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.**

Equilateral Triangle (Promorphs)

The **Vector Rosette of Actual Growth** of a regularly-triangular crystal, i.e. of any two-dimensional crystal having as its intrinsic shape (when fully developed) the **Equilateral Triangle** (which itself has D_{3} symmetry) and possessing whatever (possible) intrinsic (point) symmetry, is as follows (blue lines) **:**

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

**REMARK :**

As in the previous document, we will often refer to a "*regularly-triangular*" crystal or shape. So the reference is to the equilateral triangle. In the sequel we will use the term "*triangular*" while this term should be taken to mean "*regularly-triangular"*, unless indicated otherwise.

As we saw in the previous document, an **intrinsically triangular crystal shape ** (with respect to two-dimensional crystals) is supported by the following plane groups **:**

P3m1, P3, Pm, Pg, Cm 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 triangular crystal in each case (of point symmetry).

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

Intrinsically triangular two-dimensional crystals with intrinsic D

The next Figure gives the

Figure above **:** Symmetry elements of a two-dimensional intrinsically triangular crystal with D_{3} intrinsic symmetry **:** Three mirror lines ( **m _{1} , m_{2} , m_{3}** ), and one 3-fold rotation axis at their point of intersection.

Figure above **:** Only one crystallographic Form is needed to construct a triangular D_{3} two-dimensional crystal **:** One initially given crystal face implies two others in virtue of the 3-fold rotation axis, resulting in one Form consisting of three faces, together making up the triangular crystal.

For a two-dimensional D

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

And the Vector Rosette of Actual Growth added

Figure above **:** The three antimers (blue, green, yellow) of a two-dimensional triangular crystal with intrinsic D_{3} symmetry. Radial (R) and interradial (IR) directions indicated. Vector Rosette of Actual Growth added. The vectors **a, b,** and **c** are equivalent.

The next Figure depicts and names the

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

The next Figure gives the three-dimensional analogue of the promorph of the previous Figure.

Figure above **:** The regular 3-fold pyramid (oblique top-view) representing the *Homostaura Anisopola triactinota* (as a category of three-dimensional promorphs). The three antimers are indicated by colors.

The symmetry of the motif as translation-free residue could be such that the intrinsic symmetry of the triangular two-dimensional crystal is according to the Cyclic Group

The next Figure shows the set of

Figure above **:** The only symmetry element of a two-dimensional triangular C_{3} crystal **:** a three-fold rotation axis in the center of the triangle.

Figure above **:** Only one crystallographic Form is needed to construct a triangular C_{3} two-dimensional crystal **:** One initially given crystal face implies two others in virtue of the 3-fold rotation axis, resulting in one Form consisting of three faces, together making up the triangular crystal.

For a two-dimensional C

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

And the Vector Rosette of Actual Growth added

Figure above **:** The three antimers (blue, green, yellow) of a two-dimensional triangular crystal with intrinsic C_{3} symmetry. Radial (R) and interradial (IR) directions indicated. Vector Rosette of Actual Growth added. The vectors **a, b,** and **c** are equivalent.

The next Figure depicts and names the

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

The next Figure gives the three-dimensional analogue of the promorph of the previous Figure.

Figure above **:** Slightly oblique top-view of the trigonal (3-fold) gyroid pyramid representing the *Homogyrostaura trimera* (as a category of three-dimensional promorphs).

The symmetry of the motif as translation-free residue could be such that the intrinsic symmetry of the triangular two-dimensional crystal is according to the Dihedral Group

The next Figure shows the set of

Figure above **:** The only symmetry element of a two-dimensional triangular D_{1} crystal is a mirror line ( **m** ).

Figure above **:** Two crystallographic Forms are needed to construct a triangular D_{1} two-dimensional crystal **:** One initially given crystal face (blue), not perpendicular or parallel to the mirror line, implies one other face (blue) in virtue of that mirror line, resulting in one open Form (blue) consisting of two faces. Another initially given face (red), perpendicular to the mirror line, does not imply other faces, so this one face constitutes an open Form (red). The two Forms together make up the whole triangular crystal.

A crystal with intrinsic D

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

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and two antimers (green, yellow). Radial (R) and interradial (IR) directions indicated.

And the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and two antimers (green, yellow). Radial (R) and interradial (IR) directions indicated. Vector Rosette of Actual Growth added. The vectors **b** and **c** are equivalent.

The next Figure depicts and names the

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

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

As we already know from earlier discussions, four antimers allow for

This configuration consists in the fact that two antimers are at one side of the mirror line while the two others are at the other side of the mirror line. See next Figures.

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and four antimers (green, yellow, and indicated by numerals). Interradial configuration. Radial (R) and interradial (IR) directions indicated.

And the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and four antimers (green, yellow). Interradial configuration. Radial (R) and interradial (IR) directions indicated. Vector Rosette of Actual Growth added. The vectors **b** and **c** are equivalent.

The symmetry of our crystal only demands that the configuration and shapes of the four antimers as a whole comply with the one mirror line of the crystal. And this means that the common point of the antimers can be positioned anywhere on the mirror line (which is vertical in the present case), dependent on the geometry of the motif as translation-free residue. The next Figure illustrates this freedom of positioning of the common point.

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and four antimers (green, yellow, and indicated by numerals). Vector Rosette of Actual Growth added. Alternative position of common point.

The next Figure depicts and names the

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

This configuration consists in the fact that two antimers lie on the mirror line, while the third antimer lies at one side of it and the fourth at the other. See next Figures.

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and four antimers (green, yellow). Radial configuration. Radial (R) and interradial (IR) directions indicated.

And the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and four antimers (green, yellow). Radial configuration. Radial (R) and interradial (IR) directions indicated. Vector Rosette of Actual Growth added. The vectors **b** and **c** are equivalent.

The next Figure depicts and names the

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

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

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and three antimers (green, yellow, and indicated by numerals).

And the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and three antimers (green, yellow). Vector Rosette of Actual Growth added. The vectors **b** and **c** are equivalent.

The symmetry of our crystal only demands that the configuration and shapes of the three antimers as a whole comply with the one mirror line of the crystal. And this means that the common point of the antimers can be positioned anywhere on the mirror line (which is vertical in the present case), dependent on the geometry of the motif as translation-free residue. The next Figure illustrates this freedom of positioning of the common point.

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and three antimers (green, yellow, and indicated by numerals). Vector Rosette of Actual Growth added. Alternative position of common point.

The next Figure depicts and names the

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

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

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and five antimers (green, yellow, and indicated by numerals).

And the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and five antimers (green, yellow). Vector Rosette of Actual Growth added. The vectors **b** and **c** are equivalent.

The next Figure depicts and names the

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

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

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and six antimers (green, yellow, and indicated by numerals).

And the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically triangular crystal with intrinsic D_{1} symmetry and six antimers (green, yellow, and indicated by numerals). Vector Rosette of Actual Growth added. The vectors **b** and **c** are equivalent.

The next Figure depicts and names the

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

The symmetry of the motif (as translation-free residue) of an intrinsically triangular 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 **:** Three crystallographic Forms are needed to construct a triangular 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 three such Forms are needed to construct an equilateral triangle , and thus the whole triangular C_{1} crystal.

C

Figure above **:** A two-dimensional intrinsically triangular 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 triangular 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,** are unique (i.e. non-equivalent).

The next Figure (two images) gives the possible

This

In the

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