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  • Hexagonal Crystal System VII
    express this by the alternate suppression and extension of lines representing faces as Figure 3e shows Figure 3e Construction of a Ditrigon representing a horizontal section through a Ditrigonal Scalenohedron Colors indicate the symmetries involved The resulting ditrigon is depicted in Figure 3f Figure 3f A Ditrigon representing a horizontal section through a Ditrigonal Scalenohedron Colors indicate the symmetries involved One can clearly see that the mirror lines coinciding with the horizontal crystallographic axes red lines are absent i e the corresponding mirror symmetry is absent while it is still present in the lines bisecting the crystallographic axes blue dashed lines This is compatible with the symmetry of the rhombohedric hemihedric Forms From Figure 2h it is evident that actually two ditrigons can be constructed The construction of the second possible ditrigon is depicted in Figure 3g Figure 3g Construction of a second Ditrigon representing a horizontal section through a second possible Ditrigonal Scalenohedron Colors indicate the symmetries involved The constructed second Ditrigon is depicted in Figure 3h Figure 3h A second Ditrigon representing a horizontal section through a second possible Ditrigonal Scalenohedron Colors indicate the symmetries involved From each of these ditrigons a triangle trigon can be derived by applying pyramidal hemihedric to them This means that in the ditrigon the remaining mirror lines representing mirror planes are suppressed as the next Figure shows The resulting Triangle represents a horizontal section of a type III rhombohedron Figure 3i Construction of a Triangle trigon from the Ditrigon of Figure 3f The resulting Triange is shown in Figure 3j Figure 3j A Triangle trigon constructed from the Ditrigon of Figure 3f It is a horizontal section through the tetartohedric Type III Rhombohedron The section must be imagined as being taken above the zig zag line consisting of the middle edges of the Rhombohedron In Figure 3j the colors symbolizing the symmetry clearly show a 3 fold rotation axis The three sides of the Triangle represent 3 faces of the rhombohedron Below these faces lie three others separated from the upper three by six zig zag going edges The lower faces can like the upper faces be represented by a triangle This second triangle is 60 0 rotated with respect to the first This means that for each upper face there is a lower face parallel to it and this is the same as saying that the whole configuration of faces has a center of symmetry Well a 3 fold rotation axis and a center of symmetry together form a 3 fold roto inversion axis and this having just a 3 fold roto inversion axis is precisely the symmetry content of our Class As can be surmised from Figure 2l actually two Triangles can be derived See Figures 3k and 3l Figure 3k Construction of a second Triangle trigon from the Ditrigon of Figure 3f Figure 3l A second Triangle trigon constructed from the Ditrigon of Figure 3f It is a horizontal section through a tetartohedric Type III Rhombohedron The section must be imagined as being taken above the zig zag line consisting of the middle edges of the Rhombohedron In exactly the same way we can derive two Triangles from the second possible ditrigon namely the one depicted in Figure 3h So all in all we can derive four type III tetartohedric rhombohedra from one holohedric dihexagonal pyramid first by applying rhombohedric hemihedric two ditrigonal scalenohedra and then applying pyramidal hemihedric to these scalenohedra four type III tetartohedric rhombohedra The derivation of the rhombohedra from scalenohedra was shown in terms of horizontal sections representing the complete Forms To complete our exposition of this derivation we will now show the complete scalenohedron and how it gives rise to rhombohedrons when subjected to pyramidal hemihedric Figure 3m Derivation of a Type III Rhombohedron from the white faces of a Ditrigonal Scalenohedron Only the shapes of the respective Forms Scalenohedron Rhombohedron are given They are not necessarily drawn at precisely the same scale From the rhombohedric hemihedric type I hexagonal prism we can derive the rhombohedric tetartohedric type I hexagonal prism by subjecting it to the pyramidal hemihedric this is equivalent to saying that we subject the holohedric protoprism to rhombohedric tetartohedric Figure 4 The rhombohedric hemihedric Type I Hexagonal Prism from which the rhombohedric tetartohedric Type I Hexagonal Prism can be derived Figure 4a Figure 4a The rhombohedric tetartohedric Type I Hexagonal Prism As can be seen from the Figure the external shape will not change the generated Form is still a hexagonal prism but with lowered symmetry From the rhombohedric hemihedric type II hexagonal prism we can derive the rhombohedric tetartohedric type II hexagonal prism when we subject it to the pyramidal hemihedric See Figures 5 and 5a Figure 5 The rhombohedric hemihedric Type II Hexagonal Prism from which the rhombohedric tetartohedric Type II Hexagonal Prism will be derived Figure 5a Figure 5a The rhombohedric tetartohedric Type II Hexagonal Prism As can be seen from the Figure the external shape will not change The generated Form remains a hexagonal prism From the rhombohedric hemihedric dihexagonal prism we can derive the rhombohedric tetartohedric type III hexagonal prism in fact two of them when we subject it to pyramidal hemihedric See Figures 6 and 6a Figure 6 The rhombohedric hemihedric Dihexagonal Prism from which the rhombohedric tetartohedric Type III Hexagonal Prism will be derived Figure 6a Figure 6a The rhombohedric tetartohedric Type III Hexagonal Prism By suppressing the yellow faces and letting the others extend till they meet in order to express the application of pyramidal hemihedric we will obtain a hexagonal prism Another such prism will be obtained when we suppress all the white faces and let the others extend till they meet To express the loss of the equatorial mirror plane by virtue of the rhombohedric hemihedric we could perhaps better depict the derivation of the rhombohedric tetartohedric type III hexagonal prism as is done in the next Figure Figure 6b By suppressing all areas except the white ones and letting these

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  • Hexagonal Crystal System VIII
    3f Figure 3f A Ditrigon representing a Ditrigonal Bipyramid We can clearly see that the mirror lines representing mirror planes going through the horizontal crystallographic axes red lines are absent and that the mirror lines representing mirror planes that bisect the angles between the horizontal crystallographic axes are still present as demanded by trigonal hemihedric In the next figure we will finally depict the derivation of the whole ditrigonal bipyramid that has a ditrigon as its equatorial plane from the holohedric dihexagonal bipyramid Figure 3g Merohedric derivation of the Ditrigonal Bipyramid from the holohedric Dihexagonal Bipyramid Two such Ditrigonal Pyramids are possible depending on which face pairs are suppressed while letting the others extend till they meet The result is depicted in Figure 3h Figure 3h A Ditrigonal Bipyramid derived from the white faces of the pyramid of Figure 3g Not necessarily drawn on the same scale which implies that we could imagine a parameter different from 1 put in front of the P of the Naumann symbol Of course the ditrigonal bipyramid is just a hexagonal bipyramid but its equatorial plane is not a regular hexagon Although its sides are equal its corners represent two angles three equal more or less obtuse angles and three equal more acute angles See Figure 3f or 3c The above figure clearly shows the equality of the horizontal edges bordering the equatorial plane The next Figures show ditrigonal bipyramids where the inequality of the corners of the ditrigon equatorial plane is clearly pronounced Recall that several dihexagons are possible and consequently several possible ditrigons and so several corresponding ditrigonal bipyramids Figure 3i Another possible Ditrigonal Bipyramid Figure 3j Another possible Ditrigonal Bipyramid Figure 3k Yet another Ditrigonal Bipyramid Left image drawn with main crystallographic axis and bisectors of the horizontal crystallographic axes The trigonal aspect is well expressed i e the ditrigonal pyramid approaches a trigonal pyramid From the holohedric protoprism hexagonal type I prism we can derive the trigonal prism when we subject it to trigonal hemihedric In fact two such trigonal prisms are possible See the following Figures Let us first expound the generation of this Form the trigonal prism in terms of horizontal sections Suppressing the mirror planes that go through the horizontal crystallographic axes can be expressed by alternate suppression of the sides of a regular hexagon representing a horizontal section through a protoprism See Figure 4 Figure 4 From a horizontal section through the holohedric Hexagonal Protoprism can be constructed the corresponding horizontal section through the Trigonal Prism by alternate suppression and extension of its sides Two such horizontal sections can in fact be constructed depending on which sides are allowed to survive and extend The red diagonal lines converging in a common point are the horizontal crystallographic axes Figure 4a Generation of the Trigonal Prism from the holohedric Hexagonal Protoprism Two Trigonal Prisms are possible depending on which faces are suppressed and the others allowed to extend till they meet Figure 4b Construction of a Trigonal Prism from the red faces of the Protoprism Figure 4c A Trigonal Prism from the red faces of the Protoprism of Figure 4a From the holohedric deuteroprism hexagonal type II prism we can derive the trigonal hemihedric type II hexagonal prism when we subject it to trigonal hemihedric As can be seen in the next Figure the new Form will have the same external shape as the holohedric deuteroprism Figure 5 The holohedric Deuteroprism retains its shape when subjected to trigonal hemihedric From the holohedric dihexagonal prism we can derive the ditrigonal prism when we apply trigonal hemihedric to it The next Figure illustrates the derivation in terms of a horizontal section Figure 6 Derivation of two Ditrigons from a horizontal section through the holohedric Dihexagonal Prism The red diagonal lines are the horizontal crystallographic axes The next Figure shows the whole holohedric Dihexagonal Prism and its partition according to trigonal hemihedric Figure 6a Partition of the holohedric Dihexagonal Prism Two possible Ditrigonal Prisms can be derived depending on which face pairs are suppressed and the remaining ones allowed to extend till they meet Figure 6b A Ditrigonal Prism derived from the red faces of the Dihexagonal Prism of Figure 6a From the holohedric basic pinacoid we can derive the trigonal hemihedric basic pinacoid when we subject it to trigonal hemihedric The external shape will remain the same i e our new Form still consists of two parallel horizontal faces Like the prisms it is an open Form which means that it can exist in real crystals only in combination with other Forms of this Class that cause it to be closed Figure 7 depicts this pinacoid Figure 7 The trigonal hemihedric Basic Pinacoid The colors indicate the mirror symmetries involved This concludes our derivation of all the Forms of the Ditrigonal bipyramidal Crystal Class by the merohedric approach All these Forms can enter in combinations with each other in real crystals FACIAL APPROACH We will now derive those same Forms by subjecting the basic faces compatible with the Hexagonal Crystal System one by one to the symmetry operations of the present Class the Ditrigonal bipyramidal Crystal Class In Part One we found the following seven basic faces compatible with the Hexagonal Crystal System a a a c 2a 2a a c 3 2 a 3a a c 3 2 a 3a a c a a a c 2a 2a a c a a a c The stereographic projection of the symmetry elements of this Class is given in Figure 8 Figure 8 Stereogram of the symmetry elements of the Ditrigonal bipyramidal Crystal Class and of all the faces of the most general Form The face a a a c generates a trigonal bipyramid when we subject it to the symmetry elements of the present Class These symmetry elements are listed above When we subject the face to the 3 fold rotation axis and reflect the result in the equatorial plane of symmetry we obtain the trigonal bipyramid The other symmetry

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  • Hexagonal Crystal System IX
    hemimorphous trigonal hemihedric Ditrigonal Prism from the trigonal hemihedric Ditrigonal Prism From the trigonal hemihedric basic pinacoid we can derive the pedion The latter is a single horizontal face by suppressing the equatorial mirror plane of the basic pinacoid we get in fact two such faces i e two independent horizontal faces an upper one and a lower one Such a pedion can close a monopyramid by forming its base Two such pedions an upper and a lower one could close a prism See Figure 7 7a and 7b Figure 7 The trigonal hemihedric Basic Pinacoid derived from the holohedric Basic Pinacoid will be subjected to hemimorphy resulting in two Pedions Monohedrons which are depicted in the next two Figures The system of four crystallographic axes is indicated red lines Figure 7a Upper Pedion derived from the trigonal hemihedric Basic Pinacoid by suppression of the equatorial mirror plane expressed by the removal of the lower face Figure 7b Lower Pedion derived from the trigonal hemihedric Basic Pinacoid by suppression of the equatorial mirror plane expressed by the removal of the upper face This concludes our derivation of all the Forms of the Ditrigonal pyramidal Crystal Class by means of the merohedric approach All these Forms can enter in combinations with each other in real crystals FACIAL APPROACH We will now derive those same Forms by subjecting the basic faces compatible with the Hexagonal Crystal System one by one to the symmetry operations of the present Class the Ditrigonal pyramidal Crystal Class In Part One we found the following seven basic faces compatible with the Hexagonal Crystal System a a a c 2a 2a a c 3 2 a 3a a c 3 2 a 3a a c a a a c 2a 2a a c a a a c The stereographic projection of the symmetry elements of this Class is given in Figure 8 Figure 8 Stereogram of the symmetry elements of the Ditrigonal pyramidal Crystal Class and of all the faces of the most general Form The face a a a c generates a trigonal monopyramid when subjected to the symmetry elements of the present Class The action of the 3 fold rotation axis multiplies the face by three Then the mirror plane symmetry is already implied i e it does not add any new faces anymore See Figure 9 Figure 9 1 Position of the face a a a c in the stereographic projection of the symmetry elements of the Ditrigonal pyramidal Crystal Class 2 Stereogram of the generated Trigonal Monopyramid The face 2a 2a a c generates a hemimorphous trigonal hemihedric Type II hexagonal monopyramid when subjected to the symmetry elements of the present Class The action of the 3 fold rotation axis multiplies this face by three Each of these faces is then duplicated by a nearby mirror plane See Figure 10 Figure 10 1 Position of the face 2a 2a a c in the stereographic projection of the symmetry elements of the Ditrigonal pyramidal Crystal

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  • Hexagonal Crystal System X
    and present in the Class under consideration the symmetry of which is 6 which is equivalent to 3 m Figure 3b Construction of a Trigonal Bipyramid ABCDE from a Ditrigonal Bipyramid by alternately suppressing faces and allowing to extend the others Figure 3c Result of the construction in Figure 3b of a Trigonal Tetartohedric Type III Trigonal Bipyramid From the trigonal prism we can derive the trigonal tetartohedric type I trigonal prism when we subject it to pyramidal hemihedric As can be seen in the next Figure the external shape does not change but of course the symmetry is lowered Figure 4 The trigonal hemihedric Trigonal Prism does not change its shape when subjected to pyramidal hemihedric So the resulting trigonal tetartohedric Form is still a Trigonal Prism From the trigonal hemihedric type II hexagonal prism we can derive the trigonal tetartohedric type II trigonal prism As can be seen in the next Figure there is a halving of the number of faces involved which here implies a transformation of a hexagonal prism into a trigonal prism Figure 5 Derivation of the trigonal tetartohedric Type II Trigonal Prism from the trigonal hemihedric Type II Hexagonal Prism Two such prisms can be derived depending on which category of faces the yellow white or the green blue faces will be suppressed and the others allowed to extend till they meet For one such trigonal prism see Figure 5a and 5b Figure 5a Construction of a trigonal tetartohedric Type II Trigonal Prism from the green blue faces of the prism of the right image of Figure 5 Figure 5b Result of the construction in Figure 5a of a trigonal tetartohedric Type II Trigonal Prism From the ditrigonal prism we can derive a trigonal tetartohedric type III trigonal prism when we apply pyramidal hemihedric to it In Figure 6 we can see that the result is a halving of the number of faces so from a ditrigonal prism a trigonal prism is generated Figure 6 Derivation of the trigonal tetartohedric Type III Trigonal Prism from the trigonal hemihedric Ditrigonal Prism Two such prisms can be derived depending on which faces are suppressed the ones marked red or the ones marked white and the others allowed to extend till they meet Figure 6a Construction of a trigonal tetartohedric Type III Trigonal Prism from the red faces of the Ditrigonal Prism in the right image of Figure 6 Figure 6b Result of the construction in Figure 6a of a trigonal tetartohedric Type III Trigonal Prism From the trigonal hemihedric basic pinacoid we can derive the trigonal tetartohedric basic pinacoid Removal of the remaining vertical mirror planes executing pyramidal hemihedric does not have any effect concerning the shape So the new Form is still one consisting of two horizontal parallel faces and thus a basic pinacoid but its symmetry is lowered accordingly See Figures 7 and 7a Figure 7 The trigonal hemihedric Basic Pinacoid from which the trigonal tetartohedric Basic Pinacoid will be derived Figure 7a Figure

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  • Hexagonal Crystal System XI
    can derive the ogdohedric type I trigonal prism by applying hemimorphy See Figure 4 Figure 4 An ogdohedric Type I Trigonal Prism The application of hemimorphy is indicated by means of the coloring of the front face It is devided in two sections an upper and lower one expressing the suppression of the equatorial mirror plane This suppression does not however entail the face s disappearance The same goes for the other two faces so the external shape is not affected by hemimorphy From the trigonal tetartohedric type II trigonal prism we can derive the ogdohedric type II trigonal prism by applying hemimorphy See Figure 5 Figure 5 An ogdohedric Type II Trigonal Prism Like the type I prism the hemimorphy does not effect a different shape but the symmetry content is lowered accordingly From the trigonal tetartohedric type III trigonal prism we can derive the ogdohedric type III trigonal prism by applying hemimorphy See Figure 6 Figure 6 An ogdohedric Type III Trigonal Prism Also in this case hemimorphy does not affect the external shape but lowers the symmetry content accordingly From the trigonal tetartohedric basic pinacoid Figure 7a Part Ten we can derive the ogdohedric pedion when applying hemimorphy In this case the trigonal tetartohedric basic pinacoid which consists of two horizontal parallel faces loses its equatorial mirror plane resulting in two independent pedions A pedion is one horizontal face Like the prisms the basic pinacoid not belonging to this Crystal Class and the monopyramids it is an open Form See Figure 7 Figure 7 An ogdohedric Pedion derived by hemimorphy from the trigonal tetartohedric Basic Pinacoid Two such pedions are possible an upper one as depicted here and a lower one independent of each other This concludes our derivation of all the Forms of the Trigonal pyramidal Crystal Class by means of the merohedric approach All these Forms can enter in the present case SHOULD enter in combinations with each other in real crystals FACIAL APPROACH We will now derive those same Forms by subjecting the basic faces compatible with the Hexagonal Crystal System one by one to the symmetry operations of the present Class the Trigonal pyramidal Crystal Class In Part One we found the following seven basic faces compatible with the Hexagonal Crystal System a a a c 2a 2a a c 3 2 a 3a a c 3 2 a 3a a c a a a c 2a 2a a c a a a c The stereographic projection of the symmetry elements of this Class is given in Figure 8 Figure 8 Stereogram of the symmetry elements of the Trigonal pyramidal Crystal Class and of all the faces of the most general Form The face a a a c generates an ogdohedric type I trigonal monopyramid when subjected to the symmetry elements of the present Class The only symmetry element of this class is a 3 fold rotation axis coincident with the vertical crystallographic axis the c axis This 3 fold rotation axis triples

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  • Hexagonal Crystal System XII
    holohedric Dihexagonal Prism does not alter its shape when subjected to rhombohedric hemihedric but loses some symmetry From this rhombohedric hemihedric Dihexagonal Prism we will derive the trapezohedric tetartohedric Ditrigonal Prism See figure 5a Figure 5a Partition of the rhombohedric hemihedric Dihexagonal Prism indicating the derivation of the trapezohedric tetartohedric Ditrigonal Prism The coloring indicates the symmetries invoved Two face categories 1 the green blue faces and 2 the yellow white faces indicate the partition that expresses the trapezohedric hemihedric being applied to the rhombohedric hemihedric Dihexagonal prism The next Figure shows the derivation of a trapezohedric tetartohedric Ditrigonal Prism Figure 5b Initial stage of the construction indicated by the blue lines of a trapezohedric tetartohedric Ditrigonal Prism by the extention of the faces of the green blue face category while suppressing the faces of the yellow white face category Figure 5c Result of the construction of a trapezohedric tetartohedric Ditrigonal Prism From the rhombohedric hemihedric basic pinacoid we can derive the trapezohedric tetartohedric Basic Pinacoid No change in external shape is involved See Figure 6 and 6a Figure 6 The holohedric Basic Pinacoid does not alter its shape when subjected to rhombohedric hemihedric but loses some symmetry From this rhombohedric hemihedric Basic Pinacoid will be derived the trapezohedric tetartohedric Basic Pinacoid Figure 6a The trapezohedric tetartohedric Basic Pinacoid The coloring symbolizes the symmetries invoved This concludes our derivation of all the Forms of the Trigonal trapezohedric Crystal Class by means of the merohedric approach All these Forms can enter in combinations with each other in real crystals FACIAL APPROACH We will now derive those same Forms by subjecting the basic faces compatible with the Hexagonal Crystal System one by one to the symmetry operations of the present Class the Trigonal trapezohedric Crystal Class In Part One we found the following seven basic faces compatible with the Hexagonal Crystal System a a a c 2a 2a a c 3 2 a 3a a c 3 2 a 3a a c a a a c 2a 2a a c a a a c The stereographic projection of the symmetry elements of this Class is given in Figure 7 Figure 7 Stereogram of the symmetry elements of the Trigonal trapezohedric Crystal Class and of all the faces of the most general Form There are no mirror planes but there are three 2 fold rotation axes coincident with the three horizontal crystallographic axes indicated by straight dashed lines Also a 3 fold rotation axis is present coincident with the vertical crysallographic axis The face a a a c generates a rhombohedron when subjected to the symmetry elements of the present Class The generation can be imagined as follows The face will be duplicated by the nearby 2 fold rotation axis i e an image of it will be found when we rotate the face about this axis by 180 0 Then this face pair consisting of an upper and a lower face that are related to each other like the upper and lower

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  • Orthorhombic Crystal System I
    of the Forms of the Holohedric Division of the Orthorhombic Crystal System There are two types of Forms to be derived closed Forms and open Forms Closed Forms pyramids From the above described Basic Form the primary proto pyramid a number of secondary pyramids can be derived whose axes stand in a rational relationship with those of the primary pyramid And this does not like it does in the Tetragonal and Hexagonal Systems concern only one axis the vertical axis but all three axes because there is no main axis to be found in the Orthorhombic Crystal System This implies that we have three series of derived pyramids Vertical Series Protopyramids The members of this Series are obtained by varying the relative c axis cut off distance Brachy diagonal Series Brachypyramids The members of this Series are obtained by varying the relative a axis cut off distance Macro diagonal Series Macropyramids The members of this Series are obtained by varying the relative b axis cut off distance Let us illustrate these three series Varying the relative c axis cut off distance which is equivalent to varying the derivation coefficient m in a b mc and in mP gives us the Vertical Series the Protopyramids which can be denoted by the Weissian symbol a b mc the Naumann symbol mP and the Miller symbol hhl Figure 4 From the Primary Pyramid P can be derived the Vertical Series the Protopyramids mP by varying the derivation coefficient m Two possible non primary i e secondary Protopyramids are indicated blue and green respectively The Primary Pyramid is a Protopyramid in which m 1 Varying the relative a axis cut off distance which is equivalent to varying the derivation coefficient n in na b c and in Pn gives us the Brachydiagonal Series Brachypyramids Because we can derive such brachypyramids from each protopyramid including the primary protopyramid the general Weissian symbol for a brachypyramid is na b mc and the corresponding Naumann symbol is The Miller symbol is khl In the next Figure I give a brachypyramid derived from the primary protopyramid where m 1 Figure 5 From the Primary Pyramid can be derived a Brachy diagonal Series the primary Brachypyramids by varying the derivation coefficient n of the Brachy axis Also from any non primary Protopyramid i e a non primary member of the Vertical Series a Brachypyramid can be derived in the same way yielding the Brachy diagonal Series Varying the relative b axis cut off distance which is equivalent to varying the derivation coefficient n in a nb c and in Pn gives us the Macrodiagonal Series Macropyramids Because we can derive such macropyramids from each protopyramid including the primary protopyramid the general Weissian symbol for a macropyramid is a nb mc and the corresponding Naumann symbol is The Miller symbol is hkl In the next Figure I give a macropyramid derived from the primary protopyramid where m 1 Figure 6 A primary Macropyramid blue derived from the primary Protopyramid Also from any Protopyramid of the Vertical Series can a Macropyramid be derived by extention of the macro axis Open Forms vertical and horizontal prisms and pinacoids From the above pyramids protopyramids brachypyramids macropyramids we can to begin with derive the corresponding vertical rhombic prisms The Rhombic Protoprism The Brachyprisms The Macroprisms When we let the derivation coefficient m in the protopyramids a b mc become infinite i e when we make the c axis cut off distance of infinite length the result will be a vertical prism the protoprism or as it can be called a primary prism that can be denoted by a b c in which stands for infinity P or 110 There is only one such protoprism its faces cutting off unit pieces from the brachy axis as well as from the macro axis while being parallel to the vertical axis Its horizontal section is a rhombus this is because the unit pieces of the brachy and macro axis do not have equal lengths The next Figure depicts the protoprism Figure 7 From any Rhombic Protopyramid a b mc can be derived the Rhombic Protoprism a b c by making m infinitely large From the pyramids of the Brachydiagonal Series na b mc a second category of vertical prisms the brachyprisms na b c Miller symbol kh0 can be derived when we again make the derivation coefficient m infinitely large Figure 8 From any Rhombic Brachypyramid na b mc can be derived a Rhombic Brachyprism na b c by making m infinitely large Here we derive such a Brachyprism directly from the Rhombic Protoprism by extention of the cut off distances of the brachy axis Like in the Rhombic Protoprism the horizontal section of the Brachyprisms is a rhombus See Figure 9 Figure 9 Horizontal sections through a Rhombic Protoprism yellow and a possible Rhombic Brachyprism blue Both are rhombi From the pyramids of the Macrodiagonal Series a nb mc a third category of vertical prisms the macroprisms a nb c hk0 can be derived when we again make the derivation coefficient m infinitely large Figure 10 From any Rhombic Macropyramid a nb mc can be derived a Rhombic Macroprism a nb c by making m infinitely large Here we derive such a Macroprism directly from the Rhombic Protoprism by extention of the cut off distances of the macro axis Also the horizontal sections of the macroprisms are rhombi as the next Figure illustrates Figure 11 Horizontal sections through the Rhombic Protoprism yellow and a possible Macroprism blue Further we can derive two categories of horizontal rhombic prisms that often are also called domes From the brachypyramids na b mc we can derive the corresponding brachydomes a b mc 0hl by making n infinite which means that we get horizontal prisms with their faces parallel to the brachy axis A primary brachydome a b c 011 is derived from the primary rhombic protopyramid a b c From the macropyramids a nb mc we can derive the corresponding macrodomes a b mc h0l by making n infinite which means that we get horizontal prisms with their faces parallel to the macro axis A primary macrodome a b c is derived from the primary rhombic protopyramid a b c The next Figures illustrate the derivation of the primary brachydome Figure 12 and of the primary macrodome Figure 13 from the primary rhombic protopyramid Figure 12 Derivation of the primary Brachydome in which m 1 from the primary Protopyramid It is an open Form front side and back side open the straight lines bordering its front and rear ends should not suggest a front and back face consisting of four faces parallel to the brachy axis It is a horizontal rhombic prism Figure 13 Derivation of the primary Macrodome in which m 1 from the primary Protopyramid It is an open Form left side and right side open the straight lines bordering its left and right ends should not suggest a left and right face consisting of four faces parallel to the macro axis It is a horizontal rhombic prism Finally we can derive three more Forms of this Crystal Class the pinacoids They are Forms in which each face intersects only one and the same crystallogaphic axis The Brachy Pinacoid The Macro Pinacoid The Basic Pinacoid From the brachydomes a b mc or from the primary brachydome a b c for that matter we can derive the brachy pinacoid a b c 010 by letting m which equals 1 in the primary brachydome become infinite It consists of two vertical faces parallel to each other and to the brachy axis See Figure 14 15 and 16 To show the derivation let us first depict the above constructed primary brachydome as it is all by itself The Figure shows a shortened version of it which is immaterial because the length of the horizontal prism is not determined Figure 14 The primary Brachydome The red solid line is the brachy axis From this brachydome we can now derive the brachy pinacoid Figure 15 Derivation of the Brachy Pinacoid from the primary Brachydome The red solid line is the brachy axis Figure 16 The Brachy Pinacoid This Form consists of two vertical faces parallel to the brachy axis Like the other pinacoids and all the prisms the brachy pinacoid is an open Form and can only exist in real crystals when combined with Forms of this Crystal Class such that the combination is a closed structure From the macrodomes a b mc or from the primary macrodome a b c for that matter we can derive the macro pinacoid a b c 100 by letting m which is equal to 1 in the primary macrodome become infinite It consists of two vertical faces parallel to each other and to the macro axis See Figure 17 18 and 19 Figure 17 Derivation of the Macro Pinacoid from the primary Macrodome The red lines indicate the crystallographic axes Figure 18 The Macro Pinacoid It consists of two vertical faces parallel to the macro axis Finally the basic pinacoid can be derived from either the brachydome or the macrodome by letting m which is equal to 1 in the primary domes become zero The result will be a horizontal face pair parallel to the horizontal crystallographic axes the brachy axis and the macro axis The Weissian symbol for this Form is a b c We have let the derivation coefficient m become zero resulting in the faces a b c to become parallel to the plane of the horizontal crystallographic axes So in fact these faces should coincide resulting in just one face with a zero coefficient of c But because m 0 only means that the face is horizontally oriented without its position being determined therewith we can still see them as two faces horizontal faces and place them at unit distance above and below the origin of the system of crystallographic axes causing the Weissian symbol of this Form to be a b c and not a b 0c Only then the resulting Form two horizontal equivalent faces complies with the symmetry content of the present Class The Naumann symbol is 0P and the Miller symbol is 001 In the next Figure we will derive the basic pinacoid from the primary brachydome Figure 19 Derivation of the Basic Pinacoid from the primary Brachy Dome Figure 20 The Basic Pinacoid It consists of two horizontal faces parallel to the brachy and macro axes We now have derived all the Forms of the Rhombic bipyramidal Crystal Class Holohedric Division These Forms all comply with the symmetry of this Class 2 m 2 m 2 m And all these Forms can enter in combinations with each other in real Crystals The faces that represent each Form are with respect to the Weissian symbolism placed between brackets as is done above for example a b c As such they are Forms These faces themselves thus without brackets i e as faces are then the eleven Basic Faces compatible with the Orthorhombic Crystal System To sum up these basic faces we get a b mc na b mc a nb mc a b c na b c a nb c a b mc a b mc a b c a b c a b c They represent all possible configurations of derivation coefficients among the three crystallographic axes Subjecting each of these basic faces to the symmetry elements of the present Crystal Class which means generating new faces according to the symmetry demands of that Class imposed on the resulting face configuration Form i e the symmetry that this configuration should have according to those demands will yield the above Forms i e the Forms of the Holohedric Division Subjecting those same basic faces to the symmetry elements of the other orthorhombic Crystal Classes will result in the Forms of those Classes The face a b c is the primary face yielding the primary rhombic protopyramid a b c when subjected to the symmetry elements of the present Class This face from which all listed faces are derivations is taken from a conspicuous face of some real crystal belonging to the present Crystal Class That the above list of basic faces is complete can be shown by the location of their poles in the stereographic projection of the symmetry elements of the present Class Figure 21 The lower right quadrant of the stereographic projection of the symmetry elements of the Rhombic bipyramidal Class and the possible locations of faces face categories a b mc na b mc a nb mc a b c na b c a nb c a b mc a b mc a b c a b c a b c In the above Figure the straight solid black lines are vertical mirror planes perpendicular to each other The solid circumference of the the circle one quarter shown represents the equatorial mirror plane The black solid ellipses signify 2 fold rotation axes Two of them are horizontal and coincide with the two horizontal crystallogaphic axes the vertically drawn axis is the brachy axis the horizontally drawn axis is the macro axis The third 2 fold rotation axis is vertical and coincides with the c axis it is perpendicular to the plane of the drawing 1 represents the faces face category a b mc They can vary along the line bisecting the quadrant in two equal halves They all cut off unit distances of the brachy and macro axes and can vary with respect to the cut off distance of the c axis All these distances are given in the form of a ratio of the three axial cut off distances If m 1 then we have the primary basic face that generates a primary rhombic protopyramid a bipyramid when subjected to the symmetry elements of the present Class Other finite non zero values of m will yield derived protopyramids 2 represents the faces face category na b mc Such a face can be everywhere inside the upper sector of the quadrant When subjected to the symmetry elements of the present Class such a face will yield a brachy pyramid 3 represents the faces face category a nb mc Such a face can be everywhere in the lower sector of the quadrant When subjected to the symmetry elements of the present Class such a face will yield a macro pyramid 4 represents the face face category a b c It is vertical and cuts off unit distances from the brachy and macro axes Only one such face is possible When it is subjected to the symmetry elements of the present Class it will yield a protoprism 5 represents the faces face category na b c They are vertical When such a face is subjected to the symmetry elements of the present Class it will generate a brachy prism It can vary along the circle segment bordering the upper sector of the quadrant 6 represents the faces a nb c They are also vertical When such a face is subjected to the symmetry elements of the present Class it will generate a macro prism It can vary along the circle segment bordering the lower sector of the quadrant 7 represents the faces a b mc They are parallel to the brachy axis When such a face is subjected to the symmetry elements of the present Class it will generate a brachydome It can vary along the the horizontal line in the drawing 8 represents the faces a b mc They are parallel to the macro axis When such a face is subjected to the symmetry elements of the present Class it will generate a macrodome It can vary along the the vertical line in the drawing 9 represents the face a b c It is vertical and parallel to the brachy axis Only one such face is possible When this face is subjected to the symmetry elements of the present Class it will generate a brachy pinacoid 10 represents the face a b c It is vertical and parallel to the macro axis Only one such face is possible When this face is subjected to the symmetry elements of the present Class it will generate a macro pinacoid 11 represents the face a b c It is horizontal Only one such face is possible When this face is subjected to the symmetry elements of the present Class it will generate a basic pinacoid Remark about the stereographic projection of Orthorhombic crystals In contrast with the Isometric Tetragonal and Hexagonal Systems all the crystallographic axes of the Orthorhombic System are non equivalent This means that the absolute magnitude of the unit intersection distances are not necessarily the same for all three axes So when we have a face a b c 111 of an orthorhombic crystal then the absolute cut off distances with respect to the three axes are not necessarily the same although the equality of the derivation coefficients and also of the corresponding Miller indices might suggest so Figure 21a shows the stereographic projection of some faces of an orthorhombic crystal where indeed we have to do with unequal cut off distances connected with equal derivation coefficients and Miller indices Figure 21a Stereographic projection of the faces a b c 111 a b c 111 where 1 means a negative Miller index written as a 1 with a score above it in the literature a b c 1 1 1 a b c 1 1 1 a b c 110 and a b c 1 1 0 of an orthorhombic crystal These faces all have equal derivation coefficients respectively Miller indices with a value equal to 1 negative or positive The absolute cut off distances are however not the same in this case as is to be expected for orthorhombic crystals So those faces do not lie on the bisector given in blue of the relevant quadrants of the projection plane They do lie however on

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  • Orthorhombic Crystal System II
    holohedric Brachy Pinacoid can be derived the hemimorphous Brachy Pinacoid when applying hemimorphy to it The external shape remains the same From the holohedric macro pinacoid can be derived the hemimorphous macro pinacoid when hemimorphy is applied to it The external shape remains the same See Figure 10 Figure 10 From the holohedric Macro Pinacoid can be derived the hemimorphous Macro Pinacoid when applying hemimorphy to it The external shape remains the same From the holohedric basic pinacoid can be derived two pedions that are single faces monohedra an upper one and a lower one when hemimorphy is applied to it So here we have a Form consisting of just one face See Figure 11 Figure 11 From the holohedric Basic Pinacoid which itself consists of two horizontal faces can be derived the hemimorphous Pedion Monohedron when applying hemimorphy to it This Pedion is an open Form it is just one horizontal face This concludes our derivation of all the Forms of the Rhombic pyramidal Crystal Class by the merohedric approach All these Forms can enter in combinations with each other in real crystals FACIAL APPROACH We will now derive those same Forms by subjecting the basic faces compatible with the Orthorhombic Crystal System one by one to the symmetry operations of the present Class the Rhombic pyramidal Crystal Class In Part One we found the following eleven basic faces compatible with the Orthorhombic Crystal System a b mc na b mc a nb mc a b c na b c a nb c a b mc a b mc a b c a b c a b c The stereographic projection of the symmetry elements of this Class is given in Figure 12 Figure 12 Stereogram of the symmetry elements of the Rhombic pyramidal Crystal Class and of all the faces of the most general Form There are two mirror planes perpendicular to each other indicated by straight solid lines each containing one of the two horizontal crystallographic axes Further there is one vertical 2 fold rotation axis indicated by a small solid ellips There is no horizontal i e perpendicular to the 2 fold rotation axis mirror plane indicated by the dashing of the circumference of the projection plane The face a b mc generates the upper half of a rhombic protopyramid a rhombic monopyramid of the Vertical Series when subjected to the symmetry elements of the present Class The face is duplicated by one of the two mirror planes and the resulting face pair is again duplicated yielding four faces that make up a monopyramid The 2 fold rotation axis is then implied See Figure 13 and 13a Figure 13 1 Position of the face a b mc in the stereographic projection of the symmetry elements of the Rhombic pyramidal Crystal Class 2 The face is reflected in one of the two mirror planes giving two faces and these are in turn reflected in the other mirror plane resulting in four faces making up a monopyramid Figure 13a In this Figure it is explained why the 2 fold rotation axis is implied by the other symmetry elements We have drawn two vertical mirror planes m1 and m2 perpendicular to each other Their line of intersection coincides in our drawing with a vertical 2 fold rotation axis perpendicular to the plane of the drawing When motif a in the form of a comma is relected in m1 the image a is generated When this newly generated motif is reflected in m2 a motif a is generated But when we take motif a and subject it to the action of the 2 fold rotation axis a rotation about this axis by 180 0 we directly obtain motif a So we see that this 2 fold rotation axis is already implied by the presence of the two mirror planes The face na b mc generates the upper half of a rhombic brachypyramid i e a monopyramid of the Brachydiagonal Series when subjected to the symmetry elements of the present Class The face is duplicated by one of the mirror planes yielding two faces This face pair is then duplicated by the other mirror plane resulting in four faces making up a monopyramid The 2 fold rotation axis is then implied See Figure 14 Figure 14 1 Position of the face na b mc in the stereographic projection of the symmetry elements of the Rhombic pyramidal Crystal Class 2 The face is reflected in one of the two mirror planes giving two faces and these are in turn reflected in the other mirror plane resulting in four faces making up a monopyramid The face a nb mc will generate the upper half of a rhombic macropyramid i e a monopyramid of the Macrodiagonal Series when subjected to the symmetry elements of the present Class The face will be duplicated by one of the two mirror planes yielding a face pair which is in turn duplicated by the other mirror plane resulting in four faces making up a monopyramid The 2 fold rotation axis is then implied See Figure 15 Figure 15 1 Position of the face a nb mc in the stereographic projection of the symmetry elements of the Rhombic pyramidal Crystal Class 2 The face is reflected in one of the two mirror planes giving two faces and these are in turn reflected in the other mirror plane resulting in four faces making up a monopyramid The face a b c is vertical It generates the hemimorphous rhombic protoprism when subjected to the symmetry elements of the present Class The face is duplicated by one of the mirror planes and the resulting face pair is in turn duplicated by the other mirror plane giving a Form consisting of four vertical faces the rhombic protoprism The 2 fold rotation axis is implied See Figure 16 Figure 16 1 Position of the face a b c in the stereographic projection of the symmetry elements of the Rhombic pyramidal Crystal Class 2 The

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