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 Ditrigon

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

Figure above **:** Hexagonal net consisting of identical equilateral triangles, non-periodically repeated. But every two such triangles, base to base, form a 60/120^{0} rhombic building block (red, green) that generates the hexagonal net (lattice) by its being *periodically* stacked. Because the triangles are equilateral we can rephrase "*base to base*" as "*side to side*".

Figure above **:** Construction of a two-dimensional C_{3} crystal with intrinsic shape according to a regular gyroid ditrigon, from a regular trigon (equilateral triangle) built up by the periodic stacking of 60/120^{0} rhombic building blocks.

Figure above **:** The emerging outline ('faces', dark blue lines) of the two-dimensional C_{3} crystal with intrinsic shape according a regular gyroid ditrigon, under construction from a regular trigon built up by the periodic stacking of 60/120^{0} rhombic building blocks.

Figure above **:** The resulting two-dimensional C_{3} crystal with intrinsic shape according a regular gyroid ditrigon.

Removing all auxiliary line segments in the blank areas (representing incomplete building blocks) gives

Figure above **:** The resulting two-dimensional C_{3} crystal with intrinsic shape according a regular gyroid ditrigon.

What we have here (Figure above) is a microscopic view of the crystal. It is supposed to be built up by the periodic stacking of (60/120

Figure above **:** The two-dimensional C_{3} crystal with intrinsic shape according a regular gyroid ditrigon. It is drawn such that only whole (rhombic) building blocks (marked red, or green) are going to participate in the formation of the crystal. Incomplete building blocks (white) -- half building blocks (equilateral triangles) in this case -- have therefore been erased from the drawing. See also next two Figures, completing the construction.

Figure above **:** The two-dimensional C_{3} crystal with intrinsic shape according a regular gyroid ditrigon. It is drawn such that only whole (rhombic) building blocks participate in the formation of the crystal. Faces are indicated by lines (which together form a regular gyroid ditrigon).

Without the indication of the faces the microscopic view of the crystal looks as follows

Figure above **:** The two-dimensional C_{3} crystal with intrinsic shape according a regular gyroid ditrigon. It is drawn such that only whole (rhombic) building blocks participate in the formation of the crystal.

The rhombi of the hexagonal net consist of two (equilateral) triangles, connected base to base (side to side). In the previous Figures, a triangle was taken, and then another (side to side) adjacent triangle was added to constitute a rhombus. And this rhombus was considered to be periodically repeated resulting in the hexagonal net.

If, on the other hand, with respect to the initially chosen triangle -- and again taking this same triangle -- we had chosen to add one of the

In fact this doesn't make a difference, because the same effect can be obtained when our original crystal is just rotated 120

Figure above **:** Construction of the above two-dimensional C_{3} crystal with intrinsic shape according a regular gyroid ditrigon. Rhombi as composed of two (side to side) adjacent triangles, differently conceived in virtue of an alternative choice of the second triangle, forming a rhombus with a first triangle, as explained just above. Compare with the original choice of adjacent triangle, as was done **above** .

Figure above **:** Result of the construction, as begun in the previous Figure. The effect of the alternative choice of the second triangle to form a rhombus, is the same as that obtained by a (clockwise) rotation of the original figure by 120^{0} about its center. Compare with the corresponding result obtained **earlier** .

The determination of the direction of the faces marked A and B, as two of the possible faces sustained by the hexagonal net, is shown in the next two Figures.

The determination of the direction of the face

Figure above **:** The face **A** as indicated in the previous Figure, as one of the possible faces allowed and sustained by the hexagonal net.

The determination of the direction of the face

Figure above **:** The face **B** as indicated in the Figure referred to earlier, as one of the possible faces allowed and sustained by the hexagonal net.

Although the last seven Figures depict the constitution of the crystal fully correctly by allowing only

The next Figure shows a macroscopic view of our regular gyroid ditrigonal two-dimensional crystal.

Figure above **:** Macroscopic view of the two-dimensional C_{3} crystal with intrinsic shape according a regular gyroid ditrigon, obtained by the removal of all lattice connection lines (i.e. including the lines that divide the rhombi into equilateral triangles, lines, which are also (point) lattice connection lines).

The next Figure shows the pattern of

Figure above **:** Pattern of symmetry elements of the two-dimensional C_{3} crystal with intrinsic shape according a regular gyroid ditrigon. It consists of one 3-fold rotation axis only, passing through its center (See also next Figure).

Figure above **:** Clarification of the 3-fold rotation axis of the crystal under investigation.

Figure above **:** Two crystallographic Forms are needed to construct the faces (and with them the outline) of our C_{3} regular gyroid ditrigonal crystal **:** An initially given face (red) implies two more faces in virtue of the 3-fold rotation axis. The result is a closed Form consisting of three faces (equilateral triangle). A second initially given face (dark blue), not parallel to one of the faces of the first Form, also implies two other faces in virtue of that same 3-fold rotation axis, also resulting in a closed Form (equilateral triangle). The two Forms combine to give our regular gyroid ditrigonal crystal.

Our crystal is supposed to have an intrinsic shape according to a regular gyroid ditrigon. The symmetry of this shape is according to the group C

Figure above **:** The empty rhombic building blocks of our regular gyroid ditrigonal crystal have been explicitly provided with C_{3} motifs **:** Each rhombic building block receives one C_{3} motif in the center of its lower half, or, in other words **:** In the center of each lower equilateral triangle (which is half a rhombic building block) a C_{3} motif has been placed. The plane group to which our crystal belongs is now P3. See also next Figure.

Figure above **:** Same as previous Figure. The distribution of the C_{3} motifs is clarified.

And when the lattice connection lines (which as such do not belong to the crystal) are removed

Figure above **:** Same as previous Figures. Lattice connection lines removed.

The next Figure highlights the three-fold rotation axis of our C

Figure above **:** The 3-fold rotation axis of the crystal under investigation (as its only symmetry element) is highlighted.

The next Figure shows the

Figure above **:** The regular gyroid ditrigonal C_{3} two-dimensional crystal of the previous Figures. The case of __ t h r e e__ equal (and in themselves asymmetric) antimers (green, yellow, blue). 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 in the above Figure, results in a

Figure above **:** Macroscopic view of the regular gyroid ditrigonal C_{3} two-dimensional crystal of the previous figure, with three equal (and in themselves asymmetric) antimers (green, yellow, blue).

The

Figure above **:** The promorph of the regular gyroid ditrigonal C_{3} crystal with three equal antimers. It is a regular gyroid three-fold polygon with three antimers, and as such the two-dimensional analogue of the regular gyroid three-fold pyramid, which represents the promorph of corresponding three-dimensional crystals or other objects. Note the similar shapes of the promorph (regular gyroid three-fold polygon) and the crystal (regular gyroid ditrigon). In fact the **above drawing of the crystal** could be used to represent this promorph. In the present drawing, however, the gyroid nature is more conspicuously displayed. Radial (R) and interradial (IR) directions are indicated.

Figure above **:** Microscopic view of a regular gyroid ditrigonal C_{3} crystal. Its C_{3} motifs have six similar antimers (three by three equal). Two types of antimer alternate with each other.

Figure above **:** The regular gyroid ditrigonal C_{3} two-dimensional crystal of the previous Figure. The case of __ s i x__ similar (and in themselves asymmetric) 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 in the above Figure, results in a

Figure above **:** Macroscopic view of the regular gyroid ditrigonal C_{3} two-dimensional crystal of the previous Figure, with six similar (and in themselves asymmetric) antimers (green, yellow).

The

Figure above **:** The promorph of the regular gyroid ditrigonal C_{3} crystal with six similar antimers. It is a regular gyroid three-fold polygon with six similar antimers and as such the two-dimensional analogue of the regular gyroid three-fold pyramid, which represents the promorph of corresponding three-dimensional crystals or other objects. Note the similar shapes of the promorph (regular gyroid 3-fold polygon) and the crystal (regular gyroid ditrigon). In fact the drawing of the crystal itself already expresses the promorph well. Nevertheless we have chosen the present Figure to represent this promorph, because it clearly expresses its gyroid nature. Radial (R) and interradial (IR) directions are indicated.

In the

**e-mail :**

To continue click HERE