This document continues the investigation of special categories (If / Then constants), and compares crystals with organisms.

Crystals and Organisms, Shape, Symmetry and Promorph.

Sequel to the investigation of some (intrinsic) **shapes** of two-dimensional crystals regarding their relationship to intrinsic **point symmetry** and **promorph.**

Amphitect Hexagon

We will now investigate two-dimensional crystals with an intrinsic shape according to an

Figure above **:** Microscopic view of a two-dimensional amphitect hexagonal D_{2} crystal consisting of the periodic stacking of rectangular building blocks, explicitly provided with D_{2} motifs. These D_{2} motifs represent the translation-free residue of the crystal (all the same whether it belongs to the plane group P2mm, P2mg, P2gg or C2mm), and in this example the residue has six antimers. So the crystal of this example itself has six antimers and is thus eupromorphic (because the six antimers are already evident in the crystal's intrinsic shape). Because the building blocks are in fact very small (i.e. in crystals they have microscopic dimensions), all the crystal faces are macroscopically smooth.

The next Figure presents a

Figure above **:** Macroscopic view of the two-dimensional amphitect hexagonal D_{2} crystal of the previous Figure.

The pattern of

Figure above **:** Pattern of symmetry elements of the above given amphitect hexagonal D_{2} two-dimensional crystal. Its consists of two mirror lines (red) perpendicular to each other, and a 2-fold rotation axis (small yellow ellipse) at their point of intersection.

Figure above **:** Two crystallographic Forms are needed to construct the faces of our amphitect hexagonal D_{2} two-dimensional crystal **:** An initially given oblique face (dark blue, not parallel to either mirror line) implies three more faces in virtue of the symmetry elements, resulting in one Form consisting of four faces (dark blue). This is a closed Form, and as such can represent a crystal. However, this crystal has the shape of a rhombus, and not that of an amphitect hexagon. So one more Form is needed **:** An initially given face (red) parallel to one of the mirror lines, implies one more face in virtue of the symmetry elements, resulting in a Form (red) consisting of two parallel faces. Combined with the first Form it yields our amphitect hexagonal crystal.

Figure above **:** An amphitect hexagonal D_{2} two-dimensional crystal. The case of __ s i x__ similar antimers (green, yellow). Note the correspondence between the morphology of the (microscopic) motif (as translation-free residue of the crystal) and the arrangement of the (macroscopic) antimers of the crystal. In this way the promorph, and in particular the number of antimers, is based on the morphology of the translation-free residue of the crystal. This residue is explicitly given in the form of a D

Figure above **:** Same as previous Figure. The six antimers are indicated by numerals.

Or, using a different coloration (to indicate the six antimers)

Figure above **:** Same as previous Figure. The six antimers are indicated by the colors green, yellow and blue.

The next Figure gives a

Figure above **:** Macroscopic view of the amphitect hexagonal D_{2} two-dimensional crystal of the previous Figure.

The next Figure depicts the six antimers in an equivalent way (In fact the internal structure of the crystal is turned by 90

Figure above **:** An amphitect hexagonal D_{2} two-dimensional crystal. (Still) the case of six similar antimers (green, yellow, blue). Note again the correspondence between the morphology of the motif (as translation-free residue) and the arrangements of the (macroscopic) antimers of the crystal. So again, in this way the promorph, and in particular the number of antimers, is based on the morphology of the (mentioned) translation-free residue of the crystal. This residue is explicitly given in the form of a D_{2} motif (black) inside each rectangular building block. It is -- or represents -- an atomic configuration such that six antimers can be distinguished in it. The crystal is eupromorphic because its intrinsic shape suggests six antimers, which indeed are present.

The next Figure gives a

Figure above **:** Macroscopic view of the amphitect hexagonal D_{2} two-dimensional crystal of the previous Figure.

The

Figure above **:** The promorph of the amphitect hexagonal crystal with six antimers. It is a 6-fold amphitect polygon (amphitect hexagon) and as such the two-dimensional analogue of the 6-fold amphitect pyramid, which represents the promorph of corresponding three-dimensional crystals or other objects. Radial (R) and interradial (IR) directions (seen from the center of the polygon) are indicated.

Figure above **:** A two-dimensional amphitect hexagonal crystal with intrinsic D_{2} symmetry. Its D_{2} motifs (black) have four antimers in radial configuration. Microscopic view.

Figure above **:** The amphitect hexagonal D_{2} two-dimensional crystal of the previous Figure. The case of __ f o u r__ similar antimers (green, yellow). Note the correspondence between the morphology of the (microscopic) motif (as translation-free residue) and the arrangement of the (macroscopic) antimers of the crystal. In this way the promorph, and in particular the number of antimers is based on the morphology of the translation-free residue of the crystal. This residue is explicitly given in the form of a D

The next Figure gives the

Figure above **:** Macroscopic view of the amphitect hexagonal D_{2} two-dimensional crystal with four radially arranged antimers, of the previous Figure.

The

Figure above **:** The promorph of the amphitect hexagonal crystal with four antimers. It is a 4-fold amphitect polygon (rhombus) and as such the two-dimensional analogue of the rhombic pyramid, which represents the promorph of corresponding three-dimensional crystals or other objects. Note the difference in shape between this promorph (rhombus, amphitect tetragon) and that of the crystal (amphitect hexagon) of which it is the promorph.

The above promorph (

Figure above **:** Slightly different representation of the promorph (*Autopola Orthostaura Tetraphragma radialia*) of the amphitect hexagonal crystal with four antimers in radial configuration. It also is a 4-fold amphitect polygon (rhombus), but now expresses the unequality of the antimers as they are in the above amphitect hexagonal crystal.

Figure above **:** A two-dimensional amphitect hexagonal crystal with intrinsic D_{2} symmetry. Its D_{2} motifs (black) have four antimers in interradial configuration. Microscopic view.

Figure above **:** The amphitect hexagonal D_{2} two-dimensional crystal of the previous Figure. The interradial case of __ f o u r__ congruent (two by two equal) antimers (green, yellow). Note the correspondence between the morphology of the (microscopic) motif (as translation-free residue) and the arrangement of the (macroscopic) antimers of the crystal. In this way the promorph, and in particular the number of antimers is based on the morphology of the translation-free residue of the crystal. This residue is explicitly given in the form of a D

The next Figure gives the

Figure above **:** Macroscopic view of the amphitect hexagonal D_{2} two-dimensional crystal with four interradially arranged antimers, of the previous Figure.

The

Figure above **:** The promorph of the amphitect hexagonal crystal with four antimers. It is a rectangle and as such the two-dimensional analogue of the rectangular pyramid, which represents the promorph of corresponding three-dimensional crystals or other objects. Note the difference in shape between this promorph (rectangle, rectangular amphitect tetragon) and that of the crystal (amphitect hexagon) of which it is the promorph.

The above promorph (

Figure above **:** Slightly different representation of the promorph (*Autopola Orthostaura Tetraphragma interradialia*) of the amphitect hexagonal crystal with four antimers in interradial configuration. It is now a rhombus (not a rectangle), and expresses the overall non-equality (but still congruity -- two by two equal, and two by two symmetric) of the four antimers (green, yellow) as they are in the above amphitect hexagonal crystal. Compare with the representation of the Autopola Orthostaura Tetraphragma *radialia* as depicted **above** .

Figure above **:** A two-dimensional amphitect hexagonal crystal with intrinsic D_{2} symmetry. Its D_{2} motifs (black) have two equal antimers (They relate to each other by a half-turn). Microscopic view.

Figure above **:** The amphitect hexagonal D_{2} two-dimensional crystal of the previous Figure. The case of __ t w o__ equal antimers (green, yellow). Note the correspondence between the morphology of the (microscopic) motif (as translation-free residue) and the arrangement of the (macroscopic) antimers of the crystal. In this way the promorph, and in particular the number of antimers is based on the morphology of the translation-free residue of the crystal. This residue is explicitly given in the form of a D

The next Figure gives the

Figure above **:** Macroscopic view of the amphitect hexagonal D_{2} two-dimensional crystal of the previous Figure with two equal antimers.

The

Figure above **:** The promorph of the amphitect hexagonal crystal with two antimers. It is a 2-fold amphitect polygon (rhombus, in the present case -- as '2-fold' -- expressing two antimers) and as such the two-dimensional analogue of the 2-fold rhombic pyramid, which represents the promorph of corresponding three-dimensional crystals or other objects. Note the difference in shape between this promorph (rhombus, amphitect tetragon) and that of the crystal (amphitect hexagon) of which it is the promorph.

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