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.**

Regular Gyroid Ditetragon

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

Figure above **:** Microscopic view of a two-dimensional crystal having as its intrinsic shape that of a *regular gyroid ditetragon*. Only empty building blocks are shown. Each building block is a square (which, as such, i.e. as empty square, can represent a D_{4} motif). If each one of them is provided with a motif that has C_{4} symmetry, the total symmetry is then according to the plane group P4. In that case the crystal is holomorphic, because its intrinsic point symmetry (C_{4} ) is the same as the symmetry (C_{4} ) of its intrinsic shape (regular gyroid ditetragon). A meromorphic crystal having this shape (intrinsically) could have an intrinsic symmetry according to the group C_{2} (which is a subgroup of C_{4} ) if the motifs of the building blocks have C_{2} symmetry (and not C_{4} symmetry). The plane group is then P2. If the motifs only have C_{1} symmetry (i.e. if they are totally asymmetric), then the intrinsic point symmetry of such a meromorphic crystal is according to the group C_{1} (Asymmetric Group) (which also is a subgroup of the group C_{4} ), and the plane group is then P1. However, in the present investigation we consider *holomorphic* crystals only. See next Figures.

Figure above **:** Construction of a two-dimensional crystal (microscopic view) having as its intrinsic shape that of a *regular gyroid ditetragon*. C_{4} motifs (black) are inserted, resulting in a total symmetry according to the plane group P4, which in turn implies a point symmetry according to the cyclic group C_{4} .

Figure above **:** Result of the above construction of a two-dimensional crystal (microscopic view) having as its intrinsic shape that of a *regular gyroid ditetragon* and possessing an intrinsic symmetry according to the group C_{4} .

The next Figure indicates the 'faces' of the constructed crystal.

Figure above **:** Microscopic view of the two-dimensional crystal having as its intrinsic shape that of a *regular gyroid ditetragon*, as was constructed above. Crystal 'faces' indicated (dark blue lines).

In the next Figure the lattice lines (indicating building blocks) are removed, leaving the crystal with its microscopic motifs (which are periodically repeated according to the square point lattice, which had been indicated by connection lines in the previous Figure).

Figure above **:** The two-dimensional regular gyroid octogonal C_{4} crystal of the previous Figure. Lattice lines omitted.

By also removing the motifs, we obtain a

Figure above **:** Macroscopic view of the regular gyroid octogonal crystal of the Figures above.

The pattern of

Figure above **:** Pattern of symmetry elements of the above given regular gyroid octogonal C_{4} two-dimensional crystal. Its consists of one 4-fold rotation axis (yellow) only.

Figure above **:** Two crystallographic Forms (red, dark blue) are needed to construct the faces of our regular gyroid octogonal C_{4} two-dimensional crystal **:** An initially given face (red) implies three more faces in virtue of the 4-fold rotation axis, resulting in one closed Form (a square) consisting four faces (red). In the same way a second initially given face (dark blue) also gives rise to a closed Form (a second Form), which also consists of four faces together forming a square. These two Forms combine to give our regular gyroid octogonal crystal.

Figure above **:** A regular gyroid ditetragonal C_{4} two-dimensional crystal. The case of __ f o u r__ 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 C

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

Figure above **:** Macroscopic view of the regular gyroid octogonal C_{4} two-dimensional crystal of the previous figure, with four equal antimers (green, yellow).

The

Figure above **:** The promorph of the regular gyroid octogonal C_{4} crystal with four antimers. It is a regular gyroid 4-fold polygon and as such the two-dimensional analogue of the regular gyroid 4-fold pyramid (See next Figure) which represents the promorph of corresponding three-dimensional crystals or other objects. Note the similarity of shapes between this promorph (regular gyroid 4-fold polygon) and that of the crystal (regular gyroid ditetragon) of which it is the promorph. In fact the depicted shape of our crystal can represent the promorph perfectly well. However, in our present drawing of the promorph (representing all *Homogyrostaura tetramera*) the gyroid nature is expressed more suggestively.

Radial (R) and interradial (IR) directions are indicated.

If we omit the four buckled lines (that only serve to emphasize the gyroid nature) ending up in the corners, we have this same promorph, where its four equal antimers are indicated by the colors green and yellow

Figure above **:** The promorph of the regular gyroid octogonal C_{4} crystal with four antimers. Buckled auxiliary lines omitted.

The three-dimensional counterpart of this promorph is depicted in the next Figure.

Figure above **:** Regular gyroid 4-fold pyramid (slightly oblique top view), representing all three-dimensional *Homogyrostaura tetramera* (i.e. all regular 4-fold gyroid promorphs of three-dimensional objects).

A

As long as the intrinsic symmetry of our crystal is according to the group C

All this means that a number of antimers, different from 4n cannot occur, because then there is no four-fold repetition, which means that the symmetry has changed. The latter has become different from C

**Meromorphic. Two antimers.**

Figure above **:** A regular gyroid octogonal C_{2} two-dimensional *meromorphic* crystal. Its C_{2} motifs (black) have two equal antimers.

Figure above **:** The regular gyroid octogonal C_{2} two-dimensional *meromorphic* crystal of the previous Figure. The case of __ t w o__ equal antimers (green, yellow), related to each other by a 2-fold rotation axis. 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 C

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

Figure above **:** Macroscopic view of the meromorphic regular gyroid octogonal C_{2} two-dimensional crystal of the previous figure, with two equal antimers (green, yellow).

The

Figure above **:** The promorph of the meromorphic non-eupromophic regular gyroid octogonal C_{2} crystal with two antimers. It is a 2-fold (and therefore amphitect) gyroid polygon (i.e. an amphitect polygon meant to express the presence of two antimers) and as such the two-dimensional analogue of the 2-fold gyroid pyramid, which represents the promorph of corresponding three-dimensional crystals or other objects. Note the difference in shape between this promorph (2-fold gyroid polygon) and that of the crystal (regular gyroid ditetragon) of which it is the promorph.

As has been explained above,

Figure above **:** Construction of a two-dimensional crystal (microscopic view) having as its intrinsic shape that of a *regular gyroid ditetragon*. Its C_{4} motifs (black) possess eight similar antimers. A pair of two similar antimers is repeated four times around the center of the motif, effecting that center to be a 4-fold rotation axis of the motif.

Figure above **:** Result of the above construction of a two-dimensional crystal (microscopic view) having as its intrinsic shape that of a *regular gyroid ditetragon* and possessing an intrinsic symmetry according to the group C_{4} .

The next Figure indicates the 'faces' of the constructed crystal.

Figure above **:** Microscopic view of the two-dimensional crystal having as its intrinsic shape that of a *regular gyroid ditetragon*, as was constructed above. Crystal 'faces' indicated (dark blue lines). The motifs (black) possess eight similar (not equal) antimers.

Figure above **:** The regular gyroid ditetragonal C_{4} two-dimensional crystal of the previous Figure. The case of __ e i g h t__ s i m i l a r (not equal) antimers (green, yellow). Although the antimers are not equal just like that, the set of eight antimers consists of two subsets of equal antimers (i.e. each subset consists of four equal antimers), such that the two types of antimers alternate. The degree of similarity of the antimers is considered to be sufficient for recognizing

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

Figure above **:** Macroscopic view of the regular gyroid ditetragonal C_{4} two-dimensional crystal of the previous figure, with eight similar antimers (green, yellow).

The

Figure above **:** The promorph of the regular gyroid octogonal C_{4} crystal with eight similar antimers. It is a regular gyroid 4-fold polygon (here meant to express the presence of 8 (2x4) similar antimers) and as such the two-dimensional analogue of the regular gyroid 4-fold pyramid (here expressing eight similar three-dimensional antimers) which represents the promorph of corresponding three-dimensional crystals or other objects. Note the similarity of shapes between this promorph (regular gyroid 4-fold polygon with eight similar antimers) and that of the crystal (regular gyroid ditetragon) of which it is the promorph. In fact the depicted shape of our crystal can represent the promorph perfectly well as soon as the eight similar antimers are indicated in it (as was done in the previous Figure). However, in our present drawing of the promorph (representing all *Homogyrostaura ditetramera*) the gyroid nature is expressed more suggestively.

Radial (R) and interradial (IR) directions are indicated. Compare with the promorph that belongs to the *Homogyrostaura tetramera* depicted **above**.

The next Figures again depict a two-dimensional regular gyroid octogonal crystal of which the translation-free residue (motif) possesses eight very

Figure above **:** A regular gyroid ditetragonal C_{4} two-dimensional crystal. Its C_{4} motifs seemingly have eight similar antimers. In fact, however, they are very dissimilar. So they must be conceived as pairs, consisting of two dissimilar structures, and only such a pair should be considered to represent an antimer. And the antimer, so conceived, is repeated four times around the center of the motif.

Figure above **:** The regular gyroid ditetragonal C_{4} two-dimensional crystal of the previous Figure. The case of __ f o u r__ 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 C

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

Figure above **:** Macroscopic view of the regular gyroid octogonal C_{4} two-dimensional crystal of the previous figure, with four equal antimers (green, yellow).

The

Figure above **:** The promorph of the regular gyroid octogonal C_{4} crystal with four antimers, where the presence of these four antimers was based on a complex translation-free residue seemingly possessing eight similar antimers, but in fact possessing only four antimers. This promorph is a regular gyroid 4-fold polygon and as such the two-dimensional analogue of the regular gyroid 4-fold pyramid which represents the promorph of corresponding three-dimensional crystals or other objects.

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