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.**

Bilateral Octagon

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

Figure above **:** Microscopic view of a two-dimensional bilaterally octagonal D_{1} crystal. It consists of the periodic stacking of rectangular building blocks, and is, in the present Figure, provided with D_{1} motifs (black). These D_{1} motifs represent the translation-free residue of the crystal (all the same whether it belongs to the plane group Pm, Pg or Cm), and in this example the residue has two antimers, and, drawn as such, the plane group to which the crystal belongs is Pm. So the crystal of this example itself has two antimers and is thus non-eupromorphic (because the crystal's intrinsic shape suggests eight antimers). 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).

The intrinsic shape of our crystal is that of a a bilateral octagon. The grammatical adjective of the latter would be "bilaterally octagonal" as in "bilaterally octagonal crystal" (where "bilaterally" is an adverb). But because such a crystal itself is octagonal as well as bilateral, we can legitimately use the expression "

Figure above **:** Macroscopic view of the two-dimensional bilateral octogonal crystal of the previous Figure with intrinsic symmetry according to the group D_{1} . This macroscopic view is obtained by removing the lattice connection lines (indicating building blocks) and the motifs.

The pattern of

Figure above **:** Pattern of symmetry elements of the above given bilateral octagonal D_{1} two-dimensional crystal. Its consists of one mirror line (red) only.

Figure above **:** Five crystallographic Forms (green, black, red, dark blue, light blue) are needed to construct the faces of our bilateral octagonal D_{1} two-dimensional crystal **:** An initially given oblique face (green, not parallel, neither perpendicular to the mirror line) implies one more face in virtue of the mirror line, resulting in one open Form consisting of two faces (green). In the same way a second Form can be generated from another initially given face (black) also not parallel, neither perpendicular to the mirror line. A third initially given face (red), parallel to the mirror line will yield one more such face, resulting in a third Form. Then a fourth initially given face (dark blue) perpendicular to the mirror line will not imply yet another face, so we then have a fourth Form consisting of one face only. Finally, in the same way a fifth initially given face (light blue) directly represents one Form. These five Forms combine to give our bilateral octagonal crystal.

Figure above **:** A bilateral octagonal D_{1} two-dimensional crystal. The case of __ t w o__ congruent (symmetric) 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 is the same as the previous Figure, but now with the lattice lines and motifs omitted and in this way presenting a

Figure above **:** Macroscopic view of the bilateral octagonal D_{1} two-dimensional crystal with two antimers (green, yellow).

The

Figure above **:** The promorph of the bilateral octagonal crystal with two antimers. It is an isosceles triangle (half a rhombus) and as such the two-dimensional analogue of the isosceles pyramid (half a rhombic pyramid), which represents the promorph of corresponding three-dimensional crystals or other objects. Note the difference in shape between this promorph (isosceles triangle ( = isosceles trigon) and that of the crystal (bilateral octagon) of which it is the promorph. Radial (R) and interradial (IR) directions are indicated.

Figure above **:** A two-dimensional bilateral octagonal crystal with intrinsic D_{1} symmetry. Its D_{1} motifs (black) have six antimers. Microscopic view

Figure above **:** The bilateral octagonal D_{1} two-dimensional crystal of the previous Figure. The case of __ s i x__ similar antimers (green, yellow, blue). 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

The next Figure is the same as the previous Figure, but now with the lattice lines and motifs omitted and in this way presenting a

Figure above **:** Macroscopic view of the just depicted bilateral octagonal D_{1} two-dimensional crystal with six antimers.

The division of the crystal into areas representing the six antimers, i.e. the allocation of the the boundaries of these antimers, as presented in the above drawings, has been partly arbitrary. As always, the only

Figure above **:** Alternative partition into antimers (green, yellow, blue) of the bilateral octagonal D_{1} two-dimensional crystal inder investigation. Compare with the earlier version above .

The

Figure above **:** The promorph (two images) of the bilateral octagonal crystal with six antimers. It is half a 12-fold amphitect polygon and as such the two-dimensional analogue of half a 12-fold amphitect pyramid, which represents the promorph of corresponding three-dimensional crystals or other objects.

Figure above **:** A two-dimensional bilateral octagonal crystal with intrinsic D_{1} symmetry. Its D_{1} motifs (black) have eight antimers. Microscopic view

Figure above **:** The bilateral octagonal D_{1} two-dimensional crystal of the previous Figure. 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

The next Figure is the same as the previous Figure, but now with the lattice lines and motifs omitted and in this way presenting a

Figure above **:** Macroscopic view of the bilateral crystal under investigation. The non-congruity of the eight antimers is clearly visible.

The next Figure depicts an alternative division (into antimers) of this crystal (still) having eight antimers.

Figure above **:** Alternative partition into antimers (green, yellow) of the above discussed bilateral octagonal D_{1} crystal with eight antimers. Compare with the earlier version .

The

Figure above **:** The promorph (two images) of the bilateral octagonal crystal with eight antimers. It is half a 16-fold amphitect polygon and as such the two-dimensional analogue of half a 16-fold amphitect pyramid, which represents the promorph of corresponding three-dimensional crystals or other objects.

Figure above **:** A two-dimensional bilateral octagonal crystal with intrinsic D_{1} symmetry. Its D_{1} motifs (black) have four radially arranged antimers. Microscopic view

Figure above **:** The bilateral octagonal D_{1} two-dimensional crystal of the previous Figure. The case of __ f o u r__ similar antimers (green, yellow) in radial configuration. 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

And the

Figure above **:** Macroscopic view of the bilateral octagonal D_{1} two-dimensional crystal, with four similar antimers (green, yellow) in radial configuration.

The

Figure above **:** The promorph of the bilateral octagonal crystal with four radially arranged antimers (previous Figures). It is a bi-isosceles triangle and as such the two-dimensional analogue of the bi-isosceles pyramid which is the promorph of corresponding three-dimensional crystals or other objects.

Figure above **:** A two-dimensional bilateral octagonal crystal with intrinsic D_{1} symmetry. Its D_{1} motifs (black) have four interradially arranged antimers. Microscopic view

Figure above **:** The bilateral octagonal D_{1} two-dimensional crystal of the previous Figure. The case of __ f o u r__ similar antimers (green, yellow) in interradial configuration. 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

And the

Figure above **:** Macroscopic view of the bilateral octagonal D_{1} two-dimensional crystal, with four similar antimers (green, yellow) in interradial configuration.

The

Figure above **:** The promorph of the bilateral octagonal crystal with four interradially arranged antimers (previous Figures). It is an isosceles trapezium and as such the two-dimensional analogue of the isoscelesly trapezoid pyramid which is the promorph of corresponding three-dimensional crystals or other objects.

Figure above **:** A two-dimensional bilateral octagonal crystal with intrinsic D_{1} symmetry. Its D_{1} motifs (black) have three similar antimers, symmetrically arranged. Microscopic view

Figure above **:** The bilateral octagonal D_{1} two-dimensional crystal of the previous Figure. The case of __ t h r e e__ similar antimers (green, yellow, blue). 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

And the

Figure above **:** Macroscopic view of the bilateral octagonal D_{1} two-dimensional crystal, with three similar antimers (green, yellow, blue).

The

Figure above **:** The promorph of the bilateral octagonal crystal with three similar antimers (previous Figures). It is half a six-fold amphitect polygon and as such the two-dimensional analogue of half a six-fold amphitect pyramid which is the promorph of corresponding three-dimensional crystals or other objects.

In the

**e-mail :**

To continue click HERE