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Wholeness and the Implicate Order
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The Crystallization Process and the Implicate Order
(Part Eighteen)

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We  continue our investigation concerning the generation of group elements of the Plane Groups.
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The Plane Group P6mm (Sequel)

The next Figure shows the generation of all the remaining group elements of the composed motif s.l. at the lattice point  R .

Figure 1.  Generation of the remaining group elements represented by the basic units of the motif s.l. at the lattice point  R .

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The second row of motifs s.l. of the  P6mm  pattern can now be completed by means of translations :

Figure 2.  Completion of the filling-in of group elements of the second row of motifs s.l. of the  P6mm  pattern.

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The next Figure indicates how the first and third rows of the motifs s.l. of our  P6mm  pattern can be reached by rotations.

Figure 3.  The first and third rows of motifs s.l. can be reached by rotations about the point  R  of the elements making up the composed motif s.l. at the lattice point right next to the lattice point  R  :
To reach the first row, we use an anticlockwise rotation of 600 about the point
R ,  represented by the transformation  p .
To reach the third row, we can use an anticlockwise rotation of 3000 about the point
R ,  represented by the transformation  p5 .

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Figure 4.  The first and third rows have been reached, and the involved group elements generated.

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The fourth row can also be reached by a rotation of already generated elements, namely by rotating the elements that make up the last composed motif s.l. of the second row. When we rotate these elements anticlockwise by 3000 about the point  R ,  then we will reach the fourth row. The next Figure indicates this rotation.

Figure 5.  Indication how to reach the fourth row by applying the transformation  p5  to the elements of the last motif s.l. of the second row.

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Figure 6.  The fourth row has been reached, and the involved group elements are generated.

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The first, third, and fourth rows can now be completed by means of translations.
In order to do so we're going to assign concise labels to the areas representing group elements, because graphically they do not provide enough space for the group element symbols to be insered (And if we magnify the pattern still more, then we cannot have a convenient overview). We begin with labeling the lattice nodes (lattice points) with the letters

A,  B,  C,  D,  E,  F,  R,  G,  H,  I,  J,  K,  L,  M,  N,  O,  P.

Each such lattice node is associated with twelve triangular areas representing group elements. These areas are numbered

1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,

for every lattice node.
We now explain how to read the labels of these areas.

The twelve areas associated with the lattice node  A  must be read as

A1,  A2,  A3,  A4,  A5,  A6,  A7,  A8,  A9,  A10,  A11,  A12.

The twelve areas associated with the lattice node  B  must be read as

B1,  B2,  B3,  B4,  B5,  B6,  B7,  B8,  B9,  B10,  B11,  B12.

And so on.
An exception to this procedure is represented by the fact that we label the generators  p, m  and  t  with their proper symbols,  p, m  and  t . Also the initial element will be adressed by its proper symbol  1  (instead of R1). So the twelve areas associated with the lattice point  R  are given as

1  (initial element, identity element),  R2,  p,  m,  R5,  R6,  R7,  R8,  R9,  R10,  R11,  R12.

The areas associated with the lattice point  G  are then given by

t,  G2,  G3,  G4,  G5,  G6,  G7,  G8,  G9,  G10,  G11,  G12..

The next Figure illustrates this labeling, and is followed by a table that connects the labels with the corresponding generative symbols for the group elements.

Figure 7.  Labeling of the areas of the  P6mm  pattern, representing group elements. The tabel below relates these labels with the true group element symbols. The initial element and the generator elements have as their label directly their true group element symbol.

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In the next Table we give all group elements (and their corresponding labels) of the displayed part of our  P6mm  pattern, thus including the ones that have already been determined above. These latter will be displayed with white symbols, while the newly generated group elements -- obtained by completing the first, third and fourth rows by applying translations -- will be given in red.

Table of Labels and Group Element Symbols, referring to Figure 7

 Label Group element Label Group element Label Group element A1 t-2ptp5 A2 t-2ptp4m A3 t-2pt A4 t-2ptp5m A5 t-2ptp A6 t-2ptm A7 t-2ptp2 A8 t-2ptpm A9 t-2ptp3 A10 t-2ptp2m A11 t-2ptp4 A12 t-2ptp3m B1 t-1ptp5 B2 t-1ptp4m B3 t-1pt B4 t-1ptp5m B5 t-1ptp B6 t-1ptm B7 t-1ptp2 B8 t-1ptpm B9 t-1ptp3 B10 t-1ptp2m B11 t-1ptp4 B12 t-1ptp3m C1 ptp5 C2 ptp4m C3 pt C4 ptp5m C5 ptp C6 ptm C7 ptp2 C8 ptpm C9 ptp3 C10 ptp2m C11 ptp4 C12 ptp3m D1 tptp5 D2 tptp4m D3 tpt D4 tptp5m D5 tptp D6 tptm D7 tptp2 D8 tptpm D9 tptp3 D10 tptp2m D11 tptp4 D12 tptp3m E1 t2ptp5 E2 t2ptp4m E3 t2pt E4 t2ptp5m E5 t2ptp E6 t2ptm E7 t2ptp2 E8 t2ptpm E9 t2ptp3 E10 t2ptp2m E11 t2ptp4 E12 t2ptp3m F1 t-1 F2 t-1p5m F3 t-1p F4 t-1m F5 t-1p2 F6 t-1pm F7 t-1p3 F8 t-1p2m F9 t-1p4 F10 t-1p3m F11 t-1p5 F12 t-1p4m 1 1 R2 p5m p p m m R5 p2 R6 pm R7 p3 R8 p2m R9 p4 R10 p3m R11 p5 R12 p4m G1 t G2 tp5m G3 tp G4 tm G5 tp2 G6 tpm G7 tp3 G8 tp2m G9 tp4 G10 tp3m G11 tp5 G12 tp4m H1 t2 H2 t2p5m H3 t2p H4 t2m H5 t2p2 H6 t2pm H7 t2p3 H8 t2p2m H9 t2p4 H10 t2p3m H11 t2p5 H12 t2p4m I 1 t-2p5tp I 2 t-2p5tm I 3 t-2p5tp2 I 4 t-2p5tpm I 5 t-2p5tp3 I 6 t-2p5tp2m I 7 t-2p5tp4 I 8 t-2p5tp3m I 9 t-2p5tp5 I 10 t-2p5tp4m I 11 t-2p5t I 12 t-2p5tp5m J1 t-1p5tp J2 t-1p5tm J3 t-1p5tp2 J4 t-1p5tpm J5 t-1p5tp3 J6 t-1p5tp2m J7 t-1p5tp4 J8 t-1p5tp3m J9 t-1p5tp5 J10 t-1p5tp4m J11 t-1p5t J12 t-1p5tp5m K1 p5tp K2 p5tm K3 p5tp2 K4 p5tpm K5 p5tp3 K6 p5tp2m K7 p5tp4 K8 p5tp3m K9 p5tp5 K10 p5tp4m K11 p5t K12 p5tp5m L1 tp5tp L2 tp5tm L3 tp5tp2 L4 tp5tpm L5 tp5tp3 L6 tp5tp2m L7 tp5tp4 L8 tp5tp3m L9 tp5tp5 L10 tp5tp4m L11 tp5t L12 tp5tp5m M1 t-3p5t2p M2 t-3p5t2m M3 t-3p5t2p2 M4 t-3p5t2pm M5 t-3p5t2p3 M6 t-3p5t2p2m M7 t-3p5t2p4 M8 t-3p5t2p3m M9 t-3p5t2p5 M10 t-3p5t2p4m M11 t-3p5t2 M12 t-3p5t2p5m N1 t-2p5t2p N2 t-2p5t2m N3 t-2p5t2p2 N4 t-2p5t2pm N5 t-2p5t2p3 N6 t-2p5t2p2m N7 t-2p5t2p4 N8 t-2p5t2p3m N9 t-2p5t2p5 N10 t-2p5t2p4m N11 t-2p5t2 N12 t-2p5t2p5m O1 t-1p5t2p O2 t-1p5t2m O3 t-1p5t2p2 O4 t-1p5t2pm O5 t-1p5t2p3 O6 t-1p5t2p2m O7 t-1p5t2p4 O8 t-1p5t2p3m O9 t-1p5t2p5 O10 t-1p5t2p4m O11 t-1p5t2 O12 t-1p5t2p5m P1 p5t2p P2 p5t2m P3 p5t2p2 P4 p5t2pm P5 p5t2p3 P6 p5t2p2m P7 p5t2p4 P8 p5t2p3m P9 p5t2p5 P10 p5t2p4m P11 p5t2 P12 p5t2p5m

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So we have now generated all group elements of the displayed part of the  P6mm  pattern.
And this was the last Plane Group of the total of 17.
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In the next -- and last -- document devoted to the 'noetic crystallization process' of two-dimensional crystals, we will summarize these 17 basic patterns as they are exhaustively partitioned in areas that represent group elements.

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