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 Hexagon (Promorphs)

The **Vector Rosette of Actual Growth** of a regularly-hexagonal crystal, i.e. of any two-dimensional crystal having as its intrinsic shape (when fully developed) the **Regular Hexagon** (which itself has D_{6} symmetry) and possessing whatever (possible) intrinsic (point) symmetry, is as follows (blue lines) **:**

This Vector Rosette, viz. a vector rosette of a regularly-hexagonal two-dimensional crystal, in fact consists of six vectors,

**REMARK :**

As in the previous document, we will use the term "*hexagonal*" , where this term should be taken to mean "*regularly-hexagonal"*, unless indicated otherwise.

As we saw in the previous document, an **intrinsically hexagonal crystal shape ** (with respect to two-dimensional crystals) is supported by all plane groups except the groups P4, P4mm, and P4gm. So the supporting plane groups are **:**

P6mm, P6, P31m, P3m1, P3, P2mm, P2mg, P2gg, C2mm, Pm, Pg, Cm, P2 and P1.

See also the **table in the previous document** where, in addition to the supporting plane groups, the corresponding point groups are indicated, and the number of crystallographic Forms needed to construct a hexagonal crystal in each case (of point symmetry).

We will now consider several possible cases of the given example of a hexagonal two-dimensional crystal as depicted just above, with respect to possible promorphs.

Two-dimensional crystals with intrinsic D

The next Figure indicates the **symmetry elements** of an intrinsic D_{6} crystal.

Figure above **:** Symmetry elements (with respect to point symmetry) of a two-dimensional crystal with intrinsic D_{6} symmetry **:** six mirror lines, and a 6-fold rotation axis at their point of intersection.

Only

Figure above **:** Only one crystallographic Form is needed to conceptually construct a hexagon-shaped intrinsic D_{6} two-dimensional crystal **:** An initially given crystal face perpendicular to one of the mirror lines implies five others in virtue of the 6-fold rotation axis, resulting in one closed Form (dark blue) having the shape of a regular hexagon.

All intrinsic

Figure above **:** The six antimers (yellow, green) of an intrinsically hexagon-shaped two-dimensional crystal with intrinsic D_{6} symmetry. Radial (R) and interradial (IR) directions indicated.

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagonal D_{6} crystal with Vector Rosette of Actual Growth added. The six vectors, **a, b, c, d, e, f,** coincide with the median lines of the antimers, and thus fully express the promorph.

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal D_{6} crystal. The six antimers are indicated (green, yellow).

The next two Figures depict the corresponding three-dimensional analogue of our just established promorph.

Figure above **:** Slightly oblique top view of a 6-fold regular pyramid as a representative of the *Homostaura Isopola hexactinota* .

Figure above **:** Slightly oblique top view of a 6-fold regular pyramid as a representative of the *Homostaura Isopola hexactinota* . One antimer is taken out.

Two-dimensional crystals with intrinsic C

The next Figure indicates the **symmetry elements** of an intrinsic C_{6} crystal.

Figure above **:** Symmetry elements (with respect to point symmetry) of a two-dimensional crystal with intrinsic C_{6} symmetry **:** a 6-fold rotation axis at the center of the crystal.

Also in the present case only

Figure above **:** Only one crystallographic Form is needed to conceptually construct a hexagon-shaped intrinsic C_{6} two-dimensional crystal **:** An initially given crystal face implies five others in virtue of the 6-fold rotation axis, resulting in one closed Form (dark blue) having the shape of a regular hexagon.

All intrinsic

Figure above **:** The six (asymmetric) antimers (yellow, green) of an intrinsically hexagon-shaped two-dimensional crystal with intrinsic C_{6} symmetry. Radial (R) and interradial (IR) directions indicated.

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagonal C_{6} crystal with Vector Rosette of Actual Growth added. The six vectors, **a, b, c, d, e, f,** coincide with the antimers (i.e. each one corresponds to an antimer), and thus partially express the promorph (partially, because they do not express the asymmetry of the antimers).

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal C_{6} crystal. The six antimers are indicated (green, yellow).

The next Figure depicts the three-dimensional analogue of our just established promorph.

Figure above **:** Three-dimensional analogue of the promorph of the above discussed two-dimensional hexagonal C_{6} crystal **:** A regular gyroid pyramid (slightly oblique top view) with six antimers ( *Homogyrostaura hexamera* ).

An intrinsically hexagon-shaped two-dimensional crystal can have a true (i.e. intrinsic) symmetry according to the group

The next Figure depicts a possible configuration of the

Figure above **:** Possible configuration of symmetry elements of a two-dimensional hexagon-shaped D_{3} crystal **:** Three mirror lines (red) connecting opposite centers of sides of the hexagon, and a three-fold rotation axis at their point of intersection.

Within this configuration of symmetry elements

Figure above **:** Given the above configuration of symmetry elements, two Forms are needed to conceptually construct a hexagonal D_{3} two-dimensional crystal **:** An initially given crystal face (red) implies two others in virtue of the mirror lines ( The action of the 3-fold rotation axis does not create more faces in addition to the ones already obtained so far). The resulting three faces constitute one open Form. A second initially given crystal face (blue) also implies two more faces, resulting in a second open Form (three faces). The two Forms together constitute the whole hexagon-shaped crystal.

Apart from the above depicted configuration of symmetry elements of a two-dimensional hexagon-shaped intrinsic D

Figure above **:** Alternative configuration of symmetry elements of a two-dimensional hexagon-shaped D_{3} crystal **:** Three mirror lines (red) connecting opposite corners of the hexagon, and a three-fold rotation axis at their point of intersection.

Within this configuration of symmetry elements only

Figure above **:** Given the above alternative configuration of symmetry elements (mirror lines connecting opposite corners), only one Form is needed to conceptually construct a hexagonal D_{3} two-dimensional crystal **:** An initially given crystal face (blue) implies, to begin with, one other face in virtue of the nearest mirror line. But these two faces directly imply four others in virtue of the 3-fold rotation axis, resulting in one closed Form, consisting of six faces, together constituting the whole hexagonal crystal. And because with this configuration of symmetry elements (mirror lines connecting opposite corners of hexagon) the minimum number of crystallographic Forms (viz. one) is needed to conceptually construct our two-dimensional hexagon-shaped D_{3} crystal, we will use this configuration in our discussions regarding promorphs.

Crystals with intrinsic D

Figure above **:** The three antimers (yellow, green, blue) of an intrinsically hexagon-shaped two-dimensional crystal with intrinsic D_{3} symmetry. Radial (R) and interradial (IR) directions indicated.

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagonal D_{3} crystal with Vector Rosette of Actual Growth added. The six vectors, **a, b, c, d, e, f,** alternately coincide with the median lines of the antimers and the lines separating adjacent antimers. The three antimers (green, yellow, blue) are indicated by numerals.

The next Figure depicts and names the

As promorph it is the two-dimensional analogue of an instance of the (three-dimensional)

Figure above **:** Promorph of the above discussed two-dimensional hexagon-shaped D_{3} crystal. The three antimers are indicated (green, yellow, blue).

The next Figure gives the three-dimensional analogue of our just established (two-dimensional) promorph.

Figure above **:** Slightly oblique top view of a regular pyramid with three antimers (indicated by colors) as the stereometric basic form representing the *Homostaura Anisopola triactinota* .

Two-dimensional crystals with intrinsic C

The next Figure indicates the **symmetry elements** of an intrinsic C_{3} crystal.

Figure above **:** Symmetry elements (with respect to point symmetry) of a two-dimensional crystal with intrinsic C_{3} symmetry **:** only a 3-fold rotation axis at the center of the crystal.

Figure above **:** Two crystallographic Forms are needed to conceptually construct a hexagon-shaped intrinsic C_{3} two-dimensional crystal **:** An initially given crystal face (red) implies two others in virtue of the 3-fold rotation axis, resulting in one open Form consisting of three faces. A second initially given crystal face (blue) also implies two other faces, also in virtue of the 3-fold rotation axis, resulting in a second open Form, also consisting of three faces. The two Forms together constitute the whole hexagon-shaped crystal.

All intrinsic

Figure above **:** The three (asymmetric) antimers (yellow, green, blue) of an intrinsically hexagon-shaped two-dimensional crystal with intrinsic C_{3} symmetry. Radial (R) and interradial (IR) directions indicated.

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagonal C_{3} crystal with Vector Rosette of Actual Growth added. Each of the three vector pairs, **(a, b), (d, f),** and **(e, c),** of the six vectors, **a, b, c, d, e, f,** coincides with an antimer (i.e. lies within an antimer).

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal C_{3} crystal. The three antimers are indicated (green, yellow, blue).

The next Figure depicts the three-dimensional analogue of our just established promorph.

Figure above **:** Three-dimensional analogue of the promorph of the above discussed two-dimensional hexagonal C_{3} crystal **:** A regular gyroid pyramid (slightly oblique top view) with three antimers ( *Homogyrostaura trimera* ).

Two-dimensional crystals with intrinsic D

The next Figure indicates the **symmetry elements** of an intrinsic D_{2} crystal.

Figure above **:** Symmetry elements (with respect to point symmetry) of a two-dimensional crystal with intrinsic D_{2} symmetry **:** Two mirror lines perpendicular to each other, and a 2-fold rotation axis at their point of intersection.

Figure above **:** Two crystallographic Forms are needed to conceptually construct a hexagon-shaped intrinsic D_{2} two-dimensional crystal **:** An initially given crystal face (red) implies, to begin with, one other face in virtue of the nearest mirror line, and then the two faces are reflected by the other mirror line, all this resulting in an open Form consisting of four faces. A second initially given crystal face (blue) implies one other face, in virtue of the 2-fold rotation axis, resulting in a second open Form, consisting of two faces. The two Forms together constitute the whole hexagon-shaped crystal.

Any even number of antimers can occur in two-dimensional crystals with intrinsic D

As we saw already in earlier documents, there are two different configurations of four antimers in D

**Interradial configuration.**

Figure above **:** Two-dimensional hexagon-shaped D_{2} crystal with four antimers (green, yellow) in interradial configuration.

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagon-shaped D_{2} crystal with four antimers (green, yellow) in interradial configuration. Vector Rosette of Actual Growth with its six vectors **a, b, c, d, e, f,** added.

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal D_{2} crystal. The four antimers are indicated (green, yellow).

The next Figure depicts the three-dimensional analogue of our just established promorph.

Figure above **:** Three-dimensional analogue of the promorph of the above discussed two-dimensional hexagonal D_{2} crystal **:** A rectangular pyramid (slightly oblique top view) with four antimers ( *Autopola Orthostaura Tetraphragma interradialia* ).

Figure above **:** Two-dimensional hexagon-shaped D_{2} crystal with four antimers (green, yellow) in radial configuration.

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagon-shaped D_{2} crystal with four antimers (green, yellow, and indicated by numerals) in radial configuration. Vector Rosette of Actual Growth with its six vectors **a, b, c, d, e, f,** added.

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal D_{2} crystal. The four antimers are indicated (green, yellow).

The next Figure depicts the three-dimensional analogue of our just established promorph.

Figure above **:** Three-dimensional analogue of the promorph of the above discussed two-dimensional hexagonal D_{2} crystal **:** A rhombic pyramid (slightly oblique top view) with four antimers ( *Autopola Orthostaura Tetraphragma radialia* ).

The motif, as translation-free residue, of a two-dimensional hexagon-shaped D

Figure above **:** Two-dimensional hexagon-shaped D_{2} crystal with two antimers (green, yellow).

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagon-shaped D_{2} crystal with two antimers (green, yellow). Vector Rosette of Actual Growth with its six vectors **a, b, c, d, e, f,** added.

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal D_{2} crystal. The two antimers are indicated (green, yellow).

The motif, as translation-free residue, of a two-dimensional hexagon-shaped D

Figure above **:** Two-dimensional hexagon-shaped D_{2} crystal with six antimers (green, yellow, and indicated by numerals).

The next Figure uses another color configuration to express the same crystal.

Figure above **:** Two-dimensional hexagon-shaped D_{2} crystal with six antimers (green, yellow).

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagon-shaped D_{2} crystal with six antimers (green, yellow). Vector Rosette of Actual Growth with its six vectors **a, b, c, d, e, f,** added.

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal D_{2} crystal. The six antimers are indicated (green, yellow, blue).

The next Figure depicts the three-dimensional analogue of our just established (two-dimensional) promorph.

Figure above **:** Slightly oblique top view of a six-fold amphitect pyramid as the basic form of the *Autopola Oxystaura hexaphragma* . As one can see, this pyramid is flattened parallel to two of its sides, while in the previous Figure the six-fold polygon is flattened perpendicular to two of its sides, but this difference is immaterial.

Two-dimensional crystals with intrinsic D

The next Figure indicates the **symmetry elements** of an intrinsic D_{1} crystal.

Figure above **:** Symmetry elements (with respect to point symmetry) of a two-dimensional crystal with intrinsic D_{1} symmetry **:** Only one mirror line.

Figure above **:** Three crystallographic Forms are needed to conceptually construct a hexagon-shaped intrinsic D_{1} two-dimensional crystal **:** An initially given crystal face (red) implies one other face in virtue of the mirror line, resulting in an open Form consisting of two faces. A second initially given crystal face (blue) also implies one other face, in virtue of that same mirror line, resulting in a second open Form also consisting of two faces. Finally, a third initially given face (green) also implies one other face and also results in an open Form consisting of two faces. The three Forms together constitute the whole hexagon-shaped crystal.

Any number (> 1) of antimers can occur in two-dimensional crystals with intrinsic D

The motif, as translation-free residue, of a two-dimensional hexagon-shaped D

Figure above **:** Two-dimensional hexagon-shaped D_{1} crystal with two antimers (green, yellow). A motif (black) is inserted in order to express the D_{1} symmetry of the crystal.

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagon-shaped D_{1} crystal with two antimers (green, yellow). Vector Rosette of Actual Growth with its six vectors **a, b, c, d, e, f,** added.

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal D_{1} crystal. The two antimers are indicated (green, yellow).

The next Figure depicts the three-dimensional analogue of our just established (two-dimensional) promorph.

Figure above **:** Oblique top view of a single isosceles pyramid (i.e. of half a rhombic pyramid) as the basic form of the *Allopola Zygopleura eudipleura* . The two antimers are indicated by colors.

As in D

**Interradial configuration.**

Figure above **:** Two-dimensional hexagon-shaped D_{1} crystal with four antimers (green, yellow) in interradial configuration.

The next Figure uses a different color configuration for the same crystal.

Figure above **:** Two-dimensional hexagon-shaped D_{1} crystal with four antimers (green, yellow, and indicated by numerals) in interradial configuration.

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagon-shaped D_{1} crystal with four antimers (green, yellow, and indicated by numerals) in interradial configuration. Vector Rosette of Actual Growth with its six vectors **a, b, c, d, e, f,** added.

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal D_{1} crystal. The four antimers are indicated (green, yellow).

The next Figure depicts the three-dimensional analogue of our just established promorph.

Figure above **:** Three-dimensional analogue of the promorph of the above discussed two-dimensional hexagonal D_{1} crystal **:** Oblique top view of a trapezoid pyramid ( = antiparallelogram pyramid) with four antimers, indicated by colors ( *Allopola Zygopleura Eutetrapleura interradialia* ).

Figure above **:** Two-dimensional hexagon-shaped D_{1} crystal with four antimers (green, yellow) in radial configuration.

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagon-shaped D_{1} crystal with four antimers (green, yellow) in radial configuration. Vector Rosette of Actual Growth with its six vectors **a, b, c, d, e, f,** added.

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal D_{1} crystal. The four antimers are indicated (green, yellow).

The next Figure depicts the three-dimensional analogue of our just established promorph.

Figure above **:** Three-dimensional analogue of the promorph of the above discussed two-dimensional hexagonal D_{1} crystal **:** A bi-isosceles pyramid (oblique top view) with four antimers ( *Allopola Zygopleura Eutetrapleura radialia* ). The four antimers are indicated by colors.

The motif, as translation-free residue, of a two-dimensional hexagon-shaped D

Figure above **:** Two-dimensional hexagon-shaped D_{1} crystal with three antimers (green, yellow, and indicated by numerals).

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagon-shaped D_{1} crystal with three antimers (green, yellow, indicated by numerals). Vector Rosette of Actual Growth with its six vectors **a, b, c, d, e, f,** added.

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal D_{1} crystal. The three antimers are indicated (green, yellow).

The next Figure depicts the three-dimensional analogue of our just established (two-dimensional) promorph.

Figure above **:** Oblique top view of half a six-fold amphitect pyramid as the basic form of the *Allopola Amphipleura triamphipleura* . The three antimers are indicated by colors.

The motif, as translation-free residue, of a two-dimensional hexagon-shaped D

Figure above **:** Two-dimensional hexagon-shaped D_{1} crystal with five antimers (green, yellow, and indicated by numerals).

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagon-shaped D_{1} crystal with five antimers (green, yellow, indicated by numerals). Vector Rosette of Actual Growth with its six vectors **a, b, c, d, e, f,** added.

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal D_{1} crystal. The five antimers are indicated (green, yellow, right image).

Figure above **:** Oblique top view of half a ten-fold amphitect pyramid as the basic form of the *Allopola Amphipleura pentamphipleura* . The five antimers are indicated by colors.

The motif, as translation-free residue, of a two-dimensional hexagon-shaped D

Figure above **:** Two-dimensional hexagon-shaped D_{1} crystal with six antimers (green, yellow, and indicated by numerals).

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagon-shaped D_{1} crystal with six antimers (green, yellow, indicated by numerals). Vector Rosette of Actual Growth with its six vectors **a, b, c, d, e, f,** added.

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal D_{1} crystal. The six antimers are indicated (green, yellow, right image).

Two-dimensional crystals with intrinsic C

The next Figure indicates the **symmetry elements** of an intrinsic C_{2} crystal.

Figure above **:** Symmetry elements (with respect to point symmetry) of a two-dimensional crystal with intrinsic C_{2} symmetry **:** A two-fold rotation axis only.

Figure above **:** Three crystallographic Forms are needed to conceptually construct a hexagon-shaped intrinsic C_{2} two-dimensional crystal **:** An initially given crystal face (red) implies one other face in virtue of the 2-fold rotation axis, resulting in an open Form consisting of two faces. A second initially given crystal face (blue) also implies one other face, in virtue of that same rotation axis, resulting in a second open Form also consisting of two faces. Finally, a third initially given face (green) also implies one other face and also results in an open Form consisting of two faces. The three Forms together constitute the whole hexagon-shaped crystal.

Any even number of antimers can occur in two-dimensional crystals with intrinsic C

The motif, as translation-free residue, of a two-dimensional hexagon-shaped C

Figure above **:** Two-dimensional hexagon-shaped C_{2} crystal with two antimers (green, yellow).

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagon-shaped C_{2} crystal with two antimers (green, yellow). Vector Rosette of Actual Growth with its six vectors **a, b, c, d, e, f,** added.

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal C_{2} crystal. The two antimers are indicated (green, yellow).

Figure above **:** Slightly oblique top view of a two-fold amphitect gyroid pyramid, i.e. an amphitect gyroid pyramid with two antimers, as the basic form of the *Heterogyrostaura dimera* . The two antimers are indicated by colors.

The motif, as translation-free residue, of a two-dimensional hexagon-shaped C

Figure above **:** Two-dimensional hexagon-shaped C_{2} crystal with four antimers (green, yellow).

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional hexagon-shaped C_{2} crystal with four antimers (green, yellow). Vector Rosette of Actual Growth with its six vectors **a, b, c, d, e, f,** added.

The next Figure depicts and names the

Figure above **:** Promorph of the above discussed two-dimensional hexagonal C_{2} crystal. The four antimers are indicated (green, yellow).

Figure above **:** Slightly oblique top view of a four-fold amphitect gyroid pyramid, as the basic form of the *Heterogyrostaura tetramera* . The four antimers are indicated by colors.

The geometry of the motif (as translation-free residue) of an intrinsically hexagon-shaped crystal with intrinsic C

Figure above **:** A two-dimensional intrinsically hexagon-shaped C_{2} crystal with eight antimers (green, yellow).

And with the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically hexagon-shaped C_{2} crystal with eight antimers (green, yellow). Vector Rosette of Actual Growth added.

The

The promorphs of C

The symmetry of the motif (as translation-free residue) of an intrinsically hexagon-shaped two-dimensional crystal can be according to the Asymmetric Group C

There are consequently no

Figure above **:** A (two-dimensional) C_{1} crystal does not have any symmetry elements (with respect to point symmetry).

Figure above **:** Six crystallographic Forms are needed to construct a hexagon-shaped two-dimensional C_{1} crystal **:** Any intially given crystal face does not imply other faces. So each face is itself already a crystallographic Form. Evidently six such Forms are needed to construct a hexagon, and thus the whole hexagon-shaped C_{1} crystal.

C

Figure above **:** A two-dimensional intrinsically hexagon-shaped C_{1} crystal.

A motif (black) is inserted to express the crystal's C_{1} symmetry.

And with the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically hexagon-shaped C_{1} crystal. A motif (black) is inserted to express the crystal's C_{1} symmetry. Vector Rosette of Actual Growth added. All vectors **a, b, c, d, e, f,** are unique (i.e. non-equivalent).

The next Figure (two images) gives the possible

This

While the foregoing documents investigated promorphs of two-dimensional crystals from the view point of their intrinsic

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