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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 bilateral octagon as to their possible promorphs, but, as in the foregoing, limiting ourselves to holomorphic crystals, which means that we will only investigate crystals of which their intrinsic symmetry is the same as the symmetry of their intrinsic shape. In the present case this means that our bilaterally octagonal crystals will have D1 intrinsic symmetry (while in other cases -- meromorphic crystals -- this symmetry could be according to C1 (the Asymmetric Group), depending on the crystal's internal structure).
Figure above : Microscopic view of a two-dimensional bilaterally octagonal D1 crystal. It consists of the periodic stacking of rectangular building blocks, and is, in the present Figure, provided with D1 motifs (black). These D1 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 "bilateral octagonal crystal", which then, as a convenient term, we will use in what follows.
Figure above : Macroscopic view of the two-dimensional bilateral octogonal crystal of the previous Figure with intrinsic symmetry according to the group D1 . This macroscopic view is obtained by removing the lattice connection lines (indicating building blocks) and the motifs.
The pattern of symmetry elements (with respect to the point symmetry) of our bilateral octagonal D1 two-dimensional crystal is given in the next Figure.
Figure above : Pattern of symmetry elements of the above given bilateral octagonal D1 two-dimensional crystal. Its consists of one mirror line (red) only.
Five crystallographic Forms are needed to construct the outline of the bilateral octagonal crystal, as the next Figure shows.
Figure above : Five crystallographic Forms (green, black, red, dark blue, light blue) are needed to construct the faces of our bilateral octagonal D1 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.
Possible a n t i m e r c o n f i g u r a t i o n s for holomorphic bilateral octagonal two-dimensional crystals.
Two antimers.
Figure above : A bilateral octagonal D1 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 D1 motif (black) inside each rectangular building block. It is -- or represents -- an atomic configuration such that two antimers (related to each other by a reflection) can be distinguished in it. The crystal is non-eupromorphic because its intrinsic shape suggests eight antimers, while in fact there are only two of them.
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 macroscopic view of the crystal.
Figure above : Macroscopic view of the bilateral octagonal D1 two-dimensional crystal with two antimers (green, yellow).
The promorph of the above bilateral octagonal non-eupromorphic crystal is (as two-dimensional analogue) belonging to the Allopola Zygopleura eudipleura. This promorph is depicted in the next Figure.
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.
Six antimers.
Figure above : A two-dimensional bilateral octagonal crystal with intrinsic D1 symmetry. Its D1 motifs (black) have six antimers. Microscopic view
Figure above : The bilateral octagonal D1 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 D1 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 non-eupromorphic because its intrinsic shape suggests eight antimers, while there are only six present.
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 macroscopic view of the crystal.
Figure above : Macroscopic view of the just depicted bilateral octagonal D1 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 macroscopic indication of the crystal's antimers is its intrinsic shape. And as we know, this is only a weak and more or less indirect indication. In crystals their promorph is microscopically defined (See for this, Basic Forms of Crystals in Second Part of Website ). The next Figure shows an alternative allocation of the (macroscopic) boundaries of the six antimers of our crystal.
Figure above : Alternative partition into antimers (green, yellow, blue) of the bilateral octagonal D1 two-dimensional crystal inder investigation. Compare with the earlier version above .
The promorph of the above bilateral octagonal non-eupromorphic crystal is (as two-dimensional analogue) belonging to the Allopola Amphipleura hexamphipleura. This promorph is depicted in the next Figure.
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.
Eight antimers.
Figure above : A two-dimensional bilateral octagonal crystal with intrinsic D1 symmetry. Its D1 motifs (black) have eight antimers. Microscopic view
Figure above : The bilateral octagonal D1 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 D1 motif (black) inside each rectangular building block. It is -- or represents -- an atomic configuration such that eight antimers can be distinguished in it. The crystal is eupromorphic because its intrinsic shape suggests eight antimers, which are indeed present.
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 macroscopic view of the crystal.
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 D1 crystal with eight antimers. Compare with the earlier version .
The promorph of the above bilateral octagonal eupromorphic crystal is (as two-dimensional analogue) belonging to the Allopola Amphipleura octamphipleura. This promorph is depicted in the next Figure.
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.
Four antimers, radial configuration.
Figure above : A two-dimensional bilateral octagonal crystal with intrinsic D1 symmetry. Its D1 motifs (black) have four radially arranged antimers. Microscopic view
Figure above : The bilateral octagonal D1 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 D1 motif (black) inside each rectangular building block. It is -- or represents -- an atomic configuration such that four radially arranged antimers can be distinguished in it. The crystal is non-eupromorphic because its intrinsic shape suggests eight antimers, while in fact there are only four present.
And the macroscopic view of this crystal :
Figure above : Macroscopic view of the bilateral octagonal D1 two-dimensional crystal, with four similar antimers (green, yellow) in radial configuration.
The promorph of the bilateral octagonal crystal is (as two-dimensional analogue) belonging to the Allopola Zygopleura Eutetrapleura radialia and is depicted in the next Figure.
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.
Four antimers, interradial configuration.
Figure above : A two-dimensional bilateral octagonal crystal with intrinsic D1 symmetry. Its D1 motifs (black) have four interradially arranged antimers. Microscopic view
Figure above : The bilateral octagonal D1 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 D1 motif (black) inside each rectangular building block. It is -- or represents -- an atomic configuration such that four interradially arranged antimers can be distinguished in it. The crystal is non-eupromorphic because its intrinsic shape suggests eight antimers, while in fact there are only four present.
And the macroscopic view of this crystal :
Figure above : Macroscopic view of the bilateral octagonal D1 two-dimensional crystal, with four similar antimers (green, yellow) in interradial configuration.
The promorph of the bilateral octagonal crystal is (as two-dimensional analogue) belonging to the Allopola Zygopleura Eutetrapleura interradialia and is depicted in the next Figure.
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.
Three antimers.
Figure above : A two-dimensional bilateral octagonal crystal with intrinsic D1 symmetry. Its D1 motifs (black) have three similar antimers, symmetrically arranged. Microscopic view
Figure above : The bilateral octagonal D1 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 D1 motif (black) inside each rectangular building block. It is -- or represents -- an atomic configuration such that three similar antimers can be distinguished in it. The crystal is non-eupromorphic because its intrinsic shape suggests eight antimers, while in fact there are only three present.
And the macroscopic view of this crystal :
Figure above : Macroscopic view of the bilateral octagonal D1 two-dimensional crystal, with three similar antimers (green, yellow, blue).
The promorph of the bilateral octagonal crystal is (as two-dimensional analogue) belonging to the Allopola Amphipleura triamphipleura (Allopola triamphipleura) and is depicted in the next Figure.
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 next document we will discuss the seventh crystal shape of our list (as given in Part XVI ) , viz. the isosceles trapezium, with respect to intrinsic symmetry and promorph in holomorphic two-dimensional crystals having this shape intrinsically.
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