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  • Derivation of the Crystal Classes V
    two initial motifs are related to each other by a 2 fold rotation axis When a center of symmetry is added a mirror plane perpendicular to the existing 2 fold axis is implied No new direction is created because a mirror plane is always equivalent to a 2 fold roto inversion axis perpendicular to it So the only definite direction associated with this implied mirror plane is an axis perpendicular to it But this direction was already present in the form of the existing 2 fold rotation axis In 6 we have two initial motifs related to each other by a mirror plane When a center of symmetry is added a 2 fold rotation axis is implied Also this implied axis does not represent a newly created direction because the mirror plane initially present is equivalent to a 2 fold roto inversion axis perpendicular to it and thus coinciding with the implied 2 fold rotation axis 2 fold rotation axes As has been said addition of a 2 fold rotation axis must be such that no new axial directions are created So in the basic axial combination 2 no axes can be added while in the basic combination 1 2 fold axes can be added at certain locations where there are no such axes present but only when there is an axis present in the corresponding location in the basic combination 2 Indeed we have shown above that adding 2 fold axes at other locations directions cause multiplication of the more than 2 fold axes Mirror planes In the two basic groups 1 and 2 of possible axial directions and also positions for Classes having more than one more than 2 fold axis the occurring foldnesses are 2 3 and 4 The 3 fold axes here make angles of 109 0 28 and 70 0 32 with each other The 2 fold axes situated between the 3 fold axes and connecting the midpoints opposite edges of the cube see Figure 2 two by two make angles of 90 0 between each other Around each 4 fold axis they are arranged according to a 4 fold symmetry The other possible 2 fold axes replacing the 4 fold axes as in the basic axial combination 1 are perpendicular to each other Here they represent the n fold axes imposing a 2 fold symmetry on the structure when viewed along one or another of these axes The vertical axis which is equivalent to two other horizontal axes perpendicular to it either is 2 fold or 4 fold The 4 fold axis may come in combinations yet to be derived in two varieties namely a 4 fold rotation axis and a 4 fold roto inversion axis The latter is an independent symmetry element So there 3 fold and 6 fold axes cannot occur i e axes connecting the centers of opposite faces of the cube cannot be 3 or 6 fold Thus parallel to such an axis only mirror planes can be added that make angles of 45 0 or 90 0 with existing axes perpendicular to it conserving in this way the 2 or 4 foldness of that axis If we have a vertical mirror plane such that it contains a 2 fold axis which lies between the 3 fold axes and connects two opposite edges of the cube then a mirror plane perpendicular to the first mirror plane is implied Such mirror planes which are diagonally inclined appear as arcs in stereograms This is because the intersection of such a plane with the projection sphere is an inclined circle The upper half of this circle is now projected with the south pole of the projection sphere as projection center onto the projection plane the equatorial plane of the projection sphere resulting in an arc The implied mirror plane does not create a new direction because the direction associated with that mirror plane namely a direction perpendicular to that mirror plane is already present in the form of another 2 fold axis present in the second group of basic axial directions See Figure 5 Figure 5 1 Vertical mirror plane seen as a straight line in stereograms containing a 2 fold rotation axis 2 Implied diagonally inclined mirror plane seen as an arc in stereograms containing the 2 fold rotation axis The two mirror planes are perpendicular to each other All these mirror planes horizontal vertical such that 2 fold or 4 fold symmetry is conserved and diagonally inclined when added do not generate new directions axes in other locations than those specified in the two basic groups 1 and 2 of possible axial directions When they are added they map the axes of the plagihedric hemihedric cube of 2 Figure 2 onto each other Because they map the cube onto itself Also the axes of the tetartohedric cube of 1 Figure 1 are mapped onto each other So addition of such mirror planes is legitimate We now have prepared ourselves for the derivation of the remaining Crystal Classes i e the Classes beyond the 27 Classes already derived in the previous Parts They belong to the group of symmetry combinations having more than one more than 2 fold rotation axis It can be mathematically proved that only two axial combinations are possible in crystals the above described basic axial combinations 1 and 2 These two axial combinations as such i e without having mirror planes themselves already represent two new crystal classes namely the Classes 23 and 432 In the next Figures we depict them in terms of the stereographic projections of their symmetry elements together with the projections of the face poles of the most general Form of each of those two Classes We also depict these general Forms themselves Figure 6 Stereogram of the symmetry elements of the Class 23 and of the face poles of its most general Form 2 fold rotation axes are indicated by small solid ellipses 3 fold rotation axes are indicated by small solid triangles The projections of upper face poles are indicated by small red solid circles while those of lower face poles are indicated by small red open circles This symmetry configuration does not have mirror planes as is indicated by the dashing of lines representing possible mirror planes i e possible in other classes It is the symmetry of the first basic axial combination 1 Figure 6a The Tetrahedric Pentagondodecahedron as the general Form of the Class 23 Figure 7 Stereogram of the symmetry elements of the Class 432 and of the face poles of its most general Form 4 fold rotation axes are indicated by small solid squares Also this Class does not possess mirror planes It has the symmetry of the second basic axial combination 2 Figure 7a The Gyroid or Pentagonicositetrahedron as the general Form of the Class 432 To the symmetry content of these two Classes we re now going to add the above mentioned symmetry elements one at a time under the above determined conditions in order to derive further Classes Addition of a center of symmetry to the symmetry content of Class 23 yields Class 2 m 3 in which 3 signifies a 3 fold roto inversion axis in the literature the is written as a score above the numeral preceding it in our case the numeral 3 See Figure 8 Figure 8 Derivation of Class 2 m 3 from Class 2 3 by adding a center of symmetry Three mirror planes are implied indicated by continuous lines two straight lines representing two vertical mirror planes one circular line the perimeter of the projection plane representing a horizontal mirror plane These three mirror planes are equivalent In 5 of Figure 4 we can see that indeed these mirror planes are implied and are each perpendicular to a 2 fold rotation axis The number of motifs represented by the projections of face poles is doubled Addition of a center of symmetry to the symmetry content of Class 432 yields Class 4 m 3 2 m See Figure 9 Figure 9 Derivation of Class 4 m 3 2 m from Class 432 by adding a center of symmetry Nine mirror planes are implied indicated by continuous lines four straight lines representing four vertical mirror planes one circular line the perimeter of the projection plane representing a horizontal mirror plane and four diagonally inclined mirror planes indicated by continuous arcs In 5 of Figure 4 we can see that indeed all these mirror planes are implied and are each perpendicular to a 2 fold rotation axis which may also be a 4 fold rotation axis because such an axis is at the same time consistent with 2 fold symmetry The number of motifs represented by the projections of face poles is doubled In the above derivation addition of a center of symmetry implies mirror planes perpendicular to existing 2 fold axes Four of those axes are inclined axes We see them in the central part between the 3 fold axes in the stereogram of Class 432 These inclined axes also get their mirror planes perpendicular to them Each of these implied mirror planes is an inclined mirror plane and appears in the stereogram of Class 4 m 3 2 m as a continuous arc opposite to the corresponding inclined 2 fold axis Recall that in a stereogram a crystal face is represented by the projection onto the projection plane of the piercing point through the projection sphere of the face s perpendicular An axis is represented by the projection onto the projection plane of its piercing points through the projection sphere But a mirror plane is represented by the projection onto the projection plane of its intersection which is a circle with the projection sphere When a mirror plane is vertical or inclined we take the upper part of its intersection with the projection sphere and project that part onto the projection plane resulting respectively in a continuous straight line or a continuous arc When a mirror plane is horizontal its intersection with the projection sphere coincides with the equator of the projection sphere The projection onto the projection plane of this equator coincides with the equator itself resulting in a continuous circle outlining the projection plane Two such inclined 2 fold axes and the corresponding inclined mirror planes are depicted in Figure 9a Figure 9a Addition of a center of symmetry implies mirror planes perpendicular to existing 2 fold axes Addition of a 2 fold rotation axis can only be such that it will end up precisely between two existing axes in order to comply with the allowed axial directions The total of these allowed directions is seen in the basic axial combination 2 i e that of Class 432 In fact there are two types of possible 2 fold rotation axes Diagonal 2 fold axes Non diagonal 2 fold axes See Figure 9b Figure 9b Left Two instances of a diagonal 2 fold rotation axis Right Two instances of a non diagonal 2 fold rotation axis Class 23 already possesses all possible non diagonal axes Class 432 possesses all possible non diagonal 4 fold rotation axes and those axes necessarily also have 2 fold symmetry They repeat an image every 90 0 so also every 180 0 Class 2 m 3 already possesses all possible non diagonal 2 fold axes Class 4 m 3 2 m also already possesses all possible non diagonal 2 fold rotational symmetry contained in its non diagonal 4 fold rotation axes So for the time being we only have to investigate the addition of a diagonal 2 fold rotation axis Addition of a diagonal 2 fold rotation axis to the symmetry content of Class 23 yields Class 432 First we depict the stereogram of Class 23 See Figure 10 Figure 10 Stereographic projection of the symmetry elements of Class 23 and of the face poles of its most general Form In the next Figure we re going to add a diagonal 2 fold rotation axis and see what happens as a consequence Figure 11 Derivation of Class 432 from Class 23 by adding a diagonal 2 fold rotation axis 1 Addition of a diagonal 2 fold rotation axis to the symmetry content of Class 23 Existing horizontal 2 fold axes generate yet another 2 fold axis 2 Because of the two new 2 fold rotation axes the number of motifs here represented by face poles is doubled resulting in a pattern that exhibits a 4 fold rotational symmetry about a vertical axis the axis perpendicular to the plane of the drawing So this vertical axis which was originally a 2 fold rotation axis becomes a 4 fold rotation axis indicated as a small blue solid square in the center of the stereogram 3 The existing 3 fold rotation axes multiply this 4 fold axis resulting in a total of three 4 fold rotation axes perpendicular to each other indicated by small solid squares See Figure 11c for an explanation 4 These new 4 fold axes multiply the remaining 2 fold axes resulting in four new axes These new axes are inclined They are indicated as small blue solid ellipses in the central region of the stereogram The resulting symmetry configuration is that of Class 432 It is the symmetry of the basic axial combination 2 In 4 the upper and lower face poles are drawn swapped with respect to those in 3 but this is immaterial insofar symmetry is concerned In 2 of the above Figure we stated that because of the action of the new 2 fold axes one added one implied the already existing vertical 2 fold axis becomes a 4 fold axis We concluded this on the basis of the emerging new pattern of motifs represented by face poles The next two Figures show the second one by means of real motifs that such a conclusion is indeed correct Figure 11a Stereogram of Class 23 The next Figure deals in terms of real motifs with the elements in the shaded area of the stereogram Figure 11b Implication of 4 fold rotational symmetry about the vertical axis as a result of the addition of a horizontal 2 fold rotation axis to the symmetry content of Class 23 1 Two motifs an upper one black and a lower one red related to each other by an existing 2 fold rotation axis 2 Addition of a 2 fold rotation axis and the implication of yet another one create two units each consisting of two motifs which are exact repetitions of each other For reasons of clarity the added and implied axes are drawn parallel to the existing 2 fold axis In reality they make angles of 45 0 with it This shows that the whole upper left unit in 2 of Figure 11 consisting of six motifs is repeated after a rotation of it by 90 0 about the vertical axis in the center of the stereogram The same applies to the other three units which effects that the four units of 2 of Figure 11 are mapped onto each other every 90 0 rotation about the vertical axis This means that this vertical axis has become a 4 fold rotation axis In 3 of Figure 11 we stated that the existing 3 fold axes multiply the 4 fold axis resulting in three such axes perpendicular to each other and replacing initially existing 2 fold axes The next Figure will make this evident Figure 11c The 3 fold axis maps the 4 fold axis onto axis a and onto axis b which were originally 2 fold axes turning them into 4 fold axes Addition of a 2 fold rotation axis to Class 432 is not possible because all the allowed locations for axes are already occupied in the symmetry configuration of this Class Addition of a diagonal 2 fold rotation axis to the symmetry content of Class 2 m 3 yields Class 4 m 3 2 m Let us first depict the stereogram of Class 2 m 3 Figure 12 Stereographic projection of the symmetry elements of Class 2 m 3 and of the face poles of its most general Form The 3 fold roto inversion axes 3 are indicated by small black open triangles Upper and lower face poles coincide on the projection plane and are as such indicated by small red centered circles The next Figure shows the derivation of Class 4 m 3 2 m This Class was already derived earlier so nothing new is actually generated Figure 13 Derivation of Class 4 m 3 2 m from Class 2 m 3 by adding a diagonal 2 fold rotation axis 1 Addition of a diagonal 2 fold rotation axis to the symmetry content of Class 2 m 3 Existing horizontal 2 fold axes generate yet another 2 fold axis 2 A 2 fold axis contained in a mirror plane implies a second mirror plane perpendicular to the first one So two new mirror planes are generated indicated as continuous straight lines colored blue The number of motifs represented by face poles is doubled by the action of the new symmetry elements 3 The resulting motif pattern shows 4 fold rotational symmetry about the vertical axis in the center of the projection plane and perpendicular to it meaning that this axis which originally was a 2 fold axis becomes a 4 fold rotation axis indicated by a small solid blue square The existing 3 fold roto inversion axes indicated by small black open triangles multiply this 4 fold axis such that now the originally existing horizontal 2 fold axes become 4 fold rotation axes indicated by small solid blue squares 4 The horizontal 4 fold axes generate four diagonally inclined 2 fold axes indicated by small blue solid ellipses in the central part of the stereogram See also Figure 2 5 A 2 fold rotation axis contained in a mirror plane implies yet another mirror plane perpendicular to the first

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  • Derivation of the Crystal Classes VI
    ellipses in the central region of the stereogram The resulting summetry configuration is that of Class 4 m 3 2 m Addition of a non diagonal mirror plane to Class 4 m 3 2 m is not possible because all the permissible locations for such mirror planes are already occupied with non diagonal mirror planes Still left to investigate is the addition of a center of symmetry to the symmetry content of Class 4 3m the addition of a horizontal diagonal 2 fold rotation axis Horizontal diagonal 2 fold axes are equivalent to the inclined 2 fold axes to that of the same Class and the addition of a non diagonal 2 fold rotation axis also to the symmetry content of that Class The latter does not to be considered because that Class has already 4 fold roto inversion axes at the corresponding locations Recall that a 4 fold roto inversion axis as well as a 4 fold rotation axis is at the same time also a 2 fold rotation axis Addition of a center of symmetry to the symmetry content of Class 4 3m yields Class 4 m 3 2 m See Figure 7 Figure 7 Addition of a center of symmetry to the symmetry elements of the Class 4 3m doubles the number of motifs The symmetry of the resulting motif pattern then demands a the 4 fold roto inversion axes to become 4 fold rotation axes See Remark below and b the presence of three non diagonal mirror planes Two vertical and one horizontal Further we have the coming into being of 2 fold rotation axes where none were there where there is an intersection of two mirror planes perpendicular to each other At some of these places there is already a 4 fold axis and such an axis already implies 2 fold symmetry The situation of a vertical mirror plane perpendicular to a diagonally inclined mirror plane is depicted in Figure 7a Because of the presence of a center of symmetry the 3 fold axes become 3 fold roto inversion axes open triangles The resulting symmetry configuration is that of Class 4 m 3 2 m Figure 7a A vertical mirror plane perpendicular to a diagonally inclined mirror plane Remark In Figure 7 we stated the following The symmetry of the resulting motif pattern then demands a the 4 fold roto inversion axes to become 4 fold rotation axes Let us explain this by means of the next Figure Figure 7b The presence of a 2 fold rotation axis and a center of symmetry implies a mirror plane perpendicular to that 2 fold axis As can be seen from Figure 7b a horizontal mirror plane is implied perpendicular to the 4 fold roto inversion axis Recall that such an axis is at the same time also a 2 fold rotation axis and by the same reason two vertical non diagonal mirror planes are implied This means that all permissible mirror planes are now present This in

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  • Isometric Crystal System
    remains unchanged We then obtain a face which can be symbolized as a na a Letting this face multiply according to the symmetry of our Class we obtain the tetrakishexahedron The Weiss symbol for this Form is accordingly a na a the Naumann symbol is On and the Miller symbol is hk0 In the Figure we see a tetrakishexahedron for the case of n 2 Figure 8 Another basic Form of the Cubic Hexakisoctahedral Crystal Class the Tetrakishexahedron The last basic Form of the Cubic Hexakisoctahedral Crystal Class is the hexakisoctahedron which is the most general Form an d which gave the name for the Class Figure 9 The most general Form of the Cubic Hexakisoctahedral Crystal Class the Hexakisoctahedron It possesses 48 faces To derive this Form we tilt the basic face of Figure 1 in the most general way which means that we turn it to the effect that the cut off pieces of the Y axis and the Z axis increase but to an unequal extent while the cut off piece of the X axis remains the same The symbol for the resulting face then becomes a na ma When we now multiply this face according to the symmetry of the Class we obtain a hexakisoctahedron The Weiss symbol for this Form is consequently a na ma the Naumann symbol is mOn while the Miller symbol is hkl The one that is depicted in Figure 9 has m 3 and n 3 2 Each of these described Forms can occur as a crystal because they are closed but also combinations of them can constitute crystal shapes This is because they are all obedient to the same symmetry bundle See Figure 10 Figure 10 Some simple Combinations of Forms of the Cubic Hexakisoctahedral Class 1 O O The corners of the octahedron are cut off by the cube 2 O O The corners of the cube are cut off by the octahedron 3 Cuboctahedron Equal development of cube and octahedron 4 O O The edges of the octahedron are cut off by the rhombic dodecahedron Often complicated combinations are encountered consisting of many faces As real representatives of the Cubic Hexakisoctahedral Class we could mention the minerals Copper Cu Gold Au Halite NaCl and Fluorite CaF 2 To see the larger picture s click on the smaller one s Fluorite Val Sarentina Italy Crystals 5 15 mm This concludes our exposition of the Cubic Hexakisoctahedral Crystal Class The Hexakistetrahedral Class Tetrahedric Hemihedric Division 4 3 m All the lower symmetrical Classes of the Isometric Crystal System can be derived from the most symmetric one i e from the Hexakisoctahedral Class or equivalently Holohedric Division treated of above We can and will do this by means of the concepts holohedric hemihedric etc See for some general information regarding this approach the relevant sections of the Essay on The Morphology of Crystals In order to derive these lower Classes we drop certain mirror planes generally resulting in new Forms possessing a smaller number of faces and a corresponding lower symmetry Because the Holohedric Division of the Isometric System has several types of mirror planes several types of hemihedric are possible To obtain the Forms of the present Class we take the holohedric Forms and drop i e suppress their main mirror planes There are three such planes in crystals of the Holohedric Division each containing two of the three crystallographic axes The suppression of these mirror planes results in a Crystal Class the Hexakistetrahedral Class possessing the following symmetry bundle Six mirror planes corresponding to the secondary mirror planes of the Holohedric Division Seven rotation axes namely three two fold rotation axes coinciding with the crystallographic axes and four polar 3 fold rotation axes which are perpendicular to the octahedron faces of the Holohedric Division There is no center of symmetry implying that in these new Forms not every face has a parallel counter face In order to obtain the Forms of this Class deriving them from the holohedric ones i e from the octahedron cube rhombic dodecahedron etc we start with the holohedric octahedron This octahedron is divided into eight faces by the main mirror planes When we drop those planes half of these faces will disappear When we then let the remaining faces extend we ll end up with a tetrahedron In fact we can derive two such tetrahedra depending on which set of octahedral faces is suppressed See Figure 11 Figure 11 The derivation of two correlate Forms tetrahedra from the holohedric octahedron 1 Octahedron with its faces divided by the main mirror planes into two sets each growing into a tetrahedron 2 Extension of the non striped faces of the octahedron and suppression of the striped faces 3 Tetrahedron developed from the non striped faces of the octahedron 4 Tetrahedron developed from the striped faces of the octahedron These two tetrahedra each bounded by four faces are thus correlate Forms They are congruent and are distinguished only by their orientation with respect to the crystallographic axes This difference can be signified by a plus or minus sign The faces of the tetrahedron are equilateral triangles The line connecting the two upper corners is perpendicular to the line connecting the two lower corners The orientation of all hemiedric Forms always corresponds in drawings with the one of the holohedric Form from which they are derived So in our case the crystallographic axes go through the middle of the edges of the tetrahedron i e each axis connects two opposite edges by going through their mid points We can symbolize such a tetrahedron as 1 2 a a a The Naumann symbol reads and the Miller symbols are K 111 and K 11 1 In fact the K is written as a greek letter and 1 is written as a 1 with a horizontal score above it The cube the rhombic dodecahedron and the tetrakishexahedron do not change their outer shape because the suppressed mirror planes are perpendicular

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  • Tetragonal Crystal System I
    the Figure is the face a a c It can become a face of a basic pinacoid Now it is easy to derive the Forms of the present Crystal Class the Holohedric Division of the Tetragonal System The primary protopyramid primary bipyramid of type I is derived from the upper right front face the unit face of the Tetragonal Crystal System a a c number 2 of Figure 2a of the image in Figure 3 below considered as just being present on its own without other faces and which is in fact a certain conspicuous face on some tetragonal crystal chosen to serve as a unit face The primary protopyramid is derived from this face by demanding that the generated Form possesses the complete symmetry bundle of the present Class The face is multiplied by the 4 fold rotation axis resulting in a total of four faces Then this quadruplet will be doubled by the horizontal mirror plane resulting in the protopyramid Figure 3 The primary Tetragonal Protopyramid Tetragonal Bipyramid of type I Each face of this Form cuts off a finite piece of each crystallographic axis and considering it as the basic Form we can denote it by the Weiss symbol a a c in which the two a s express the equivalence of the two horizontal crystallographic axes and the one c expresses the fact that the vertical crystallographic axis is not equivalent to either horizontal one The protopyramid is a bipyramid bounded by eight isosceles triangles Its equatorial plane has the shape of a square The Naumann symbol is P and the Miller symbol is 111 the derivation coefficients of the Weiss symbol and the indices of the Miller symbol are set equal to 1 because all other Forms are considered to be derivations of this one Form The stereographic projection of the faces i e all the faces of the tetragonal protopyramid is depicted in Figure 3a Figure 3a Stereogram of the Tetragonal Protopyramid An upper face pole is represented by a red dot A lower face pole is represented by a small circle In the present case their positions on the projection plane coincide So we see four sets of two faces each an upper and a lower one The first derived Forms are bipyramids like the protopyramid but with a different a c ratio when compared with the primary protopyramid of Figure 3 The general Weiss symbol for such pyramids is consequently a a mc the Naumann symbol is mP the letter before P relates to the cut off piece of the vertical crystallographic axis the Miller symbol is hhl In figure 1 we already saw such a derived protopyramid assuming m having there some rational value and in the figure below we see two examples of derived protopyramids with m 2 and m 1 2 respectively If we would set up a stereogram of such a derived protopyramid we would get the same picture as Figure 3a but with the projections of the faces closer to the perifery of the projection plane or closer to the center of it respectively Figure 4 Derived Tetragonal Protopyramids with m 2 left and m 1 2 right In the next Figure we see two more Forms namely 1 the deuteropyramid and the ditetragonal pyramid Figure 5 1 primary Deuteropyramid Tetragonal Bipyramid of type II 2 Ditetragonal bipyramid While a protopyramid is generated by subjecting the face a a mc to all the symmetry operations of the present Crystal Class when m 1 we get the primary protopyramid a deuteropyramid is generated by subjecting the face a a mc to all those symmetry operations when m 1 3 in Figure 2a we get the primary deuteropyramid The shape of the deuteropyramid tetragonal bipyramid of type II does not differ from that of the protopyramid It differs in its orientation with respect to the crystallographic axes Each of its faces cuts off pieces only from the main axis and from one of the secondary i e horizontal axes and is parallel to the other secondary axis So the Weiss symbol is a a c or generally because also derived pyramids are possible a a mc and the Naumann symbol is mP the sign before P relates to the main axis the sign after P to the secondary axis Recall that the sign stands for infinity in Figures denoted by a horizontal 8 The Miller symbol is h0l The stereographic projection of the deuteropyramid is depicted in Figure 5a Figure 5a Stereogram of the Tetragonal Deuteropyramid When we subject the face a na mc i e a face of which the orientation with respect to the crystallographic axes is the most general to all the symmetries of the present Crystal Class we obtain a ditetragonal bipyramid as yet another Form of the Holohedric Division The ditetragonal bipyramid has 16 faces which are oriented such that each of them intersects all three axes at different distances The equatorial plane is a ditetragon i e an octagon figure with eight sides having all sides of equal length but with alternating equal angles See Figure 6 Figure 6 A Ditetragon the equatorial plane of the Ditetragonal bipyramid The general Weiss symbol is a na mc 1 in Figure 2a the Naumann symbol is mPn and the Miller symbol is hkl Limiting Forms are the protopyramid when n 1 and the deuteropyramid when n The stereographic projection of the ditetragonal bipyramid is depicted in Figure 5b Figure 5b Stereogram of the Ditetragonal Bipyramid When for the described pyramids the derivation coefficient m increases the pyramids become more and more sharp When the coefficient finally is equal to infinity the sides of the pyramids have become vertical i e parallel to the main crystallographic axis and the Forms become prisms These are open Forms which in their conventional oriention do not enclose space completely Their top and bottom are open From the primary protopyramid we derive the protoprism tetragonal prism of type I by

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  • Tetragonal Crystal System II
    derivation of the Forms of the Diteragonal pyramidal Crystal Class by the Merohedric approach FACIAL APPROACH The Forms of this Class can also be derived from the basic faces face types relevant and compatible with the Tetragonal Crystal System These Faces are a a c a a c a na mc a a c a a c a na c a a c We will subject these faces one by one to the symmetries of the Class The stereographic projection of the symmetry elements of this Class will help us visualize the subjection of those faces to these symmetry elements Figure 23 Stereogram of the symmetry elements of the Ditetragonal pyramidal Class and the stereographic projection of the face poles of the Form derived from the face with a general orientation namely the face a na mc The dashed circle is the projection plane and the dashing indicates that this plane is not a mirror plane Solid straight lines are vertical mirror planes The position of all the basic faces together in the projection plane of the stereographic projection is indicated in Figure 2a and separately below The symmetry elements of the present Class as depicted in Figure 23 will multiply some faces and leave unchanged others depending on their orientation The face a a c indicated in Figure 24 will when subjected to the symmetry elements of the present Class yield the type I tetragonal monopyramid of Figure 16 consisting of four faces in virtue of the effect of the 4 fold rotation axis Figure 24 The position of the face pole a a c red dot in the stereographic projection of the elements of the Ditetragonal pyramidal Class The multiplication of the general face a na mc black dots is also indicated in order to clearly show the effect of the symmetry operations The stereographic projection of the type I tetragonal monopyramid is depicted in Figure 24aI which shows the multiplication of the face a a c by the symmetry operations of the present Class Ditetragonal pyramidal Class Figure 24aI Stereogram of the Type I Tetragonal Monopyramid From the face a a c which is just a negative variant of a a c we can derive the other half i e also a tetragonal pyramid but with its apex directed downward This Form is depicted in Figure 17 It is the lower half of the holohedric protopyramid Its stereographic projection is depicted in Figure 24aII Figure 24aII Stereogram of the other Type I Tetragonal Monopyramid The face poles are depicted by small red circles to indicate that the faces are below the equatorial plane of the Form i e below the plane of the secondary crystallographical axes The face a a c indicated in Figure 25 will when subjected to the symmetry elements of the present Class yield a type II tetragonal monopyramid i e a same pyramid as the one derived from a a c but differing in its orientation Figure 18 The face a a c

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  • Tetragonal Crystal System III
    the structure onto itself so we have to do with a special 2 fold rotation axis Further we see in the stereogram Figure 8 two horizontal 2 fold rotation axes perpendicular to each other and indicated by two small solid ellipses for each axis connected by a dashed line if this line were solid than such a line would indicate a vertical mirror plane in which the axis lies Next we see two vertical mirror planes represented by solid lines They bisect the angles between the 2 fold rotation axes The circumference of the projection plane is dashed which means that the projection plane is not also a mirror plane We are now ready to successively derive the Forms of the present Class by subjecting the above listed basic faces one by one to the symmetry elements of the Class Because these symmetry elements are indicated in the stereogram Figure 8 we can insert a face in that stereogram i e we can insert the projection of a face pole representing the face concerned into the stereogram and see what happens when we let the symmetry elements do their work on it We start with the face a a c Its position in the stereogram is indicated by a red dot in Figure 9 Figure 9 The position of the projected face pole a a c in the stereogram of the symmetry elements of the Tetragonal scalenohedric Crystal Class is indicated by a red dot Also indicated black dots and small circles are all the face poles representing the complete Form generated from the general face a na mc in order to clearly see the effect of the symmetry elements Dots represent upper faces small circles represent lower faces It is a face that is not perpendicular to any crystallographic axis It is the unit face for the Tetragonal Crystal System and can be seen as the upper right front face of a tetragonal bipyramid When subjected to the 4 axis a copy of it is rotated 90 0 clockwise about the axis in this case rotated about the main crystallographic axis and then inverted through the origin of the system of crystallographic axes the result will be a lower face When this latter face is in turn subjected to the 4 axis i e be rotated 90 0 and then inverted through the origin the result will be another upper face Subjecting this face in turn to the 4 axis will generate another lower face When we finally subject this face in turn to the 4 axis we will obtain a face which we already had namely our initial face The configuration of faces so obtained is a sphenoid Figure 1 also called a disphenoid The mirror planes and the 2 fold rotation axes are implied by the 4 axis if the four motifs involved are themselves symmetric The 4 axis itself see Figure 8a does not demand such a symmetry of the motifs When indeed the motifs are themselves symmetric then their configuration according to the 4 axis looks as in the next Figure and this configuration complies with the symmetry content of the present Class Figure 8b A configuration of symmetrical motifs according to a 4 fold roto inversion axis Because of the symmetry of the motifs themselves additional symmetries for the whole object consisting of four motifs are generated i e implied namely two mirror planes one that goes through a b c and one that goes through a b c and two 2 fold rotation axes going through the inversion point one lying in the plane f g c and the other in the plane h i c If we rotate the whole object vertically by 45 0 then we get the same orientation of the symmetry elements as depicted in the corresponding stereographic projection Let us develop in stages the stereographical projection of the resulting sphenoidic hemihedric sphenoid in the next Figures We ll start with the stereographic projection of the initial face namely the face a a c Figure 9a Position of the face pole a a c in the stereographic projection of the symmetry elements of the Tetragonal scalenohedric Crystal Class We do not depict here as in the next three Figures the face poles of the complete general Form which were after all only included for reasons of clarity with respect to the workings of the symmetry elements of the Class which could then easily be read off from their configuration From the initial face a second face is generated see Figure 9b Figure 9b Stereogram of the initial face a a c and of the second generated face generated by the action of the 4 axis A dot represents an upper face a small circle a lower one In generating this second face by the action of the 4 axis we rotatate it 90 0 clockwise about the axis which coincides with the crystallographic main axis and then invert it through the inversion point the middle of the crystallographic main axis This results in a lower face meaning that its face pole lies on the lower hemisphere of the projection sphere and is then upwardly projected i e it is projected from below onto the projection plane Next we will generate the third face See Figure 9c Figure 9c When we rotate the above generated lower face small red circle 90 0 clockwise about the main crystallographic axis and then invert it through the inversion point a third face is generated which is an upper face Next we will generate the fourth face See Figure 9d Figure 9d When we rotate this third face red dot in the upper left quadrant of the stereographic projection of Figure 9c clockwise 90 0 about the main crystallographic axis and then invert it through the inversion point then a fourth face a lower face will be generated When we repeat this operation on that fourth face we ll get the first face the

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  • Tetragonal Crystal System IV
    The pyramidal hemihedric Basic Pinacoid The coloring of the areas symbolizes the suppression of all the vertical mirror planes Both faces remain however after applying this hemihedric The non suppression of the vertical mirror plane is also symbolized by the colors Vertically below each yellow area of the upper face we find a yellow area of the lower face and below each brown area of the upper face we find a brown area of the lower face This concludes the derivation of the Forms of the Tetragonal bipyramidal Class by means of the Merohedric Approach 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 bipyramidal 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 29 Stereographic projection of the symmetry elements of the Tetragonal bipyramidal Crystal Class and of the faces of the most general Form a Tetragonal Bipyramid The face a a c is the unit face of the Tetragonal Crystal System It therefore cuts off equal pieces by definition unit distances from the origin of the axes from the horizontal crystallographic axes and is inclined in such a way by definition as to cut off a piece of unit length from the vertical axis Its stereographic projection is depicted in Figure 30 Figure 30 Position of the face a a c in the stereographic projection of the symmetry elements of the Tetragonal bipyramidal Crystal Class This face is multiplied four times in virtue of the 4 fold rotation axis and the result four faces forming an open monopyramid is reflected with respect to the mirror plane coinciding with the equatorial plane resulting in a pyramidal hemihedric Type I tetragonal bipyramid Figure 16 Its stereographic projection is depicted in Figure 30a Figure 30a Stereogram of the pyramidal hemihedric Type I Tetragonal Bipyramid Recall that a mirror plane in a stereographic projection is always depicted as a solid line either a circle or a straight line In the stereogram of this figure the only mirror plane is the equatorial plane solid circle The north south and east west dashed lines as they appear in the stereogram are in this case not symmetry elements i e also not rotation axes They represent the equatorial crystallographic axes The other dashed lines are just visual aids The face a a c is parallel to the east west crystallographic axis It cuts off unit distances from the north south crystallographic axis and from the vertical crystallographic axis Its position in the stereographic projection is depicted in Figure 31 Figure 31 The position of the face a a c in the stereogram of the Tetragonal bipyramidal Crystal Class This face

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  • Tetragonal Crystal System V
    the absence of vertical mirror planes as done in the next Figure Figure 6 The hemimorphic Type III Tetragonal Pyramid Absence of vertical mirror planes is indicated 4 Derivation from the pyramidal hemihedric Type I Tetragonal Prism The suppression of the equatorial mirror plane applying hemimorphy does not have any effect on the external shape of the prism but of course it looses some symmetry So we obtain again a type I tetragonal prism See Figure 7 Figure 7 Derivation of the hemimorphous Type I Tetragonal Prism from the pyramidal hemihedric Type I Tetragonal Prism The absence of vertical as well as of horizontal mirror planes is indicated 5 Derivation from the pyramidal hemihedric Type II Tetragonal Prism Applying hemimorphy to the pyramidal hemihedric type II tetragonal prism gives again rise to a type II tetragonal prism but with lower symmetry See Figure 8 Figure 8 Getting a hemimorphous Type II Tetragonal Prism from a pyramidal hemihedric Type II Tetragonal Prism The absence of all the mirror planes is indicated 6 Derivation from the pyramidal hemihedric Type III Tetragonal Prism the Tetragonal Tritoprism Applying hemimorphy to the tritoprism does not change its external shape but lowers its crystallographic symmetry accordingly See Figure 9 Figure 9 The pyramidal hemihedric Tritoprism has become a hemimorphous pyramidal hemihedric Type III Tetragonal Prism having no mirror planes whatsoever despite the fact that the Figure suggests some vertical mirror planes 7 Derivation from the pyramidal hemihedric Basic Pinacoid The basic pinacoid consists of two parallel faces When hemimorphy is applied to this Form we obtain two independent halves of it two pedions So this new Form the pedion is just one face The derivation of one such pedion is depicted in Figure 10 and 10a Figure 10 The pyramidal hemihedric Basic Pinacoid from which the Pedion is derived by hemimorphy Figure 10a A Pedion monohedron derived from the pyramidal hemihedric Basic Pinacoid All these Forms can engage in combinations As an example could serve the combination of Forms in crystals of the mineral Wulfenite BRUHNS 1912 p 63 See Figure 11 On top we see the primary protopyramid here a monopyramid p in the Figure of which the corners are cut off in an oblique fashion by the tritoprism r in the Figure At the bottom the polar corner of the other monopyramid p in the Figure is cut off by a pedion Figure 11 Combination of two Tetragonal Type I Pyramids p and p and a Tritoprism r 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 pyramidal 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 12 Stereogram of the symmetry elements of the Tetragonal

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