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 Octagon

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

Figure above **:** Microscopic view of a two-dimensional amphitect octagonal 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 eight antimers. So the crystal of this example itself has eight antimers and is thus eupromorphic (because the eight 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 center of the crystal is highlighted (green).

By removing the lattice lines (indicating building blocks) and motifs, we obtain a

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

The pattern of

Figure above **:** Pattern of symmetry elements of the above given amphitect octagonal 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 **:** Three crystallographic Forms (green, red, dark blue) are needed to construct the faces of our amphitect octagonal D_{2} two-dimensional crystal **:** An initially given oblique face (green, not parallel to either mirror line) implies three more faces in virtue of the symmetry elements, resulting in one Form consisting of four faces (green). 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 octagon. Two more Forms are 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. Another initially given face (dark blue) parallel to the other mirror line also implies one more face, resulting in yet another Form (dark blue) consisting of two parallel faces. These three Forms combined finally yield our amphitect octagonal crystal.

Figure above **:** An amphitect octagonal D_{2} two-dimensional crystal. The case of __ e i g h t__ 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

By removing lattice lines and motifs we obtain a

Figure above **:** Macroscopic view of the amphitect octagonal D_{2} two-dimensional crystal of the previous Figure with eight antimers (green, yellow).

The

Figure above **:** The promorph of the amphitect octagonal crystal with eight antimers. It is an 8-fold amphitect polygon (amphitect octagon) and as such the two-dimensional analogue of the 8-fold amphitect pyramid, which represents the promorph of corresponding three-dimensional crystals or other objects.

Figure above **:** An amphitect octagonal D_{2} two-dimensional crystal. Its D_{2} motifs (black) have four radially arranged antimers. Microscopic view.

Figure above **:** The amphitect octagonal 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

By removing lattice lines and motifs we obtain a

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

The

Figure above **:** The promorph of the amphitect octagonal 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 octagon) of which it is the promorph.

As noted earlier (previous document) the above promorph (

Figure above **:** Slightly different representation of the promorph (*Autopola Orthostaura Tetraphragma radialia*) of the amphitect octagonal 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 octagonal crystal.

Figure above **:** An amphitect octagonal D_{2} two-dimensional crystal. Its D_{2} motifs (black) have four interradially arranged antimers. Microscopic view.

Figure above **:** The amphitect octagonal 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

By removing lattice lines and motifs we obtain a

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

The

Figure above **:** The promorph of the amphitect octagonal 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 octagon) of which it is the promorph.

As noted earlier (previous document), also the above promorph (

Figure above **:** Slightly different representation of the promorph (*Autopola Orthostaura Tetraphragma interradialia*) of the amphitect octagonal 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 octagonal crystal. Compare with the representation of the Autopola Orthostaura Tetraphragma *radialia* as depicted **above** .

Figure above **:** An amphitect octagonal D_{2} two-dimensional crystal. Its D_{2} motifs (black) have two equal antimers. Microscopic view.

Figure above **:** The amphitect octagonal 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

By removing lattice lines and motifs we obtain a

Figure above **:** Macroscopic view of the amphitect octagonal D_{2} two-dimensional crystal of the previous Figure with two equal antimers (green, yellow).

The

Figure above **:** The promorph of the amphitect octagonal 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 octagon) of which it is the promorph.

In the

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