1

7 Space groups and point groups

In certain cases the simple form may consist of merely one (monohedron), two parallel faces (pinacoid), or two intersecting faces (). Shubnikov [1].

In the plane or in two dimensions there are 17 groups, in space or in three dimensions there are 230 groups. The number of groups increases rapidly when adding on dimension. The permutations in space under the constraint of periodic translation give the 230 groups. From the allowed symmetry operations - thus under the constraint of periodic translation - comes the 32 point groups. Instead of 10 in 2D. A particular combination of the variables like x,y,z corresponds to coordinates for a point in space. If we add them up like x+y+z they represent a surface. In order to keep an identity of each surface you may go cyclic (cosπ[x+y+z]) which has shown to be of extreme power in mathematics or natural science. Another way is to go exponential as below and surfaces keep again their identity and collaborate to give finite shapes. We describe the fundamental permutations of variables under the constraint of the allowed symmetry –the 32 point groups – as mathematical functions using the exponential scale. These general permutations and also some special permutations (that correspond to the special positions in a group) describe the 47 simple forms which crystals may take [1]. In these simple forms, all faces are equal to one another. Each term of permutation x,y,z describes the {hkl} faces of the simple form of the crystal of the corresponding mineral. Mathematically we use the concept of how planes build symmetries. Which is normal as periodic structures – like crystals – can be described as built of planes. In the permutations below for a position of type x,x,x in crystallographic literature we use x,y,z in the calculations in the Cartesian system of Mathematica. For a position of type x,y,z in the crystallographic literature we use here x,2y,3z, whenever needed. This is particularly useful for the cubic system as will come in the next chapter.

The point groups except the cubic

In our exponential expressions for planes to describe various forms it is possible to simplify, and certainly so in the use of hyperbolic functions for the centric cases. But we have preferred to use the elaborated expressions that are to be compared with the space group co-ordinates of equivalent positions. The names are after Shubnikov[1], and also from Bob’s rockshop[2].

2

7.1 The triclinic, monoclinic and orthorhombic groups

Below we give the groups with symbols, numbers and equivalent positions after the crystallographic literature[3]. Numbers in brackets are the Shubnikov numbers for the simple forms.

P1 is group no 1 and has the co-ordinates x,y,z. We formulate the equation 7.1.1. And the figure is in 7.1.1.

P1 No.1 x,y,z;

x y z e + + = 100 7.1.1

P 1 No.2 x,y,z; -x,-y,-z;

This group has a centre of symmetry, as is obvious from figure 7.1.2.

Its equation is in 7.1.2.

x y z x y z e + + + e! ! ! = 100 7.1.2

The figures are called monohedron resp pinacoid(Shub 32).

Figure 7.1.1 Monohedron Figure 7.1.2 Pinacoid

P2 No.3 x,y,z; -x,y,-z;

x y z x y z e + + + e! + ! = 100 7.1.3

The form is called dihedron, one plane is continually bent to another.

3

Figure 7.1.3a Dihedron b Different projection

Pm No.6 x,y,z; x,-y,z;

x y z x y z e + + + e ! + = 100 7.1.4

This form in figure 7.1.4 is also called dihedron(Shub 31). Bending again, but in another direction.

Figure 7.1.4a Dihedron b Different projection

P2/m No.10 x,y,z; -x,-y,-z; x,-y,z; -x,y,-z;

x y z x y z x y z x y z e + + + e! ! ! + e ! + + e! + ! = 1000 7.1.5

The shape in figure 7.1.5 is an orthorhombic (Shub 15).

Figure 7.1.5a Orthorhombic prism b Different projection 4

P222 No.16 x,y,z; -x,-y,z; x,-y,-z; -x,y,-z;

Below is also the equation for a regular given in 7.1.6 to be compared with the next simple form 7.1.7.

x y z x y z x y z x y z e + + + e! ! + + e ! ! + e! + ! = 1000 7.1.6

x 2y 3z x 2y 3z x 2 y 3z x 2 y 3z e + + + e! ! + + e ! ! + e! + ! = 1000 7.1.7

The shape of this simple form in figure 7.1.7 is an orthorhombic tetrahedron (Shub 22).

Figure 7.1.6 Regular tetrahedron Figure 7.1.7 Orthorhombic tetrahedron

Pmm2 No.25 x,y,z; -x,-y,z; x,-y,z; -x,y,z;

x 2y 3z x 2y 3z x 2 y 3z x 2 y 3z e + + + e! ! + + e ! + + e! + + = 1000 7.1.8

The name of this simple form as shown in figure 7.1.8 is orthorhombic (Shub 1).

Figure 7.1.8 Orthorhombic pyramid.

Pmmm No.47 x,y,z; -x,-y,z; x,-y,-z; -x,y,-z; -x,-y,-z; x,y,-z; -x,y,z; x,-y,z;

x 2y 3z x 2y 3z x 2 y 3z x 2 y 3z e + + + e! ! + + e ! ! + e! + ! +

(x 2 y 3z) ( x 2 y 3z) (x 2y 3z) ( x 2y 3z) e ! + + + e! ! ! + + e! ! + + e! ! + + = 1000 7.1.9 5

The name is orthorhombic (Shub 8), which is shown in figure 7.1.9.

Figure 7.1.9a Orthorhombic bipyramid. b Different projection

7.2 Tetragonal groups

P4 No.75 x,y,z; -x,-y,z; y,-x,z; -y,x,z;

x 2y 3z x 2y 3z 2x y 3z 2x y 3z 5 e + + + e! ! + + e ! + + e! + + = 10 7.2.1

The name is tetragonal pyramid(Shub 4) and shown in figure 7.2.1.

Figure 7.2.1a Tetragonal pyramid b Different projection.

P 4 No. 81 x,y,z; -x,-y,z; y,-x,-z; -y,x,-z;

x 2y 3z x 2y 3z 2x y 3z 2x y 3z 5 e + + + e! ! + + e ! ! + e! + ! = 10 7.2.2

The name is tetragonal tetrahedron(Shub 25)and shown in figure 7.2.2.

Figure7.2.2a Tetragonal tetrahedron. b Different projection. 6

P4/m No.83 x,y,z; -x,-y,z; x,y,-z; -x,-y,-z; -y,x,z; y,-x,z; -y,x,-z; y,-x,-z;

x 2y 3z x 2y 3z x 2y 3z x 2 y 3z e + + + e! ! + + e + ! + e! ! ! 7.2.3 2x y 3z 2x y 3z 2x y 3z 2x y 3z 6 + e! + + + e ! + + e! + ! + e ! ! = 10

The name is tetragonal bipyramid (Shub 11) and shown in figure 7.2.3.

Figure 7.2.3a Tetragonal bipyramid. b Different projection.

P422 No.89 x,y,z; -x,-y,z; -x,y,-z; x,-y,-z; -y,-x,-z; y,x,-z; -y,x,-z; y,-x,-z;

x 2y 3z x 2y 3z x 2y 3z x 2 y 3z e + + + e! ! + + e! + ! + e ! ! 7.2.4 2x y 3z 2x y 3z 2x y 3z 2x y 3z 6 + e! ! ! + e + ! + e ! + + e! + + = 10

The name is the tetragonal trapezohedron (Shub 26) and shown in figure 7.2.4.

Figure 7.2.4a Tetragonal trapezohedron. b Different projection.

7

P4mm No.99 x,y,z; -x,-y,z; -x,y,z; x,-y,z; y,x,z; -y,-x,z; -y,x,z; y,-x,z;

x 2y 3z x 2y 3z x 2y 3z x 2 y 3z e + + + e! ! + + e! + + + e ! + 7.2.5 2x y 3z 2x y 3z 2x y 3z 2x y 3z 10 + e + + + e! ! + + e! + + + e ! + = 10

The name is ditetragonal pyramid(Shub 5) and shown in figure 7.2.5.

Figure 7.2.5a Ditetragonal pyramid. b Different projection.

P 4 m2 No.115 x,y,z; -x,-y,z; -x,y,z; x,-y,z; -y,x,-z; y,-x,-z; y,x,-z; -y,-x,-z;

x 2y 3z x 2y 3z x 2y 3z x 2 y 3z e + + + e! ! + + e! + + + e ! + 7.2.6 2x y 3z 2x y 3z 2x y 3z 2x y 3z 10 + e! + ! + e ! ! + e + ! + e! ! ! = 10

The name is tetragonal scalenohedron (Shub 33) shown in figure 7.2.6.

Figure 7.2.6a Tetragonal scalenohedron b Different projection.

8

There is a second mineral form to this symmetry called the tetragonal disphenoid with only four faces so we use the special position in 4j. We arrive at the equation in

2x z 2x z 2y z 2y z 10 e + + e! + + e ! + e! ! = 10 and we see the disphenoid in figure 7.2.6c. The only difference from the tetragonal tetrahedron in P 4 above is its orientation. That give it mirror planes parallel to the axes in the co-ordinate system chosen.

Figure 7.2.6c Disphenoid.

P4/mmm No.123 x,y,z; -x,-y,z; x,y,-z; -x,-y,-z; -x,y,z; x,-y,z; -x,y,-z; x,-y,-z; -y,x,z; y,-x,z; -y,x,-z; y,-x,-z; y,x,z; -y,-x,z; y,x,-z; -y,-x,-z;

ex+ 2y+3z + e!x! 2y+3z + ex+ 2y! 3z + e!x!2 y! 3z x 2 y 3z x 2 y 3z x 2 y 3z x 2y 3z +e! + + + e ! + + e! + ! + e ! ! 7.2.7 +e!2x+ y+ 3z + e2x! y+3z + e!2x+ y! 3z + e2x! y!3z 2x y 3z 2x y 3z 2x y 3z 2x y 3z 10 + e + + + e! ! + + e + ! + e! ! ! = 10

The name is ditetragonal bipyramid (Shub 12) and shown in figure 7.2.7.

9

Figure 7.2.7a Ditetragonal bipyramid

Figure 7.2.7b-c Different projections.

7.3 Trigonal and hexagonal groups

P3 No.143 x,y,z; -y,x-y,z: y-x,-x,z; or in Cartesian coordinates

1 3 3 1 1 3 3 1 x, y, z;! x ! y, x ! y, z;! x + y, ! x ! y, z; 2 2 2 2 2 2 2 2

1 3 1 3 1 3 1 3 ! x ! y +2 3(! y+ x) +2z ! x + y +2 3(- y! x) +2z x+ 2 3y+ 2z 2 2 2 2 2 2 2 2 e + e + e 7.3.1

5 = 10

The name is trigonal pyramid (Shub 2) and is shown in figure 7.3.1.

10

7.3.1 a Trigonal pyramid b Different projection

P 3 No.147 x,y,z; -y,x-y,z: y-x,-x,z; -x,-y,-z; y,y-x,-z: x-y,x,-z; or in Cartesian coordinates

1 3 3 1 1 3 3 1 x, y, z;-x, -y,-z;! x ! y, x ! y, z;! x + y, ! x ! y, z; 2 2 2 2 2 2 2 2

1 3 3 1 1 3 3 1 x ! y, x + y, -z; x + y, ! x + y, -z; 2 2 2 2 2 2 2 2

1 3 1 3 1 3 1 3 ! x ! y +2 3(! y+ x) +2z ! x + y +2 3(- y! x) +2z x+ 2 3y+ 2z 2 2 2 2 2 2 2 2 e + e + e +

1 3 1 3 1 3 1 3 ![! x! y+ 2 3 (! y + x )+2z] ! [! x + y +2 3(- y! x) +2z] ![x+ 2 3y+ 2z] 2 2 2 2 2 2 2 2 e + e + e 7.3.2

5 = 10

The name is rhombohedron (Shub 27) and is shown in figure 7.3.2.

7.3.2 a Rhombohedron b Different projecti 11

P312 No.149 x,y,z; -y,x-y,z; y-x,-x,z; -y,-x,-z; x,x-y,-z; y-x,y,-z; or in Cartesian coordinates

1 3 3 1 1 3 3 1 x, y, z;! x, y, -z;! x ! y, x ! y, z; x + y, x ! y, -z; 2 2 2 2 2 2 2 2

1 3 3 1 1 3 3 1 ! x + y, ! x ! y, z; x ! y, ! x ! y, -z; 2 2 2 2 2 2 2 2

1 3 1 3 1 3 1 3 ! x ! y +2 3[! y + x ]+2z x+ y+2 3[! y+ x] !2z x+ 2 3y+ 2z 2 2 2 2 2 2 2 2 e e e + + + 7.3.3 1 3 1 3 1 3 1 3 ! x+ y+ 2 3[! y! x] +2z x ! y +2 3[! y ! x ]!2z ! x+2 3y !2z 2 2 2 2 2 2 2 2 5 e + e + e = 10

The name is trigonal trapezohedron(Shub 24) and is shown in figure 7.3.3.

7.3.3 a Trigonal trapezohedron b Different projection

12

P3m1 No.156 x,y,z; -y,x-y,z; y-x,-x,z; -y,-x,z; x,x-y,z; y-x,y,z; or in Cartesian coordinates

1 3 3 1 1 3 3 1 x, y, z;! x, y, z;! x ! y, x ! y, z; x + y, x ! y, z; 2 2 2 2 2 2 2 2

1 3 3 1 1 3 3 1 ! x + y, ! x ! y, z; x ! y, ! x ! y, z; 2 2 2 2 2 2 2 2

1 3 1 3 1 3 1 3 (! x ! y +2 3[! y + x ]+2z) (! x + y +2 3[! y! x] +2z) (x+ 2 3y+2z ) 2 2 2 2 2 2 2 2 e e e + + + 7.3.4 1 3 1 3 1 3 1 3 ( x + y +2 3[! y + x +2z )] ( x! y+2 3[! y! x] +2z) (! x+2 3y +2z ) 2 2 2 2 2 2 2 2 6 e + e + e = 10

The name is ditrigonal pyramid (Shub 3) and is shown in figure 7.3.4.

7.3.4a Ditrigonal pyramid b Different projection

13

P 3 m1 No.164 x,y,z; -y,x-y,z; y-x,-x,z; -y,-x,z; x,x-y,z; y-x,y,z; -(x,y,z); -(-y,x-y,z); -(y-x,-x,z); -(-y,-x,z); -(x,x-y,z); -(y-x,y,z); or in Cartesian coordinates

1 3 3 1 1 3 3 1 x, y, z;! x, y, z;! x ! y, x ! y, z; x + y, x ! y, z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 ! x + y, ! x ! y, z; x ! y, ! x ! y, z; 2 2 2 2 2 2 2 2

1 3 3 1 1 3 3 1 -(x,y, z);-( !x, y, z); -(! x ! y, x ! y, z);-( x + y, x ! y, z); 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 !(! x + y, ! x ! y, z);-( x ! y, ! x ! y, z); 2 2 2 2 2 2 2 2

1 3 1 3 1 3 1 3 (! x ! y +2 3[! y + x ]+2z) ( x + y +2 3[! y + x ]+2z) (x+ 2 3y+2z ) 2 2 2 2 2 2 2 2 e + e + e +

1 3 1 3 1 3 1 3 (! x+ y+ 2 3[! y! x] +2z) ( x! y+ [! y ! x ]+2z) (! x+2 3y +2z ) 2 2 2 2 2 2 2 2 e e e + + + 7.3.5 1 3 1 3 1 3 1 3 ! (! x ! y +2 3[! y + x ]+2z) ! ( x + y +2 3[! y + x ]+2z) ! (x +2 3y +2z ) 2 2 2 2 2 2 2 2 e + e + e +

1 3 1 3 1 3 1 3 !(! x+ y+ 2 3[! y! x] +2z) ! ( x! y+[! y ! x ]+2z) ! (! x +2 3y+ 2z ) 2 2 2 2 2 2 2 2 6 e + e + e = 10

The name is ditrigonal scalenohedron (shub 35) and is shown in figure 7.3.5.

7.3.5 a Ditrigonal scalenohedron b Different projection 14

Hexagonal groups

P6 No.168 x,y,z; -y,x-y,z: y-x,-x,z; -x,-y;,z y,y-x,z: x-y,x,z; or in Cartesian coordinates

1 3 3 1 1 3 3 1 x, y, z;-x, -y,z;! x ! y, x ! y, z;! x + y, ! x ! y, z; 2 2 2 2 2 2 2 2

1 3 3 1 1 3 3 1 x ! y, x + y, z; x + y, ! x + y, z; 2 2 2 2 2 2 2 2

1 3 1 3 1 3 1 3 ! x! y+2 3( ! y+ x)+2z ! x+ y+2 3(- y! x)+2z x+2 3y+2z 2 2 2 2 2 2 2 2 e + e + e + 7.3.6 1 3 1 3 1 3 1 3 x! y+2 3( y+ x)+2z x+ y+2 3( y! x)+2z !x!2 3y+2z 8 e + e2 2 2 2 + e 2 2 2 2 = 10

The name is hexagonal pyramid (Shub 6) and is shown in figure 7.3.6.

7.3.6 a Hexagonal pyramid b Different projection

P 6 No.174 x,y,z; -y,x-y,z: y-x,-x,z; x,y,-z; -y,x-y,-z: y-x,-x,-z; or in Cartesian coordinates

15

1 3 3 1 1 3 3 1 x, y, z;! x ! y, x ! y, z;! x + y, ! x ! y, z; 2 2 2 2 2 2 2 2

1 3 3 1 1 3 3 1 x, y, -z;! x ! y, x ! y, -z;! x + y, ! x ! y, -z; 2 2 2 2 2 2 2 2

1 3 1 3 1 3 1 3 ! x! y+2 3( ! y+ x)+2z ! x+ y+2 3(- y! x)+2z x+2 3y+2z 2 2 2 2 2 2 2 2 e + e + e + 7.3.7 1 3 1 3 1 3 1 3 (! x! y)+2 3( ! y+ x)!2z (! x+ y+2 3( ! y! x)!2z x+2 3y!2z 8 e + e 2 2 2 2 + e 2 2 2 2 = 10

Trigonal bipyramid is the name (Shub 9) and shown in figure 7.3.7. Mathematically this class may exist, but to date no mineral is known to crystallise with this form. [Bob’s rockshop]. In inorganic chemistry as well as in mineralogy it is one of the co-ordination polyhedra.

7.3.7 a Trigonal bipyramid b Different projection

16

P6/m No.175 x,y,z; -y,x-y,z; y-x,-x,z; -x,-y,-z; y,y-x,-z; x-y,x,-z; -x,-y,z; y,y-x,z; x-y,x,z; x,y,-z; -y,x-y,-z; y-x,-x,-z; or in Cartesian coordinates

1 3 3 1 1 3 3 1 x, y, z;-x, -y, z;! x ! y, x ! y, z;! x + y, ! x ! y, z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 x ! y, x + y, z; x + y, ! x + y, z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 -x, -y, -z;x, y, -z; x + y, - x + y, -z; x ! y, x + y, -z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 ! x + y, - x ! y, -z;- x ! y, x ! y, -z; 2 2 2 2 2 2 2 2

1 3 1 3 1 3 1 3 ! x ! y +2 3(! y+ x) +2z ! x + y +2 3(- y! x) +2z x+ 2 3y+ 2z 2 2 2 2 2 2 2 2 e + e + e +

1 3 1 3 1 3 1 3 x ! y +2 3( y + x )+2z x+ y+ 2 3 ( y! x) +2z ! x!2 3y +2z 2 2 2 2 2 2 2 2 e e e + + + 7.3.8 1 3 1 3 1 3 1 3 ! (! x ! y +2 3(! y+ x) +2z) ! (! x+ y+ 2 3 (- y! x) + 2z) ! (x+2 3y +2z ) 2 2 2 2 2 2 2 2 e + e + e +

1 3 1 3 1 3 1 3 !( x ! y +2 3( y + x )+2z) !( x + y +2 3( y ! x )+2z) ! (! x !2 3y+ 2z) 2 2 2 2 2 2 2 2 12 e + e + e = 10

The name is hexagonal bipyramid (Shub 13) and is shown in figure 7.3.8.

7.3.8a Hexagonal bipyramid b Different projection

17

P622 No.177 x,y,z; -y,x-y,z; y-x,-x,z; -x,-y,z; y,y-x,z; x-y,x,z; y,x,-z; -x,y-x,-z; x-y,-y,-z; -y,-x,-z; x,x-y,-z; y-x,y,-z;

1 3 3 1 1 3 3 1 x, y, z;! x, y, -z;! x ! y, x ! y, z; x + y, x ! y, z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 ! x + y, ! x ! y, -z; x ! y, ! x ! y, -z; 2 2 2 2 2 2 2 2

1 3 3 1 1 3 3 1 -x,-y,+z; x, -y,-z; x + y, - x + y, +z; - x ! y, - x + y, +z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 x ! y, x + y, -z;- x + y, + x + y, -z; 2 2 2 2 2 2 2 2

The name is hexagonal trapezohedron (Shub 28) and is shown in figure 7.3.9.

1 3 1 3 1 3 1 3 ! x! y+2 3( ! y+ x)+2z ! x+ y+2 3(- y! x)+2z x+2 3y+2z 2 2 2 2 2 2 2 2 e + e + e +

1 3 1 3 1 3 1 3 x! y+2 3( y+ x)+2z x+ y+2 3( y! x)+2z !x!2 3y+2z 2 2 2 2 2 2 2 2 e + e + e + 7.3.9 1 3 1 3 1 3 1 3 x! y+2 3( ! y! x)!2z x+ y+2 3( y! x)!2z !x+2 3y!2z e + e2 2 2 2 + e2 2 2 2 +

1 3 1 3 1 3 1 3 ! x! y+2 3( y! x)!2z ! x+ y+2 3( y+ x)!2z x!2 3y!2z 12 e + e 2 2 2 2 + e 2 2 2 2 = 10

7.3.9a Hexagonal trapezohedron b Different projection 18

P6mm No.183 x,y,z; -y,x-y,z; y-x,-x,z; -x,-y,z; y,y-x,z; x-y,x,z; y,x,z; -x,y-x,z; x-y,-y,z; -y,-x,z; x,x-y,z; y-x,y,z;

1 3 3 1 1 3 3 1 x, y, z;! x, y, z;! x ! y, x ! y, z; x + y, x ! y, z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 ! x + y, ! x ! y, z; x ! y, ! x ! y, z; 2 2 2 2 2 2 2 2

1 3 3 1 1 3 3 1 -x,-y,z;x, -y,z; x + y, - x + y, z;- x ! y, - x + y, z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 x ! y, x + y, z;- x + y, + x + y, z; 2 2 2 2 2 2 2 2

The name is dihexagonal pyramid (Shub 7) and is shown in figure 7.3.10.

1 3 1 3 1 3 1 3 ! x ! y +2 3(! y+ x) +2z ! x + y +2 3(- y! x) +2z x+ 2 3y+ 2z 2 2 2 2 2 2 2 2 e + e + e +

1 3 1 3 1 3 1 3 x ! y +2 3( y + x )+2z x+ y+ 2 3 ( y! x) +2z ! x!2 3y +2z 2 2 2 2 2 2 2 2 e e e + + + 7.3.10 1 3 1 3 1 3 1 3 x ! y +2 3(! y! x) +2z x+ y+2 3( y ! x )+2z ! x+2 3y +2z 2 2 2 2 2 2 2 2 e + e + e +

1 3 1 3 1 3 1 3 ! x ! y +2 3( y ! x )+2z ! x+ y+2 3( y + x )+2z x! 2 3y+2z 2 2 2 2 2 2 2 2 25 e + e + e = 10

7.3.10a Dihexagonal pyramid 19

P 6 m2 No.187 x,y,z; -y,x-y,z; y-x,-x,z; x,y,-z; y,x-y,-z; y-x,-x,-z; -y,-x,z; x,x-y,z; y-x,y,z; -y,-x,-z; x,x-y,-z; y-x,y,-z;

1 3 3 1 1 3 3 1 x,y,z;! x ! y, x ! y,z;! x + y,! x ! y,z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 x,y,-z;! x ! y, x ! y,-z;! x + y,! x ! y,-z; 2 2 2 2 2 2 2 2

1 3 3 1 1 3 3 1 !x,y,z; x ! y,! x ! y,z; x + y, x ! y,z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 !x,y,-z; x ! y,! x ! y,-z; x + y, x ! y,-z; 2 2 2 2 2 2 2 2

The name is ditrigonal bipyramid (Shub10) and is shown in figure 7.3.11.

1 3 1 3 1 3 1 3 ! x ! y +2 3(! y+ x) +2z ! x + y +2 3(- y! x) +2z x+ 2 3y+ 2z 2 2 2 2 2 2 2 2 e + e + e +

1 3 1 3 1 3 1 3 (! x ! y )+2 3 (! y+ x) !2z (! x + y )+2 3 (! y! x) !2z x+ 2 3y!2z 2 2 2 2 2 2 2 2 e e e + + + 7.3.11 1 3 1 3 1 3 1 3 x ! y +2 3(! y! x) +2z x+ y+2 3(- y+ x) +2z ! x+2 3y +2z 2 2 2 2 2 2 2 2 e + e + e +

1 3 1 3 1 3 1 3 ( x ! y )+2 3 (! y! x) !2z ( x + y +2 3(! y+ x) !2z ! x+2 3y !2z 2 2 2 2 2 2 2 2 8 e + e + e = 10

7.3.11a Ditrigonal bipyramid b Different projection 20

P6mmm No.191 x,y,z; -y,x-y,z; y-x,-x,z; y,x,z; -x,y-x,z; x-y,-y,z; -x,-y,z; y,y-x,z; x-y,x,z; -y,-x,z; x,x-y,z; y-x,y,z; x,y,-z; -y,x-y,-y-x,-x,-z; y,x,-z; -x,y-x,-z; x-y,-y,-z; -x,-y,-z; y,y-x,-z; x-y-z; -y,-x,-z; x,x-y,-z; y-x,y,-z; or Cartesian

1 3 3 1 1 3 3 1 x, y, z;!x, y, z;! x ! y, x ! y, z; x + y, x ! y, z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 ! x + y, ! x ! y, z; x ! y, ! x ! y, z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 -x,-y,z;x, -y,z; x + y, - x + y, z;- x ! y, - x + y, z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 x ! y, x + y, z;- x + y, + x + y, z; 2 2 2 2 2 2 2 2

1 3 3 1 1 3 3 1 x, y, -z;!x, y, -z;! x ! y, x ! y, -z; x + y, x ! y, -z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 ! x + y, ! x ! y, -z; x ! y, ! x ! y, -z; 2 2 2 2 2 2 2 2

1 3 3 1 1 3 3 1 -x,-y,-z;x, -y, -z; x + y, - x + y, -z;- x ! y, - x + y, -z; 2 2 2 2 2 2 2 2 1 3 3 1 1 3 3 1 x ! y, x + y, -z;- x + y, + x + y, -z; 2 2 2 2 2 2 2 2

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1 3 1 3 1 3 1 3 ! x! y+2 3( ! y+ x)+2z ! x+ y+2 3(- y! x)+2z x+2 3y+2z e + e 2 2 2 2 + e 2 2 2 2 +

1 3 1 3 1 3 1 3 x! y+2 3( y+ x)+2z x+ y+2 3( y! x)+2z !x!2 3y+2z e + e2 2 2 2 + e 2 2 2 2 +

1 3 1 3 1 3 1 3 x! y+2 3( ! y! x)+2z x+ y+2 3( y! x)+2z !x+2 3y+2z 2 2 2 2 2 2 2 2 e + e + e +

1 3 1 3 1 3 1 3 ! x! y+2 3( y! x)+2z ! x+ y+2 3( y+ x)+2z x!2 3y+2z 2 2 2 2 2 2 2 2 e + e + e + 7.3.12 1 3 1 3 1 3 1 3 ! x! y+2 3( ! y+ x)!2z ! x+ y+2 3(- y! x)!2z x+2 3y!2z e + e 2 2 2 2 + e 2 2 2 2 +

1 3 1 3 1 3 1 3 x! y+2 3( y+ x)!2z x+ y+2 3( y! x)!2z !x!2 3y!2z e + e 2 2 2 2 + e 2 2 2 2 +

1 3 1 3 1 3 1 3 x! y+2 3( ! y! x)!2z x+ y+2 3( y! x)!2z !x+2 3y!2z e + e2 2 2 2 + e2 2 2 2 +

1 3 1 3 1 3 1 3 ! x! y+2 3( y! x)!2z ! x+ y+2 3( y+ x)!2z x!2 3y!2z 25 e + e 2 2 2 2 + e 2 2 2 2 = 10

The name is bihexagonal bipyramid (Shub14) and is shown in figure 7.3.12.

7.3.12a Bihexagonal bipyramid b Different projection

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7.4 Special positions in the hexagonal or trigonal systems to describe simple Shubnikov forms, and closed bodies important in crystal chemistry.

We start with making an open trigonal prism after group P312 with the positions 3j in the equation 7.4.1. The Shubnikov number for this form is 16, and the form is in figure 7.4.1.

1 3 3 1 1 3 3 1 x y ! x! y+ x! y ! x+ y! x! y e + + e 2 2 2 2 + e 2 2 2 2 7.4.1 5 = 10 We continue with the open ditrigonal prism using group P6m2 with the special position in 6l for the equation in 7.4.2. The shape is in figure 7.4.2 and Shubnikov number is 17.

1 3 3 1 1 3 3 1 ! x! y+ x! y ! x+ y! x! y ex+ y + e 2 2 2 2 + e 2 2 2 2 1 3 3 1 1 3 3 1 x y x y x y x y x y + + ! ! ! ! e! + + e 2 2 2 2 + e2 2 2 2 7.4.2 6 = 10

Fig 7.4.1 Fig 7.4.2

Next is the open tetragonal prism in group P4/m using 4j that gives the equation in 7.4.3. The form is in figure 7.4.3 and Shubnikov number is 18.

x 2 y x 2y 2x y 2x y 6 e + + e! ! + e! + + e ! =10 7.4.3

The ditetragonal prism is in group P4/mmm and we use 8p for the equation in 7.4.4. The form is in figure 7.4.4 and Shubnikov number is 19.

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x 2 y x 2y 2x y 2x y e + + e! ! + e! + + e ! + 7.4.4 x 2y x 2y 2x y 2x y 10 e ! + + e ! + e + + e! ! =10

Figure 7.4.3 Figure 7.4.4

We continue with the open hexagonal prism using group P6/m with the special position in 6j for the equation in 7.4.5. The Shubnikov number is 20 for this form, which is in figure 7.4.5a.

1 3 1 3 1 3 1 3 ! x! y+(! y+ x) ! x+ y+(- y! x) ex+ y + e 2 2 2 2 + e 2 2 2 2 + 1 3 1 3 1 3 1 3 x y x! y+( y+ x) x+ y+( y! x) e! ! + e 2 2 2 2 + e2 2 2 2 7.4.5 10 = 10

For all these open forms it is possible to make ordinary polyhedra important in inorganic crystal chemistry, using special positions like 2e in this group P6/m So we make the closed z z hexagonal prism adding the term e + e! to the equation above. And the prism is in figure 7.4.5b.

Figure 7.4.5a Figure 7.4.5b 24

The open dihexagonal prism is in figure 7.4.6 after equation 7.4.6 from the group P6/mmm and its special position 12p. The Shubnikov number is 21 for this form. In order to increase resolution we have multiplied the y-terms in the equation below, like we have done in the above part of this chapter.

1 3 1 3 1 3 1 3 ! x! y+2 3(! y+ x) ! x+ y+2 3(- y! x) ex+2 3y + e 2 2 2 2 +e 2 2 2 2 + 1 3 1 3 1 3 1 3 x! y+2 3(! y! x) x+ y+2 3(- y+ x) e!x +2 3y + e2 2 2 2 + e 2 2 2 2 + 1 3 1 3 1 3 1 3 x 2 3y x! y+2 3( y+ x) x+ y+2 3( y! x) e! ! +e 2 2 2 2 + e2 2 2 2 + 1 3 1 3 1 3 1 3 ! x! y+2 3( y! x) ! x+ y+2 3( y+ x) ex!2 3y + e 2 2 2 2 + e 2 2 2 2 + 30 = 10 7.4.6

Figure 7.4.6

References 13

1 A.V.Shubnikov and V.A.Koptsik, Symmetry IN SCIENCE AND ART, PLENUM,1974, page 73.

2... Bob’s Rockshop: Introduction to Crystallography and Mineral Crystal Systems. http://www.rockhounds.com.

3 INTERNATIONAL TABLES for X-RAY CRYSTALLOGRAPHY, The Kynoch Press, 1969.