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.**

Isosceles Trapezium (Bilateral Tetragon)

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

Figure above **:** Microscopic view of a two-dimensional isoscelesly trapezoid (bilaterally tetragonal) D_{1} crystal consisting of the periodic stacking of rectangular building blocks, 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. As drawn, the crystal belongs to the plane group Pm (primitive rectangular lattice, pattern of symmetry elements **:** mirror lines only, all of them of the same type). So the crystal of this example itself has two antimers and is thus non-eupromorphic (because the crystal's intrinsic shape suggests four 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 intrinsic shape of our crystal is that of an isosceles trapezium or (equivalently) a bilateral tetragon. The grammatical adjective of the latter would be "bilaterally tetragonal" as in "bilaterally tetragonal crystal" (where "bilaterally" is an adverb). But because such a crystal itself is tetragonal as well as bilateral, we can legitimately use the expression "

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

Figure above **:** Macroscopic view of the bilateral tetragonal crystal of the previous Figure.

The pattern of

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

Figure above **:** Three crystallographic Forms (red, dark blue, green) are needed to construct the faces of our bilateral tetragonal D_{1} two-dimensional crystal **:** An initially given oblique face (red, 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 (red). Then a second initially given face (dark blue) will not imply yet another face, so we then have a second Form consisting of one face only. Finally, in the same way a third initially given face (green) directly represents one Form. These three Forms combine to give our bilateral tetragonal crystal.

Figure above **:** A bilateral tetragonal 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

Removing the lattice connection lines and the motifs, results in a

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

The

Figure above **:** The promorph of the bilateral tetragonal 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 tetragon, or equivalently, isosceles trapezium) of which it is the promorph. Radial (R) and interradial (IR) directions are indicated.

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

Figure above **:** The bilateral tetragonal D_{1} two-dimensional crystal of the previous Figure. 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 the colors green, yellow and blue.

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 six antimers (green, yellow, blue) is clearly visible.

The

Figure above **:** The promorph (two images) of the bilateral hexagonal 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. Note the (slight) difference in shape between this promorph (half a 12-fold amphitect polygon) and that of the crystal (bilateral hexagon) of which it is the promorph. Radial (R) and interradial (IR) directions are indicated.

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

Figure above **:** The bilateral tetragonal 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

Removing the lattice connection lines and the motifs, results in a

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

The

Figure above **:** The promorph of the bilateral hexagonal 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 bilateral tetragonal two-dimensional D_{1} crystal. Its D_{1} motifs (black) have four interradially arranged antimers. Microscopic view.

Figure above **:** The bilateral tetragonal 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

Removing the lattice connection lines and the motifs, results in a

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

The

Figure above **:** The promorph of the bilateral tetragonal 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. Here the intrinsic shape of the crystal is the same as that of its promorph (both an isosceles trapezium, or equivalently, a bilateral tetragon).

Figure above **:** A bilateral tetragonal two-dimensional D_{1} crystal. Its D_{1} motifs (black) have three symmetrically arranged antimers. Microscopic view.

Figure above **:** The bilateral tetragonal 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

The next Figure gives an alternative distribution of the three (macroscopical) antimers

Figure above **:** The bilateral tetragonal D_{1} two-dimensional crystal under investigation. Alternative distribution of the three (macroscopic) antimers (green, yellow, blue).

Removing the lattice connection lines and the motifs, results in a

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

The

Figure above **:** The promorph of the bilateral tetragonal 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

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