure depicts the three dimensional analogue of our just established promorph Figure above Three dimensional analogue of the promorph of the above discussed two dimensional hexagonal C 3 crystal A regular gyroid pyramid slightly oblique top view with three antimers Homogyrostaura trimera D 2 symmetry Two dimensional crystals with intrinsic D 2 symmetry do not have the full symmetry of the regular hexagon so also in this case the intrinsic shape of the crystal a regular hexagon does not express the crystal s intrinsic symmetry It suggests a higher symmetry than the crystal actually possesses The next Figure indicates the symmetry elements of an intrinsic D 2 crystal Figure above Symmetry elements with respect to point symmetry of a two dimensional crystal with intrinsic D 2 symmetry Two mirror lines perpendicular to each other and a 2 fold rotation axis at their point of intersection Two crystallographic Forms are needed to conceptually construct a hexagonal crystal with intrinsic D 2 symmetry Figure above Two crystallographic Forms are needed to conceptually construct a hexagon shaped intrinsic D 2 two dimensional crystal An initially given crystal face red implies to begin with one other face in virtue of the nearest mirror line and then the two faces are reflected by the other mirror line all this resulting in an open Form consisting of four faces A second initially given crystal face blue implies one other face in virtue of the 2 fold rotation axis resulting in a second open Form consisting of two faces The two Forms together constitute the whole hexagon shaped crystal Any even number of antimers can occur in two dimensional crystals with intrinsic D 2 symmetry D 2 symmetry and four antimers As we saw already in earlier documents there are two different configurations of four antimers in D 2 crystals possible viz the interradial and radial configurations Interradial configuration Figure above Two dimensional hexagon shaped D 2 crystal with four antimers green yellow in interradial configuration And with the Vector Rosette of Actual Growth added Figure above Two dimensional hexagon shaped D 2 crystal with four antimers green yellow in interradial configuration Vector Rosette of Actual Growth with its six vectors a b c d e f added The next Figure depicts and names the promorph of our intrinsically hexagon shaped two dimensional crystal with intrinsic D 2 symmetry and with four antimers in interradial configuration It is a rectangle and as such the two dimensional analogue of the rectangular pyramid As promorph it is the two dimensional analogue of an instance of the three dimensional Autopola Orthostaura Tetraphragma interradialia Figure above Promorph of the above discussed two dimensional hexagonal D 2 crystal The four antimers are indicated green yellow The next Figure depicts the three dimensional analogue of our just established promorph Figure above Three dimensional analogue of the promorph of the above discussed two dimensional hexagonal D 2 crystal A rectangular pyramid slightly oblique top view with four antimers Autopola Orthostaura Tetraphragma interradialia Radial configuration Figure above Two dimensional hexagon shaped D 2 crystal with four antimers green yellow in radial configuration And with the Vector Rosette of Actual Growth added Figure above Two dimensional hexagon shaped D 2 crystal with four antimers green yellow and indicated by numerals in radial configuration Vector Rosette of Actual Growth with its six vectors a b c d e f added The next Figure depicts and names the promorph of our intrinsically hexagon shaped two dimensional crystal with intrinsic D 2 symmetry and with four antimers in radial configuration It is a rhombus and as such the two dimensional analogue of the rhombic pyramid As promorph it is the two dimensional analogue of an instance of the three dimensional Autopola Orthostaura Tetraphragma radialia Figure above Promorph of the above discussed two dimensional hexagonal D 2 crystal The four antimers are indicated green yellow The next Figure depicts the three dimensional analogue of our just established promorph Figure above Three dimensional analogue of the promorph of the above discussed two dimensional hexagonal D 2 crystal A rhombic pyramid slightly oblique top view with four antimers Autopola Orthostaura Tetraphragma radialia D 2 symmetry with two antimers The motif as translation free residue of a two dimensional hexagon shaped D 2 crystal can be such that only two antimers can be distinguished See next Figures Figure above Two dimensional hexagon shaped D 2 crystal with two antimers green yellow And with the Vector Rosette of Actual Growth added Figure above Two dimensional hexagon shaped D 2 crystal with two antimers green yellow Vector Rosette of Actual Growth with its six vectors a b c d e f added The next Figure depicts and names the promorph of our intrinsically hexagon shaped two dimensional crystal with intrinsic D 2 symmetry and with two antimers It is a rhombus and as such the two dimensional analogue of the rhombic pyramid As promorph it is the two dimensional analogue of an instance of the three dimensional Autopola Orthostaura diphragma Figure above Promorph of the above discussed two dimensional hexagonal D 2 crystal The two antimers are indicated green yellow D 2 symmetry and six antimers The motif as translation free residue of a two dimensional hexagon shaped D 2 crystal can be such that six antimers can be distinguished See next Figures Figure above Two dimensional hexagon shaped D 2 crystal with six antimers green yellow and indicated by numerals The next Figure uses another color configuration to express the same crystal Figure above Two dimensional hexagon shaped D 2 crystal with six antimers green yellow And with the Vector Rosette of Actual Growth added Figure above Two dimensional hexagon shaped D 2 crystal with six antimers green yellow Vector Rosette of Actual Growth with its six vectors a b c d e f added The next Figure depicts and names the promorph of our intrinsically hexagon shaped two dimensional crystal with intrinsic D 2 symmetry and with six antimers It is a 6 fold amphitect polygon and as such the two dimensional analogue of the six fold amphitect pyramid As promorph it is the two dimensional analogue of an instance of the three dimensional Autopola Oxystaura hexaphragma Figure above Promorph of the above discussed two dimensional hexagonal D 2 crystal The six antimers are indicated green yellow blue The next Figure depicts the three dimensional analogue of our just established two dimensional promorph Figure above Slightly oblique top view of a six fold amphitect pyramid as the basic form of the Autopola Oxystaura hexaphragma As one can see this pyramid is flattened parallel to two of its sides while in the previous Figure the six fold polygon is flattened perpendicular to two of its sides but this difference is immaterial D 1 symmetry Two dimensional crystals with intrinsic D 1 symmetry do not have the full symmetry of the regular hexagon so also in this case the intrinsic shape of the crystal a regular hexagon does not express the crystal s intrinsic symmetry It suggests a higher symmetry than the crystal actually possesses The next Figure indicates the symmetry elements of an intrinsic D 1 crystal Figure above Symmetry elements with respect to point symmetry of a two dimensional crystal with intrinsic D 1 symmetry Only one mirror line Three crystallographic Forms are needed to conceptually construct a hexagonal crystal with intrinsic D 1 symmetry Figure above Three crystallographic Forms are needed to conceptually construct a hexagon shaped intrinsic D 1 two dimensional crystal An initially given crystal face red implies one other face in virtue of the mirror line resulting in an open Form consisting of two faces A second initially given crystal face blue also implies one other face in virtue of that same mirror line resulting in a second open Form also consisting of two faces Finally a third initially given face green also implies one other face and also results in an open Form consisting of two faces The three Forms together constitute the whole hexagon shaped crystal Any number 1 of antimers can occur in two dimensional crystals with intrinsic D 1 symmetry D 1 symmetry and two antimers The motif as translation free residue of a two dimensional hexagon shaped D 1 crystal can be such that only two antimers can be distinguished See next Figures Figure above Two dimensional hexagon shaped D 1 crystal with two antimers green yellow A motif black is inserted in order to express the D 1 symmetry of the crystal And with the Vector Rosette of Actual Growth added Figure above Two dimensional hexagon shaped D 1 crystal with two antimers green yellow Vector Rosette of Actual Growth with its six vectors a b c d e f added The next Figure depicts and names the promorph of our intrinsically hexagon shaped two dimensional crystal with intrinsic D 1 symmetry and with two antimers It is half a rhombus i e an isosceles triangle and as such the two dimensional analogue of half a rhombic pyramid i e of an isosceles pyramid As promorph it is the two dimensional analogue of an instance of the three dimensional Allopola Zygopleura eudipleura Figure above Promorph of the above discussed two dimensional hexagonal D 1 crystal The two antimers are indicated green yellow The next Figure depicts the three dimensional analogue of our just established two dimensional promorph Figure above Oblique top view of a single isosceles pyramid i e of half a rhombic pyramid as the basic form of the Allopola Zygopleura eudipleura The two antimers are indicated by colors D 1 symmetry with four antimers As in D 2 crystals there are in D 1 crystals two different configurations of four antimers possible viz the interradial and radial configurations Interradial configuration Figure above Two dimensional hexagon shaped D 1 crystal with four antimers green yellow in interradial configuration The next Figure uses a different color configuration for the same crystal Figure above Two dimensional hexagon shaped D 1 crystal with four antimers green yellow and indicated by numerals in interradial configuration And with the Vector Rosette of Actual Growth added Figure above Two dimensional hexagon shaped D 1 crystal with four antimers green yellow and indicated by numerals in interradial configuration Vector Rosette of Actual Growth with its six vectors a b c d e f added The next Figure depicts and names the promorph of our intrinsically hexagon shaped two dimensional crystal with intrinsic D 1 symmetry and with four antimers in interradial configuration It is an isosceles trapezium and as such the two dimensional analogue of the trapezoid pyramid i e it is its base As promorph it is the two dimensional analogue of an instance of the three dimensional Allopola Zygopleura Eutetrapleura interradialia Figure above Promorph of the above discussed two dimensional hexagonal D 1 crystal The four antimers are indicated green yellow The next Figure depicts the three dimensional analogue of our just established promorph Figure above Three dimensional analogue of the promorph of the above discussed two dimensional hexagonal D 1 crystal Oblique top view of a trapezoid pyramid antiparallelogram pyramid with four antimers indicated by colors Allopola Zygopleura Eutetrapleura interradialia Radial configuration Figure above Two dimensional hexagon shaped D 1 crystal with four antimers green yellow in radial configuration And with the Vector Rosette of Actual Growth added Figure above Two dimensional hexagon shaped D 1 crystal with four antimers green yellow in radial configuration Vector Rosette of Actual Growth with its six vectors a b c d e f added The next Figure depicts and names the promorph of our intrinsically hexagon shaped two dimensional crystal with intrinsic D 1 symmetry and with four antimers in radial configuration It is a bi isosceles triangle and as such the two dimensional analogue of the bi isosceles pyramid As promorph it is the two dimensional analogue of an instance of the three dimensional Allopola Zygopleura Eutetrapleura radialia Figure above Promorph of the above discussed two dimensional hexagonal D 1 crystal The four antimers are indicated green yellow The next Figure depicts the three dimensional analogue of our just established promorph Figure above Three dimensional analogue of the promorph of the above discussed two dimensional hexagonal D 1 crystal A bi isosceles pyramid oblique top view with four antimers Allopola Zygopleura Eutetrapleura radialia The four antimers are indicated by colors D 1 symmetry with three antimers The motif as translation free residue of a two dimensional hexagon shaped D 1 crystal can be such that three antimers can be distinguished See next Figures Figure above Two dimensional hexagon shaped D 1 crystal with three antimers green yellow and indicated by numerals And with the Vector Rosette of Actual Growth added Figure above Two dimensional hexagon shaped D 1 crystal with three antimers green yellow indicated by numerals Vector Rosette of Actual Growth with its six vectors a b c d e f added The next Figure depicts and names the promorph of our intrinsically hexagon shaped two dimensional crystal with intrinsic D 1 symmetry and with three antimers It is half a 6 fold amphitect polygon and as such the two dimensional analogue of half a six fold amphitect pyramid As promorph it is the two dimensional analogue of an instance of the three dimensional Allopola Amphipleura triamphipleura Figure above Promorph of the above discussed two dimensional hexagonal D 1 crystal The three antimers are indicated green yellow The next Figure depicts the three dimensional analogue of our just established two dimensional promorph Figure above Oblique top view of half a six fold amphitect pyramid as the basic form of the Allopola Amphipleura triamphipleura The three antimers are indicated by colors D 1 symmetry with five antimers The motif as translation free residue of a two dimensional hexagon shaped D 1 crystal can be such that five antimers can be distinguished See next Figures Figure above Two dimensional hexagon shaped D 1 crystal with five antimers green yellow and indicated by numerals And with the Vector Rosette of Actual Growth added Figure above Two dimensional hexagon shaped D 1 crystal with five antimers green yellow indicated by numerals Vector Rosette of Actual Growth with its six vectors a b c d e f added The next Figure depicts and names the promorph of our intrinsically hexagon shaped two dimensional crystal with intrinsic D 1 symmetry and with five antimers It is half a ten fold amphitect polygon and as such the two dimensional analogue of half a ten fold amphitect pyramid As promorph it is the two dimensional analogue of an instance of the three dimensional Allopola Amphipleura pentamphipleura Figure above Promorph of the above discussed two dimensional hexagonal D 1 crystal The five antimers are indicated green yellow right image The next Figure depicts

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  • General Ontology XV
    Hexamphipleura D 1 Allpola amphipleura Heptamphipleura C 2 Stauraxonia gyrostaura Heterogyrostaura dimera C 2 Stauraxonia gyrostaura Heterogyrostaura tetramera C 2 Stauraxonia gyrostaura Heterogyrostaura hexamera C 2 Stauraxonia gyrostaura Heterogyrostaura octomera C 1 Anaxonia Anaxonia acentra C 1 Stauraxonia heterostaura Allopola heteropleura Symmetry Family of Promorphs Promorph C 2 Stauraxonia gyrostaura Heterogyrostaura dimera C 2 Stauraxonia gyrostaura Heterogyrostaura tetramera C 2 Stauraxonia gyrostaura Heterogyrostaura hexamera C 2 Stauraxonia gyrostaura Heterogyrostaura octomera C 1 Anaxonia Anaxonia acentra C 1 Stauraxonia heterostaura Allopola heteropleura Symmetry Family of Promorphs Promorph D 2 Heterostaura autopola orthostaura Diphragma D 2 Heterostaura autopola orthostaura Tetraphragma interradialia D 2 Heterostaura autopola orthostaura Tetraphragma radialia D 2 Heterostaura autopola oxystaura Hexaphragma D 2 Heterostaura autopola oxystaura Octophragma D 1 Allopola zygopleura Eudipleura D 1 Allopola zygopleura Eutetrapleura interradialia D 1 Allopola zygopleura Eutetrapleura radialia D 1 Allpola amphipleura Triamphipleura D 1 Allpola amphipleura Pentamphipleura D 1 Allpola amphipleura Hexamphipleura D 1 Allpola amphipleura Heptamphipleura C 2 Stauraxonia gyrostaura Heterogyrostaura dimera C 2 Stauraxonia gyrostaura Heterogyrostaura tetramera C 2 Stauraxonia gyrostaura Heterogyrostaura hexamera C 2 Stauraxonia gyrostaura Heterogyrostaura octomera C 1 Anaxonia Anaxonia acentra C 1 Stauraxonia heterostaura Allopola heteropleura Symmetry Family of Promorphs Promorph D 3 Stauraxonia homostaura Anisopola triactinota C 3 Stauraxonia gyrostaura Homogyrostaura trimera D 1 Allopola zygopleura Eudipleura D 1 Allopola zygopleura Eutetrapleura interradialia D 1 Allopola zygopleura Eutetrapleura radialia D 1 Allpola amphipleura Triamphipleura D 1 Allpola amphipleura Pentamphipleura D 1 Allpola amphipleura Hexamphipleura D 1 Allpola amphipleura Heptamphipleura C 1 Anaxonia Anaxonia acentra C 1 Stauraxonia heterostaura Allopola heteropleura Symmetry Family of Promorphs Promorph D 6 Stauraxonia homostaura Isopola hexactinota C 6 Stauraxonia gyrostaura Homogyrostaura hexamera D 3 Stauraxonia homostaura Anisopola triactinota C 3 Stauraxonia gyrostaura Homogyrostaura trimera D 2 Heterostaura autopola orthostaura Diphragma D 2 Heterostaura autopola orthostaura Tetraphragma interradialia D 2 Heterostaura autopola orthostaura Tetraphragma radialia D 2 Heterostaura autopola oxystaura Hexaphragma D 2 Heterostaura autopola oxystaura Octophragma D 1 Allopola zygopleura Eudipleura D 1 Allopola zygopleura Eutetrapleura interradialia D 1 Allopola zygopleura Eutetrapleura radialia D 1 Allpola amphipleura Triamphipleura D 1 Allpola amphipleura Pentamphipleura D 1 Allpola amphipleura Hexamphipleura D 1 Allpola amphipleura Heptamphipleura C 2 Stauraxonia gyrostaura Heterogyrostaura dimera C 2 Stauraxonia gyrostaura Heterogyrostaura tetramera C 2 Stauraxonia gyrostaura Heterogyrostaura hexamera C 2 Stauraxonia gyrostaura Heterogyrostaura octomera C 1 Anaxonia Anaxonia acentra C 1 Stauraxonia heterostaura Allopola heteropleura Below we will present a second summary of the investigation of the above intrinsic shapes Rectangle Square Parallelogram Rhombus Equilateral Triangle and Regular Hexagon of two dimensional crystals regarding their relationship to intrinsic point symmetry and promorph The enumeration of the crystals with the mentioned intrinsic shapes will in this summary be done from the viewpoint of Promorphology Things allow that in the present cases the names of the given two dimensional promorphs planimetric basic forms can be the same as those of their three dimensional counterparts The complete System of three dimensional promorphs stereometric basic forms and their systematic derivation can be found on Second Part of Website and there at BASIC FORMS It is with promorphs but only those based on i n t r i n s i c symmetry and geometrical structure that crystals here exemplified by their imaginary two dimensional counterparts connect with organisms While also individual molecules of chemical substances do have intrinsic symmetry and promorph they are still microscopic as are atoms Crystals on the other hand although in a way being just molecules are macroscopic and do as such fall into the size range of organisms In this way we have three types of intrinsic dynamical systems or their immediate products representing intrinsic things beings viz molecules crystals and organisms And crystals clearly are intermediate between molecules and organisms That s why it was so important for us to find out that crystals do have like molecules and organisms definite promorphs so that there is a certain continuity of basic structure as we go from the microscopic molecules to the macroscopic organisms via the macroscopic crystals It must be admitted however that crystals in a way resist to being promorphologically interpreted because of their periodic nature This nature lets them to be homogeneous in the sense that the same surroundings and with the same orientation are found with respect to points that are very close together and which are regularly arranged throughout the crystal So no macroscopic boundaries seem to be present in the interior of a single i e not twinned crystal as we do find such boundaries in organisms such as in regular sea urchins where we find ten radially spaced areas alternately of the same nature resulting in the fact that such a sea urchin consists of five identical counterparts antimers arranged around its main body axis Indeed we can distinguish between ambulacral and interambulacral areas within the sea urchin s body And although something like this seems not to be present in single crystals we have found Last Series of documents of Second Part of Website Basic Forms of Crystals that single crystals nevertheless do have promorphs Radially twinned crystals do have promorphs anyway but in a more or less concealed way based on the structure of the translation free residue of the crystal The symmetry of that residue is macroscopically visible as the intrinsic point symmetry of the crystal and this latter macroscopic symmetry reveals part of the promorph i e generally this point symmetry reveals only one aspect of the promorph and sometimes it reveals the whole promorph which boils down to revealing the number and arrangement of the antimers And in addition to this we found in the present series of documents that the intrinsic s h a p e of a single crystal could be related in some way to the number of antimers but as the foregoing documents have shown only very loosely so The crystal has a definite and for the particular species constant pattern of corners which expresses definite accomplished growth in certain well defined directions And these somehow co express the crystal s promorph But compared to organisms this expression is weak On the other hand the main aspect of the promorph viz the particular intrinsic point symmetry is clearly and directly expressed in single crystals while in organisms it is not All organisms are strictly speaking asymmetric Their true symmetry is more or less hidden and has to be derived from the evaluation of certain constant structural main features of the given organism So in this way also in organisms their promorph is more or less hidden i e not directly visible In the present series of documents we will concentrate on the promorphs of crystals illustrated in imaginary two dimensional crystals and later compare them with those of organisms Intrinsic shape translational symmmetry i e total symmetry or symmetry including all translational features plane group symmetry intrinsic non translational symmetry i e point group symmetry and promorph are structural features of natural intrinsic bodies such as crystals and organisms Taken in themselves they are whatness categories or entitative If Then constants determining their c o n c r e t a within the Mathematical Inorganic and Organic Layers of Being And their study here only serves to create an insight into the workings of If Then constants in general which will help to set up a general ontology as indicated in the first documents especially Part I of the present series Well let s now go ahead with our summary of promorphs as they can occur in two dimensional crystals having the above mentioned intrinsic shapes Recall that the promorph of some natural body here exemplified by two dimensional crystals is expressed by the simplest geometric figure geometrically expressing the intrinsic point symmetry and the number and arrangement of antimers in that natural body So one must not confuse the shape of the promorph and that of the relevant natural body Where necessary we have into the drawing of the crystal inserted a motif to express the intrinsic point symmetry of that crystal Let s now consider things about the above mentioned second summary of our results given below a summary relating things from the promorphological viewpoint The presented promorphs relate to crystals discussed in the forgoing and having the above mentioned intrinsic shapes We list these crystal shapes together with their own point symmetries Rectangle D 2 Square D 4 Parallelogramm C 2 Rhombus D 2 Equilateral Triangle D 3 Regular Hexagon D 6 The promorphs encountered among these shapes are now together with the corresponding two dimensional crystals themselves arranged in the same spirit as was done in our Promorphological System of Basic Forms Second Part of Website see LINK above namely according to decreasing symmetry i e increasing differentiation In the resulting presentation each large heading indicates the promorphological Section to which the relevant crystals promorphologically belong Holomorphic and Meromorphic crystals Let s say something more about the relationship between the symmetry of the crystal s intrinsic shape and that of the crystal itself i e that between the geometric and crystallographic symmetry We call a crystal h o l o m o r p h i c when the point symmetry of its intrinsic shape is the same as the intrinsic point symmetry of the crystal If the crystal s point symmetry is l o w e r than that of its intrinsic shape we call the crystal m e r o m o r p h i c So in holomorphic crystals their whole shape reflects the intrinsic symmetry of the crystal while in meromorphic crystals only a part of their shape reflects the intrinsic symmetry of the crystal If we have for example a two dimensional crystal with intrinsic C 2 symmetry and possessing as its intrinsic shape that of a rectangle then only a part of this rectangle indicates the true symmetry of the crystal namely the 2 fold rotation axis of the rectangle In different but equivalent words The symmetry of the rectangle is according to the group D 2 while the true symmetry of the crystal is according to the group C 2 and the group C 2 is a subgroup of the group D 2 what we denote as follows C 2 D 2 and which is shown in the next diagram group table Diagram above The Cyclic Group C 2 yellow is a subgroup of the Dihedral Group D 2 whole table So in general meromorphic crystals have an intrinsic point symmetry that is represented by a subgroup of the group representing the symmetry of their intrinsic shape while in holomorphic crystals their intrinsic symmetry is the same as the symmetry of their intrinsic shape For an introduction to Group Theory see Second Part of Website and there at GROUP THEORY Summary of the Promorphs of some two dimensional crystals Homostaura regular polygons Homostaura Isopola hexactinota Regular polygons with equipolar cross axes and six antimers Planimetric Basic Form representing this promorph Example of crystal Holomorphic crystal Homostaura Isopola tetractinota Regular polygons with equipolar cross axes and four antimers Planimetric Basic Form representing this promorph Example of crystal Holomorphic crystal Homostaura Anisopola triactinota Regular polygons with unequipolar cross axes and three antimers Planimetric Basic Form representing this promorph Examples of crystals Holomorphic crystal Meromorphic crystal D 3 D 6 Gyrostaura gyroid i e twisted polygons homogyrostaura regular gyroid polygons Homogyrostaura hexamera Regular gyroid polygons with six antimers Planimetric Basic Form representing this promorph Example of crystal Meromorphic crystal C 6 D 6 Homogyrostaura tetramera Regular gyroid polygons with four antimers Planimetric Basic Form representing this promorph The next two Figures illustrate this basic form still further by giving equivalent geometric figures all representing and expressing the same promorph Example of crystal Meromorphic crystal C 4 D 4 Homogyrostaura trimera Regular gyroid polygons with three antimers Planimetric Basic Form representing this promorph Examples of crystals Meromorphic crystal C 3 D 3 Meromorphic crystal C 3 D 6 Gyrostaura gyroid i e twisted polygons heterogyrostaura amphitect i e flattened gyroid polygons Heterogyrostaura octomera amphitect gyroid polygons with eight antimers Planimetric Basic Form representing this promorph Examples of crystals Meromorphic crystal C 2 D 2 Meromorphic crystal C 2 D 4 Holomorphic crystal Meromorphic crystal C 2 D 2 Meromorphic crystal C 2 D 6 Heterogyrostaura tetramera amphitect gyroid polygons with four antimers Planimetric Basic Form representing this promorph Examples of crystals Meromorphic crystal C 2 D 2 Holomorphic crystal Meromorphic crystal C 2 D 2 Meromorphic crystal C 2 D 6 Heterogyrostaura dimera amphitect gyroid polygons with two antimers Planimetric Basic Form representing this promorph Examples of crystals Meromorphic crystal C 2 D 2 Meromorphic crystal C 2 D 4 Holomorphic crystal Meromorphic crystal C 2 D 2 Meromorphic crystal C 2 D 6 Heterostaura Autopola amphitect polygons Autopola oxystaura autopola with 6 8 10 antimers therefore at least three radial cross axes present that must consequently intersect at a c u t e angles Autopola Oxystaura octophragma amphitect polygons with eight antimers Planimetric Basic Form representing this promorph Examples of crystals Holomorphic crystal Holomorphic crystal Autopola Oxystaura hexaphragma amphitect polygons with six antimers Planimetric Basic Form representing this promorph Examples of crystals Holomorphic crystal Meromorphic crystal D 2 D 4 Meromorphic crystal D 2 D 4 Holomorphic crystal Meromorphic crystal D 2 D 6 Heterostaura Autopola amphitect polygons Autopola orthostaura autopola with 4 or 2 antimers i e tetraphragma or diphragma therefore there is either only one radial cross axis present or two radial cross axes coinciding with respectively with one or two directional axes In the latter case these radial cross axes are p e r p e n d i c u l a r to each other i e they intersect at right angles Orthostaura tetraphragma orthostaura with four antimers Autopola Orthostaura Tetraphragma radialia amphitect polygons here rhombi with four antimers and with radial directional axes Planimetric Basic Form representing this promorph Examples of crystals Holomorphic crystal Meromorphic crystal D 2 D 4 Meromorphic crystal D 2 D 4 Holomorphic crystal Meromorphic crystal D 2 D 6 Autopola Orthostaura Tetraphragma interradialia amphitect polygons here rectangles or rhombi with four antimers and with interradial directional axes Planimetric Basic Form representing this promorph In fact the Autopola Orthostaura Tetraphragma interradialia can be represented by a still more general geometric Figure See next Figure Examples of crystals Holomorphic crystal Meromorphic crystal D 2 D 4 Meromorphic crystal D 2 D 4 Holomorphic crystal Meromorphic crystal D 2 D 6 Autopola Orthostaura Diphragma amphitect polygons here rhombi with two antimers Planimetric Basic Form representing this promorph Examples of crystals Holomorphic crystal Meromorphic crystal D 2 D 4 Meromorphic crystal D 2 D 4 Holomorphic crystal Meromorphic crystal D 2 D 6 Heterostaura Allopola half amphitect polygons bilateral forms Allopola amphipleura bilateral forms with 3 5 6 7 8 antimers Allopola heptamphipleura half amphitect polygons with seven antimers Planimetric Basic Form representing this promorph Example of crystal Meromorphic crystal D 1 D 2 Allopola hexamphipleura half amphitect polygons with six antimers Planimetric Basic Form representing this promorph Examples of crystals Meromorphic crystal D 1 D 2 Meromorphic crystal D 1 D 4 Meromorphic crystal D 1 D 4 Meromorphic crystal D 1 D 2 Meromorphic crystal D 1 D 3 Meromorphic crystal D 1 D 6 Allopola pentamphipleura half amphitect polygons with five antimers Planimetric Basic Form representing this promorph Examples of crystals Meromorphic crystal D 1 D 2 Meromorphic crystal D 1 D 4 Meromorphic crystal D 1 D 2 Meromorphic crystal D 1 D 3 Meromorphic crystal D 1 D 6 Allopola triamphipleura half amphitect polygons with three antimers Planimetric Basic Form representing this promorph Examples of crystals Meromorphic crystal D 1 D 2 Meromorphic crystal D 1 D 4 Meromorphic crystal D 1 D 4 Meromorphic crystal D 1 D 2 Meromorphic crystal D 1 D 3 Meromorphic crystal D 1 D 6 Heterostaura Allopola half amphitect polygons bilateral forms Allopola zygopleura half rhombic polygons i e isosceles triangles or isosceles trapezia i e bilateral tetragons bilateral forms with four or two antimers Zygopleura Eutetrapleura symmetric zygopleura with four antimers Eutetrapleura radialia zygopleura with four antimers and radial directional axes bi isosceles triangles Planimetric Basic Form representing this promorph or another but equivalent representation with straight interradial cross axes As one can see the geometric figure representing this promorph consists of two isosceles triangles connected base to base Examples of crystals Meromorphic crystal D 1 D 2 Meromorphic crystal D 1 D 4 Meromorphic crystal D 1 D 4 Meromorphic crystal D 1 D 2 Meromorphic crystal D 1 D 3 Meromorphic crystal D 1 D 6 Eutetrapleura interradialia zygopleura with four antimers and interradial directional axes isosceles trapezia Planimetric Basic Form representing this promorph or another but equivalent representation with straight radial cross axes Examples of crystals Meromorphic crystal D 1 D 2 Meromorphic crystal D 1 D 4 Meromorphic crystal D 1 D 4 Meromorphic crystal D 1 D 2 Meromorphic crystal D 1 D 3 Meromorphic crystal D 1 D 6 Heterostaura Allopola half amphitect polygons bilateral forms Allopola zygopleura half rhombic polygons i e isosceles triangles or isosceles trapezia i e bilateral tetragons bilateral forms with four or two antimers Zygopleura dipleura symmetric or asymmetric zygopleura with two antimers eudipleura or dysdipleura Zygopleura eudipleura isosceles triangles symmetric zygopleura with two antimers bilateral forms s str Planimetric Basic Form representing this promorph Examples of crystals Meromorphic crystal D 1 D 2 Meromorphic crystal D 1 D 4 Meromorphic crystal D 1 D 4 Meromorphic crystal D 1 D 2 Meromorphic crystal D 1 D 3 Meromorphic crystal D 1 D 6 Asymmetric forms Either forms without axes apart from the trivial 1 fold rotation axis and without antimers or asymmetric Allopola with antimers Asymmetric Allopola either heteropleural i e asymmetric Amphipleura or heteropleural Zygopleura In what follows we only display the Anaxonia acentra no axes no antimers and promorphologically represented by a right angled triangle and the Zygopleura dysdipleura heteropleural Zygopleura with two unequal antimers promorphologically represented

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