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  • Tetragonal Crystal System VI
    the top face and the bottom face are not symmetrical counterparts anymore I e there is no mirror plane parallel to them no equatorial mirror plane because vertically below a yellow area of the top face we find a brown area of the bottom face and below a brown area of the top face we find a yellow area of the bottom face Now all the brown faces together form again a basic pinacoid and the yellow faces together form another pinacoid but these two pinacoids are exactly identical So now we can say that the trapezohedric hemihedric generates from the holohedric basic pinacoid a trapezohedric hemihedric basic pinacoid Figure 7 The Basic Pinacoid when subjected to trapezohedric hemihedric yields again a Basic Pinacoid but one with lower symmetry All these Forms can engage in combinations FACIAL APPROACH We will now derive those same Forms by subjecting the basic faces compatible with the Tetragonal Crystal System one by one to the symmetry operations of the present Class the Tetragonal trapezohedric Crystal Class Recall that the basic faces were the following a a c a a c a na mc a a c a a c a na c a a c The stereogram of the symmetry elements of the present Class is Figure 8 Stereogram of the symmetry elements of the Tetragonal trapezohedric Crystal Class and all the faces of the most general Form The face a a c the unit face of the Tetragonal Crystal System cuts off unit pieces from all three crystallographic axes Its position in the stereographical projection of the symmetry elements of the present Class is shown in Figure 9 Figure 9 Position of the face a a c in the stereographic projection of the symmetry elements of the Tetragonal trapezohedric Crystal Class When we subject this face to the 2 fold rotation axis which goes through it a second face will be generated exactly below the initial one When we next apply the 4 fold rotation axis this pair will be multiplied four times resulting in a type I tetragonal bipyramid Figure 1 The initial face is not allowed to be symmetric because mirror planes are excluded in this Class So also their multiplication by rotation does not generate any mirror symmetry The stereographic projection of the type I tetragonal bipyramid is shown in Figure 9a Figure 9a Stereogram of the trapezohedric hemihedric Type I Tetragonal Bipyramid A small red circle centered with a red point indicates the coinciding projections of a lower face and an upper face respectively The face a a c is parallel to the east west crystallographic axis It cuts off unit pieces from the other two axes Its position in the stereographic projection in indicated in Figure 10 Figure 10 Position of the face a a c in the stereographic projection of the symmetry elements of the Tetragonal trapezohedric Crystal Class When we subject this face to the 2 fold rotation axis which goes through it a second face

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  • Tetragonal Crystal System VII
    they meet recovering the tetragonal prism but one with lower symmetry the Tetartohedric Type II Prism The sphenoidic hemihedric ditetragonal prism becomes a tritoprism type III tetragonal prism See Figure 6 Figure 6 The sphenoidic hemihedric Ditetragonal Prism looses the mirror planes that go between the horizontal crystallographic axes Two possible tetartohedral tetragonal Tritoprisms result one from the four red faces and one from the four grey faces The sphenoidic hemihedric basic pinacoid remains a basic pinacoid See Figure 7 Figure 7 Derivation of the tetartohedric Basic Pinacoid from the sphenoidic hemihedric Basic Pinacoid By suppressing the faces of one color and letting extend the faces of the other color in the right image we get this Pinacoid It consists of two parallel horizontal faces As the figure shows these faces are not mirror reflections of each other This concludes our derivation of all the Forms of the Tetragonal disphenoidic Crystal Class by the merohedric approach These Forms can engage in combinations with each other and thus it is possible that several such Forms appear in one and the same crystal FACIAL APPROACH We will now derive those same Forms by subjecting the basic faces compatible with the Tetragonal Crystal System one by one to the symmetry operations of the present Class the Tetragonal disphenoidic Crystal Class Recall that the basic faces were the following a a c a a c a na mc a a c a a c a na c a a c The only symmetry element of this Class is the 4 fold roto inversion axis 4 Because the Class does not possess any mirror plane nor any 2 fold rotation axis the motifs which figure in the stereographic projections are themselves NOT symmetrical i e each motif does not itself possess any mirror plane We could represent such a motif as a comma but in images of stereographic projections one normally does not so The motifs are represented there by dots small circles plus signs etc where the shape of such symbols is of no relevance The motifs in our case are possible crystal faces and they are not each for themselves symmetrical which is detectable by means of the shape and orientation of etch pits on their surface or by means of other physical features When indeed the motifs faces are not themselves symmetrical the actions of the 4 fold roto inversion axis 4 do not imply any other symmetries thus no mirror planes or 2 fold rotation axes appear As the only available symmetry element in this Class the 4 fold roto inversion axis can be depicted as in the next Figure in which the asymmetry of the motifs is indicated Figure 8 The four fold roto inversion axis as the only symmetry element of the Tetragonal disphenoidic Crystal Class The action of this axis is rotation by 90 0 about its axis followed by inversion through a point on that axis but recall that its action can also and totally equivalently be

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  • Hexagonal Crystal System I
    2 So the Miller symbol for this Form is 112 2 The Naumann symbol is P2 i e the c intercept has unit length expressed by the absence of any number before P while the non unit intercepts with respect to the horizontal axes are of length 2 expressed by the 2 after the P To derive the deuteropyramid from the protopyramid a rotation of the latter by 30 0 is not sufficient It is even impossible in the world of crystals because if we just rotate the protopyramid relative to the system of crystallographic axes namely rotating it about the c axis then the derivation coefficient belonging to the a 3 axis becomes irrational namely half of the square root of three 1 2 square root 3 Let us explain this by considering Figure 7 and 8 In Figure 7 the hexagonal primary protopyramid OA OB 1 the unit distance If we rotate this pyramid by 30 0 clockwise about the vertical axis and if we imagine that we then get the pyramid of Figure 8 then OC in Figure 8 would be equal to OB in Figure 7 i e it would be 1 But then OD would be 1 2 square root 3 0 866 See Figure 9 Figure 9 Horizontal crystallographic axes and the traces of three faces black one face and parts of two adjacent faces of the Deuteropyramid In Figure 9 the angle COL is 60 0 OL OC So OCL is an equilateral triangle This implies that CD 1 2 CL 1 2 OC So if OC 1 the unit distance OB in Figure 7 then 1 2 2 OD 2 1 2 and this is equivalent to 1 4 OD 2 1 So OD 2 3 4 Thus OD square root 3 square root 4 which is equal to 1 2 square root 3 0 866 Remark From Figure 9 we can also see that KOD is half an equilateral triangle which implies that OK 2OD In the same way OM 2OD holds So if OD is set equal to one i e equal to OB of Figure 7 in other words if OD is the unit distance unit intercept then OK 2 and OM 2 establishing the Weiss symbol for the face CL and giving 2a 2a a c for the deuteropyramid This means if OC 1 that the coefficient for the a 3 axis has become irrational But it is proven that the derivation coefficients can never become irrational this impossibility is a direct consequence of the periodic internal structure of crystals and so we cannot derive the deuteropyramid from the protopyramid by just rotating the latter 30 0 We must enlarge the equatorial plane such that OD in figure 8 will become equal to 1 again and thus equal to OB in Figure 7 So we must enlarge the equatorial plane by a linear factor of 1 1547 because 1 1547 times 0 866 equals 1 Thus if the intercept with respect to the c axis remains the same then this pyramid is not congruent with the protopyramid but is a little more obtuse When we would proportionally enlarge the c intercept the two pyramids still would not be congruent but they are then similar pyramids they have the same shape but differ in size However in this case the deuteropyramid with enlarged c intercept is not a primary deuteropyramid anymore because of the enlargement mentioned the distance OF in Figure 7 is the unit intercept with respect to the c axis and the corresponding intercept in the similar deuteropyramid where the intercept was enlarged is longer i e longer than the unit intercept for this axis Let s recapitulate and draw some conclusions A deuteropyramid is not similar to and thus certainly not congruent with the primary protopyramid P when this deuteropyramid is itself primary P2 in which similar means having exactly the same shape But it can be made similar to the primary protopyramid by an appropriate change of length of the c intercept causing the deuteropyramid not to be primary anymore And even when this deuteropyramid is now made similar it still is not congruent with the primary protopyramid So if a deuteropyramid combines with a primary protopyramid its faces must have a definite relationship to those of the protopyramid they cannot just be anywhere in the combination but can only occur at certain definite distances from the faces of the protopyramid with which they combine because the deuteropyramid is not congruent with the protopyramid To fully understand the relationships between the hexagonal proto and deuteropyramids and also the relationship between them and the dihexagonal pyramid a Form yet to be discussed it is useful to consider the equatorial planes of those Forms All such considerations are equally valid for all the yet to be discussed prismatic Forms the protoprism deuteroprism and the dihexagonal prism See Figure 10 11 and 12 Figure 10 Equatorial planes of the hexagonal Protopyramid and Deuteropyramid When we want to derive the Deuteropyramid from the Protopyramid we rotate the Protopyramid red 30 0 about the main crystallogaphic axis the resulting equatorial plane is given in a greenish color and enlarge that resulted equatorial plane till the a 1 intercept is equal to 1 again In the next Figure I show the relationship of the dihexagon with the hexagons equatorial planes of the proto and deuteropyramids As the hexagons are the equatorial planes of the proto and deuteropyramids as well as of the proto and deuteroprisms Forms yet to be discussed the dihexagon is the equatorial plane of the dihexagonal pyramid as well as of the dihexagonal prism Forms yet to be discussed Figure 11 When we construct a dihexagon in the way indicated black we get a special dihexagon namely a regular dodecagon i e a completely regular polygon consisting of twelve equal sides and twelve equal angles corners This is evident from the fact that the green hexagon is just

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  • Hexagonal Crystal System II
    the holohedric dihexagonal prism Seem Figures 11 and 12 Figure 11 The hemimorphic Dihexagonal Prism can be derived from the holohedric Dihexagonal Prism by suppressing the horizontal mirror plane of the latter Because this plane is perpendicular to the faces no loss of faces is effected but the resulting prism has lower symmetry See Figure 12 Figure 12 The hemimorphic Dihexagonal Prism The absence of a horizontal mirror plane is indicated by marks The final Form of this Crystal Class is the pedion It is derived from the holohedric basic pinacoid by suppressing the horizontal mirror plane of this Form Two pedions are in fact derived this way an upper one and a lower one which are independent of each other They i e two pedions can close a hemimorphic prism at its top and at its bottom or one of them can close a prism at say its top while that prism is closed at its bottom by another Form of this Class for instance a monopyramid with its top down like in Figure 6 3c or 3 Figure 13 The hemimorphic Pedion can be derived from the holohedric Basic Pinacoid by suppressing the horizontal mirror plane i e the equatorial mirror plane lying between its faces resulting in suppression of either the yellow face or the red face In this way two independent Pedions an upper one and a lower one can be generated See Figures 14 and 15 Figure 14 A hemimorphic Pedion Monohedron generated from the yellow face of the holohedric Basic Pinacoid of Figure 13 Figure 15 A hemimorphic Pedion Monohedron generated from the red face of the holohedric Basic Pinacoid of Figure 13 This concludes the derivation of all the Forms of the Dihexagonal pyramidal Crystal Class hemimorphy of holohedric by means of the Merohedric Approach 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 Dihexagonal 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 16 Figure 16 Stereogram of the symmetry elements of the Dihexagonal pyramidal Crystal Class and of all the faces of the most general Form The face a a a c is parallel to the a 2 axis Its position in the stereographic projection is given in 1 of Figure 17 and the generation of the Form the type I hexagonal monopyramid is given in 2 of Figure 17 In virtue of the 6 fold rotation axis the face is multiplied six times around the c axis resulting in a six sided monopyramid Application of the remaining symmetry elements of the

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  • Hexagonal Crystal System III
    holohedric hexagonal Deuteroprism By applying pyramidal hemihedric to the holohedric dihexagonal prism we get a hexagonal tritoprism type III hexagonal prism in the same way as in the case of the derivation of the pyramidal hemihedric tritopyramid from the holohedric dihexagonal pyramid See Figure 10 11 and 12 Figure 10 Derivation of the pyramidal hemihedric Type III Hexagonal Prism pyramidal hemihedric Hexagonal Tritoprism from the holohedric Dihexagonal Prism Figure 11 Actual construction of a pyramidal hemihedric Tritoprism by means of suppression of the red faces and extension of the white faces till they meet A similar hexagonal prism can be constructed by suppression of the white faces and extension of the red ones Figure 12 A pyramidal hemihedric Tritoprism generated from the white faces of the Dihexagonal Prism of the Figures 10 and 11 If finally we apply the pyramidal hemihedric to the holohedric basic pinacoid the result the pyramidal hemihedric basic pinacoid will have the same shape but its symmetry is lowered accordingly See Figure 13 Figure 13 Suppression of the vertical mirror planes of the holohedric Basic Pinacoid does not result in a reduction of faces because both faces are perpendicular to those suppressed mirror planes If we let dissappear the colored regions of the faces expressing the suppression of the mirror planes the subsequent extension of the surviving white regions will recover the original faces and thus the Basic Pinacoid but the latter now having a lower symmetry Indicated are the four crystallographic axes red and the bisectors of the equatorial crystallographic axes Again the p in the expression 2p of the Naumann symbol expresses the fact that we here have to do with the pyramidal hemihedric of the Hexagonal Crystal System This concludes our derivation of all the Forms of the Hexagonal Bipyramidal Crystal Class by means of the Merohedric Approach All these Forms can engage in combinations 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 Hexagonal 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 14 Figure 14 Stereogram of the symmetry elements of the Hexagonal bipyramidal Crystal Class and of all the faces of the most general Form The face a a a c cuts off a piece of unit length from the a 1 axis it is parallel to the a 2 axis and cuts off a piece of unit length from the negative end of the a 3 axis and from the c axis as well Its position in the stereographic projection of the symmetry elements of the present

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  • Hexagonal Crystal System IV
    pyramid with its apex down can be generated from the red faces of the bipyramid of Figure 9 Applying hemimorphy to the pyramidal hemihedric type I hexagonal prism results in a hemimorphous pyramidal hemihedric type I hexagonal prism i e with respect to just its shape nothing happens when the equatorial mirror plane is removed See Figure 11 Figure 11 Generation derivation of the hemimorphous pyramidal hemihedric Type I Hexagonal Prism from the pyramidal hemihedric Type I Hexagonal Prism Although its shape does not change its symmetry is lowered by loosing the equatorial mirror plane Applying hemimorphy to the pyramidal hemihedric type II hexagonal prism yields a hemimorphous pyramidal hemihedric type II hexagonal prism Also in this case the external shape remains the same but the symmetry is lowered See Figure 12 Figure 12 Generation of the hemimorphous pyramidal hemihedric Type II Hexagonal Prism from the pyramidal hemihedric Type II Hexagonal Prism Applying hemimorphy to the pyramidal hemihedric hexagonal tritoprism type III hexagonal prism yields a hemimorphous pyramidal hemihedric type III hexagonal prism Also here there is no change in outer shape but there is lowering of symmetry because of the removal of the equatorial mirror plane See Figure 13 Figure 13 Generation of the hemimorphous pyramidal hemihedric Tritoprism hemimorphous pyramidal hemihedric Type III Hexagonal Prism from the pyramidal hemihedric Type III Hexagonal Prism From the pyramidal hemihedric basic pinacoid is derived the hemimorphous pyramidal hemihedric pedion by applying hemimorphy By dropping its equatorial mirror plane the basic pinacoid is dissolved into two independent halves an upper and a lower one i e an upper pedion and a lower pedion So a pedion is a Form consisting of just one horizontal face In combining with a hemimorphous pyramidal hemihedric prism it can close that prism at its bottom or at its top Two pedions can close it at its top and bottom See Figure 14 and 15 Figure 14 The pyramidal hemihedric Basic Pinacoid is derived from the holohedric Basic Pinacoid and supplies the Form from which we will derive the Pedion by applying hemimorphy See Figure 15 Figure 15 The hemimorphous pyramidal hemihedric upper Pedion is derived from the pyramidal hemihedric Basic Pinacoid by suppression of its lower halve in virtue of the removal of the equatorial mirror plane In an analogous way a lower pedion can be derived All these Forms can engage in combinations 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 Hexagonal 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 16 Figure

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  • Hexagonal Crystal System V
    two kinds of horizontal hexagonal sections The sides of these hexagonal sections correspond to faces of the trapezohedron and if we let those faces extend then their mutual intersections determine their shapes In the next Figure we will depict a superposition of the constructions of the two types of hexagons which differ only in their being 30 0 rotated with respect to each other hexagons that both simultaneaously figure in one and the same trapezohedron Figure 7 Superposition of the two constructions of regular hexagons So we have two hexagons only differing in orientation by a 30 0 rotation One such a hexagon represents a horizontal section through the trapezohedron somewhere above the plane which contains the horizontal crystallographic axes the other represents such a section below that plane at a distance from it which is the same but in the other direction as that of the first section The first section hexagon represents the six nothern faces of the trapezohedron The other section hexagon represents the six southern faces of the trapezohedron See Figure 8 Figure 8 The two regular hexagons resulting from the superposition When these faces are allowed to extend they will meet as indicated in Figure 4 forming the hexagonal trapezohedron From the holohedric hexagonal protoprism we can derive the trapezohedric hemihedric type I hexagonal prism by applying trapezohedric hemihedric to it The prism does not change its external shape as can be seen in Figure 9 but looses symmetry because all the mirror planes are suppressed Figure 9 Derivation of the trapezohedric hemihedric Type I Hexagonal Prism from the holohedric Hexagonal Protoprism by suppression of all the mirror planes From the holohedric hexagonal deuteroprism is derived the trapezohedric hemihedric type II hexagonal prism by applying the trapezohedric hemihedric to it As can be seen from Figure 10 the external shape remains unchanged but the symmetry is lowered Figure 10 Derivation of the trapezohedric hemihedric Type II Hexagonal Prism from the holohedric Hexagonal Deuteroprism by suppression of all the mirror planes From the holohedric dihexagonal prism is derived the trapezohedric hemihedric dihexagonal prism by applying the trapezohedric hemihedric to it As can be seen from Figure 11 its shape remains unchanged but the symmetry is lowered Figure 11 Derivation of the trapezohedric hemihedric Dihexagonal Prism from the holohedric Dihexagonal Prism From the holohedric basic pinacoid is derived the trapezohedric hemihedric basic pinacoid So also in this case the new Form assumes the same shape but its symmetry is lowered Figure 12 To obtain a trapezohedric hemihedric Basic Pinacoid we must as for obtaining any trapezohedric hemihedric Form drop all mirror planes of the corresponding holohedric Form The suppression of all the vertical mirror planes is expressed by the colored areas of each face The suppression of the equatorial mirror plane is expressed by the fact that each yellow area of the upper face has a white area directly below it on the lower face and each white area a yellow one This concludes our derivations of all the Forms of the Hexagonal trapezohedric Crystal Class All these Forms can combine 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 Hexagonal 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 13 Figure 13 Stereogram of the symmetry elements of the Hexagonal trapezohedric Crystal Class and of all the faces of the most general Form The periphery of the stereogram i e the circle bordering the plane of projection is dashed meaning that there is no equatorial mirror plane The same is the case with respect to all vertical mirror planes So the 2 fold axes symbolized by small solid ellipses do not lie in any mirror plane nor are they perpendicular to any of them because there aren t any mirror planes The face a a a c will give rise to a trapezohedric hemihedric type I hexagonal bipyramid when subjected to the symmetry elements of this Class First the face will be multiplied six times by virtue of the 6 fold rotation axis The result will be a type I hexagonal monopyramid Further to each of these six faces when subjected to the 2 fold rotation axis that goes right through it will be added a counterface intersecting the c axis at its opposite end opposite when compared with the c intercept of the face that was initially subjected to such a rotation See Figure 14 Figure 14 2 fold rotation i e rotation of 180 0 about the axis of a face intersecting the c axis A second face is generated intersecting the c axis in the opposite direction This 2 fold rotation should not be confused with a reflection mirror plane as the next Figure illustrates Figure 15 2 fold rotation of a motif generates a second motif resulting in the fact that the set now consisting of two motifs has 2 fold rotational symmetry As can be seen such a rotation does not generate reflectional symmetry i e the motifs are not related to each other by a mirror plane So each of the six faces will be associated with a counterface by virtue of the 2 fold rotation axes The result will be a type I hexagonal bipyramid The position of the initial face and the generation of the corresponding Form is depicted stereographically in Figure 16 Figure 16 1 Position of the face a a a c in the stereographic projection of the symmetry elements of the Hexagonal trapezohedric Crystal Class 2 Stereogram of the trapezohedric hemihedric Type I

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  • Hexagonal Crystal System VI
    plane Because they are identical in size and orientation they coincide in the drawing We must further imagine that these dihexagons represent respectively the upper and the lower faces of the dihexagonal bipyramid Now we subject the upper dihexagon to a suppression of the mirror lines parallel to the three horizontal crystallographic axes a 1 a 2 and a 3 thin red lines in the above Figure that suppression of mirror lines being expressed by the alternate extension and suppression of side pairs representing face pairs The red sides are allowed to extend at the expence of the blue sides which are suppressed In this way a ditrigon is generated like in Figure 3c With the lower dihexagon we do the same but in such a way that the whole operation differs from the former in its orientation by 60 0 This means that now the blue sides representing faces are allowed to extend at the expence of the red ones which are suppressed This orientational difference garantees the absence of the equatorial mirror plane in the ditrigonal scalenohedron to be constructed i e to be derived In this way a ditrigon is generated like in Figure 3e The superposition of these two ditrigons was already shown in Figure 3f In the above Figure 3g we see this same superposition Now we re going to think in terms of faces In Figure 3g we can see the following refers to positions above to positions below the equatorial plane so O refers to the upper apex of the bipyramid and of the scalenohedron to be derived while O refers to the lower apex A O E is a member of the upper face pair F O A A O E It is allowed to extend at the expence of the adjacent face pairs resulting in A O C D O E is a member of the lower face pair K O D D O E It is allowed to extend at the expence of the adjacent face pairs resulting in D O B The two faces of the scalenohedron to be generated A O C and D O B seem to overlap which of course is not possible So the one must lie below the other D O B lies below A O C and their line of intersection is the line B C see Figure 3g This line cannot be horizontal because it starts at B and ends with C but must be inclined Half of it is below the equatorial plane the plane of the horizontal crystallographic axes the other half lies above that plane In fact we ve now constructed two faces of the scalenohedron one lying below the other separated by a non horizontal line of intersection If we repeat this construction of faces of the scalenohedron six times around the figure formed by the two superimposed ditrigons then we will obtain all the faces of the scalenohedron This scalenohedron consists of twelve unequilateral triangles It has six longer and more obtuse edges for example B O and C O and six shorter and sharper edges for example C O and B O These are as followed organized Directly beneath each longer more obtuse edge lies a shorter sharper edge For example directly beneath the longer and more obtuse B O lies the shorter and sharper B O In Figure 3g we see that the line B C goes from the lower ditrigon blue to the upper ditrigon red and that s of course the reason why the line B C is not horizontal It is one of the six equal middle edges of the scalenohedron The other middle edges are C G G H H I I J and J B As can be seen from their going from the upper ditrigon to the lower and then to the upper again it is clear that these six middle edges of the scalenohedron go in a zig zag way In Figure 3h we ve drawn the ditrigonal scalenohedron and its orientation with respect to the four crystallographic axes Figure 3i depicts the same scalenohedron but this time with letters inserted corresponding to those of Figure 3g Figure 3h The Ditrigonal Scalenohedron The four crystallographic axes are drawn in red Figure 3i The Ditrigonal Scalenohedron with letters inserted corresponding to those of Figure 3g The two faces treated of in Figure 3g are colored The four crystallographic axes are drawn in red When we apply rhombohedric hemihedric to the holohedric protoprism we obtain a rhombohedric hemihedric type I hexagonal prism See Figure 4 Figure 4 The holohedric Protoprism does not alter its shape when subjected to rhombohedric hemihedric but loses some symmetry From the holohedric deuteroprism we can derive the rhombohedric hemihedric type II hexagonal prism when we subject it to rhombohedric hemihedric It does not alter its shape but loses some symmetry accordingly See figure 5 Figure 5 The holohedric Deuteroprism does not alter its shape when subjected to rhombohedric hemihedric but loses some symmetry From the holohedric dihexagonal prism can be derived the rhombohedric hemihedric dihexagonal prism when we apply rhombohedric hemihedric to it The prism does not change its shape but is lowered in symmetry See Figure 6 Figure 6 The holohedric Dihexagonal Prism does not alter its shape when subjected to rhombohedric hemihedric but loses some symmetry From the holohedric basic pinacoid we can derive the rhombohedric hemihedric basic pinacoid by subjecting it to rhombohedric hemihedric Also this Form does not alter its shape but loses some symmetry accordingly See Figure 7 Figure 7 The holohedric Basic Pinacoid does not alter its shape when subjected to rhombohedric hemihedric but loses some symmetry This concludes the derivation of all the Forms of the Ditrigonal scalenohedric Crystal Class All these Forms can engage in combinations 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

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