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 Gyroid Octagon

The next two Figures give a holomorphic two-dimensional crystal having as its intrinsic shape that of an

Figure above **:** Microscopic view of a two-dimensional C_{2} crystal (plane group P2) with the intrinsic shape of an amphitect gyroid octagon. Each parallelogrammatic building block contains in its center a C_{2} motif (black, consisting of two motif units, related to each other by a half-turn about the center of the building block).

Figure above **:** Same as previous Figure. The eight crystal 'faces' are indicated (dark blue lines). Recall that the building blocks are actually very small indeed, resulting in all the crystal faces being perfectly smooth.

The next Figure gives a

Figure above **:** Macroscopic view of the two-dimensional C_{2} crystal (plane group P2) with the intrinsic shape of an amphitect gyroid octagon.

The pattern of

Figure above **:** Pattern of symmetry elements of the above two-dimensional C_{2} crystal having as its intrinsic shape an amphitect gyroid octagon. It consists of one 2-fold rotation axis (indicated by a small yellow ellipse) only.

The next Figure shows that

Figure above **:** Four Forms are needed to conceptually construct the (faces of the) above crystal **:** An initially given face implies one more face in virtue of the2-fold rotation axis. The result is one Form consisting of two faces. Four such forms are needed. They are indicated by the colors red, dark blue, green and black.

The number and arrangement of a n t i m e r s in a crystal depend on the geometry of its translation-free residue. This residue is the *motif* as it remains after having everything 'telescoped in', i.e. after elimination of all translational elements in the crystal.

In the above Figures that would mean that the translation-free residue is the following **:** **.** And here we clearly have to do with two antimers. However, in the present document (concerning C_{2} crystals) we will interpret motifs, as they are drawn in the ensuing Figures, __not__ as motifs just like that, but as only *r e p r e s e n t i n g the existence of a C _{2} motif w i t h a n y p o s s i b l e n u m b e r o f a n t i m e r s* .

The precise relation of antimers and the

Figure above **:** The case of ** e i g h t** similar antimers (green, yellow, blue) in a two-dimensional C

The next Figure gives a

Figure above **:** Macroscopic view of the above two-dimensional C_{2} amphitect gyroid octogonal crystal with eight similar antimers (green, yellow, blue).

The

Figure above **:** Possible representation of the promorph of the above discussed two-dimensional C_{2} amphitect gyroid octogonal crystal with eight similar antimers. It is an eight-fold amphitect gyroid polygon, and as such the two-dimensional analogue of an eight-fold amphitect gyroid pyramid (which represents the promorph of corresponding three-dimensional crystals or other objects). To represent this promorph geometrically, we could have chosen a somewhat simpler figure -- in fact we can use the drawing of the crystal itself, without motifs and building blocks, and thus its macroscopic view as given above. We have nevertheless decided for the present figure because it emphasizes very clearly the gyroid nature of the polygon.

Figure above **:** A two-dimensional C_{2} amphitect gyroid octogonal crystal. The case of six similar antimers (green, yellow, blue). The crystal is non-eupromorphic because its intrinsic shape suggests eight antimers while in fact there are six.

The next Figure gives a

Figure above **:** Macroscopic view of the above two-dimensional C_{2} amphitect gyroid octogonal crystal with six similar antimers (green, yellow, blue).

The

Figure above **:** Possible representation of the promorph of the above discussed two-dimensional amphitect gyroid octogonal C_{2} crystal with six antimers. It is an amphitect gyroid six-fold polygon (six antimers, green, yellow, blue), and as such the two-dimensional analogue of the amphitect gyroid six-fold pyramid, which is the promorph of corresponding three-dimensional crystals or other objects.

Figure above **:** A two-dimensional C_{2} amphitect gyroid octogonal crystal. The case of four similar antimers (yellow, blue). The crystal is non-eupromorphic because its intrinsic shape suggests eight antimers while in fact there are only four.

The next Figure gives a

Figure above **:** Macroscopic view of the above two-dimensional C_{2} amphitect gyroid octogonal crystal with four similar antimers (yellow, blue).

The

Figure above **:** The promorph of a two-dimensional amphitect gyroid octogonal C_{2} crystal with four similar antimers. It is a 4-fold amphitect gyroid polygon, and as such the 2-D analogue of a 4-fold amphitect gyroid pyramid, which is the promorph of corresponding three-dimensional crystals or other objects.

Figure above **:** A two-dimensional amphitect gyroid octogonal C_{2} crystal with two equal antimers (yellow, blue, related to each other by a half-turn) ( The inserted C_{2} motifs have -- as they are drawn here -- two antimers, and are, in the present case, also meant to represent C_{2} motifs with two antimers). The crystal is non-eupromorphic because its intrinsic shape suggests eight antimers while in fact there are only two.

The next Figure gives a

Figure above **:** Macroscopic view of the above two-dimensional C_{2} amphitect gyroid octogonal crystal with two equal antimers (yellow, blue).

The

Figure above **:** The promorph of a two-dimensional amphitect gyroid octogonal C_{2} crystal with two equal antimers. It is a 2-fold amphitect gyroid polygon (i.e. an anmphitect polygon meant to represent two antimers), and as such the 2-D analogue of a 2-fold amphitect gyroid pyramid, which is the promorph of corresponding three-dimensional crystals or other objects.

In the

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