General Ontology XII

General Ontology Cosmos and Nomos

Theory of Ontological Layers and Complexity Layers

Part XII

Crystals and Organisms

Theory of intrinsic Shape, intrinsic Symmetry and Promorph

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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 D₃ 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, a, b, c, each originating in the center of the crystal and ending up at a corner. As such it encodes the crystal's shape.

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.

D₃ symmetry.
Intrinsically triangular two-dimensional crystals with intrinsic D₃ symmetry have the full symmetry of the Equilateral Triangle, i.e. their intrinsic symmetry is the same as the symmetry of their intrinsic shape. So in the present case the intrinsic shape happens to express the crystal's intrinsic (point) symmetry.
The next Figure gives the symmetry elements of such a crystal.

Figure above : Symmetry elements of a two-dimensional intrinsically triangular crystal with D₃ intrinsic symmetry : Three mirror lines ( m₁ , m₂ , m₃ ), and one 3-fold rotation axis at their point of intersection.

One crystallographic Form is needed to conceptually construct a triangular D₃ crystal. See next Figure.

Figure above : Only one crystallographic Form is needed to construct a triangular D₃ 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.

D₃ symmetry and three antimers.
For a two-dimensional D₃ crystal only three antimers are possible. See next two Figures.

Figure above : The three antimers (blue, green, yellow) of a two-dimensional triangular crystal with intrinsic D₃ 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 D₃ 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 promorph of our two-dimensional triangular D₃ crystal. It is an equilateral triangle, and as such the two-dimensional analogue of the regular three-fold pyramid. As promorph it is the two-dimensional analogue of an instance of the (three-dimensional) Homostaura Anisopola triactinota.

Figure above : Promorph of the above discussed two-dimensional triangular D₃ 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.

C₃ symmetry.
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 C₃ .
The next Figure shows the set of symmetry elements of such a crystal, consisting of a 3-fold rotation axis only.

Figure above : The only symmetry element of a two-dimensional triangular C₃ crystal : a three-fold rotation axis in the center of the triangle.

One crystallographic Form is needed to conceptually construct a triangular C₃ crystal. See next Figure.

Figure above : Only one crystallographic Form is needed to construct a triangular C₃ 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.

C₃ symmetry and three antimers.
For a two-dimensional C₃ crystal only three antimers are possible. See next two Figures.

Figure above : The three antimers (blue, green, yellow) of a two-dimensional triangular crystal with intrinsic C₃ 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 C₃ 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 promorph of our two-dimensional triangular C₃ crystal. It is a regular three-fold gyroid polygon (trigonal gyroid polygon), and as such the two-dimensional analogue of the regular three-fold gyroid pyramid (trigonal gyroid pyramid). As promorph it is the two-dimensional analogue of an instance of the (three-dimensional) Homogyrostaura trimera.

Figure above : Promorph of the above discussed two-dimensional triangular C₃ 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).

D₁ symmetry.
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 D₁ .
The next Figure shows the set of symmetry elements of such a crystal, consisting of a mirror line only.

Figure above : The only symmetry element of a two-dimensional triangular D₁ crystal is a mirror line ( m ).

Two crystallographic Forms are needed to conceptually construct a triangular D₁ crystal. See next Figure.

Figure above : Two crystallographic Forms are needed to construct a triangular D₁ 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 D₁ symmetry can in principle have any number (> 1) of antimers depending on the geometry of the motif as translation-free residue. We will discuss several examples.

D₁ symmetry and two antimers.
The motif as translation-free residue of a two-dimensional D₁ crystal can be such that two antimers can be distinguished. See next two Figures.

Figure above : A two-dimensional intrinsically triangular crystal with intrinsic D₁ 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 D₁ 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 promorph of our two-dimensional triangular D₁ crystal. It is an isosceles triangle and as such the two-dimensional analogue of half a rhombic pyramid. As promorph it is the two-dimensional analogue of an instance of the (three-dimensional) Allopola Zygopleura eudipleura.

Figure above : Promorph of the above discussed two-dimensional triangular D₁ crystal. The two antimers are indicated (green, yellow).

D₁ symmetry with four antimers.
The motif as translation-free residue of a two-dimensional D₁ crystal can be such that four antimers can be distinguished.
As we already know from earlier discussions, four antimers allow for two different spatial configurations, viz. the interradial and the radial configuration.

Interradial configuration of the four antimers.
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 D₁ 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 D₁ 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 D₁ 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 promorph of our two-dimensional triangular D₁ crystal with an interradial configuration of its four antimers. It is an isosceles trapezium and as such the two-dimensional analogue of a trapezoid pyramid. As promorph it is the two-dimensional analogue of an instance of the (three-dimensional) Allopola Zygopleura Eutetrapleura interradialia.

Figure above : Promorph of the above discussed two-dimensional triangular D₁ crystal. The four antimers are indicated (green, yellow).

Radial configuration of the four antimers.
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 D₁ symmetry and four antimers (green, yellow). Radial configuration. Radial (R) and interradial (IR) directions indicated.

And the Vector Rosette of Actual Growth added :

The next Figure depicts and names the promorph of our two-dimensional triangular D₁ crystal with radial configuration of its four antimers. It is a bi-isosceles triangle and as such the two-dimensional analogue of a bi-isosceles pyramid. As promorph it is the two-dimensional analogue of an instance of the (three-dimensional) Allopola Zygopleura Eutetrapleura radialia.

Figure above : Promorph of the above discussed two-dimensional triangular D₁ crystal. The four antimers are indicated (green, yellow).

D₁ symmetry and three antimers.
The motif as translation-free residue of a two-dimensional D₁ crystal can be such that three antimers can be distinguished. See next Figures.

Figure above : A two-dimensional intrinsically triangular crystal with intrinsic D₁ 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 D₁ 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 D₁ 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 promorph of our two-dimensional triangular D₁ crystal with three antimers. It is half a six-fold amphitect polygon, and as such the two-dimensional analogue of half a six-fold amphitect pyramid. As promorph it is the two-dimensional analogue of an instance of the (three-dimensional) Allopola Amphipleura triamphipleura.

Figure above : Promorph of the above discussed two-dimensional triangular D₁ crystal. The three antimers are indicated (green, yellow).

D₁ symmetry and five antimers.
The motif as translation-free residue of a two-dimensional D₁ crystal can be such that five antimers can be distinguished. See next Figures.

Figure above : A two-dimensional intrinsically triangular crystal with intrinsic D₁ 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 D₁ 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 promorph of our two-dimensional triangular D₁ crystal with five antimers. It is half a ten-fold amphitect polygon, and as such the two-dimensional analogue of half a ten-fold amphitect pyramid. As promorph it is the two-dimensional analogue of an instance of the (three-dimensional) Allopola Amphipleura pentamphipleura.

Figure above : Promorph of the above discussed two-dimensional triangular D₁ crystal. In the right image the five antimers are indicated by colors (green, yellow).

D₁ symmetry and six antimers.
The motif as translation-free residue of a two-dimensional D₁ crystal can be such that six antimers can be distinguished. See next Figures.

Figure above : A two-dimensional intrinsically triangular crystal with intrinsic D₁ 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 D₁ 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 promorph of our two-dimensional triangular D₁ crystal with six antimers. It is half a twelve-fold amphitect polygon, and as such the two-dimensional analogue of half a twelve-fold amphitect pyramid. As promorph it is the two-dimensional analogue of an instance of the (three-dimensional) Allopola Amphipleura hexamphipleura.

Figure above : Promorph of the above discussed two-dimensional triangular D₁ crystal. In the right image the six antimers are indicated by colors (green, yellow).

C₁ symmetry and no antimers.
The symmetry of the motif (as translation-free residue) of an intrinsically triangular two-dimensional crystal can be according to the Asymmetric Group C₁ , i.e. the crystal then does not have any (point) symmetry at all, despite its intrinsic regularly-triangular shape (which has itself D₃ symmetry).
There are consequently no symmetry elements apart from the trivial one, viz. the 1-fold rotation axis (0⁰ or 360⁰ rotation about any axis) :

Figure above : A (two-dimensional) C₁ crystal does not have any symmetry elements (with respect to point symmetry).

Three crystallographic Forms are needed to conceptually construct a two-dimensional triangular crystal with intrinsic C₁ symmetry. See next Figure.

Figure above : Three crystallographic Forms are needed to construct a triangular two-dimensional C₁ 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 C₁ crystal.

C₁ crystals cannot have genuine antimers. See next Figures.

Figure above : A two-dimensional intrinsically triangular C₁ crystal.
A motif (black) is inserted to express the crystal's C₁ symmetry.

And with the Vector Rosette of Actual Growth added :

Figure above : A two-dimensional intrinsically triangular C₁ crystal. A motif (black) is inserted to express the crystal's C₁ 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 promorphs (and their names) of our triangular C₁ crystal. The promorph is either (left image) half an isosceles triangle (and, equivalently, a quarter of a rhombus), and is as such the two-dimensional analogue of a quarter of a rhombic pyramid, or (right image) an irregular triangle, indicating two unequal antimers (yellow, green), and is as such the two-dimensional analogue of an irregular pyramid (or, equivalently, a 1-fold pyramid).

This concludes our investigation concerning the relation of intrinsic triangular (2-dimensional) crystal shape to intrinsic point symmetry and to promorph.

In the next document we will investigate the sixth and last shape from our above given list : rectangle, square, parallelogram, rhombus, equilateral triangle, regular hexagon.

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