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.**

Parallelogram / Rhombus (Promorphs)

We shall start with the Parallelogram

The **Vector Rosette of Actual Growth** of a parallelogrammatic crystal, i.e. of any two-dimensional crystal having as its intrinsic shape (when fully developed) the **Parallelogram** (which itself has C_{2} symmetry) and possessing whatever (possible) intrinsic (point) symmetry, is as follows (blue lines) **:**

This Vector Rosette, viz. a vector rosette of a parallelogrammatic two-dimensional crystal, in fact consists of four vectors,

Only two plane groups, viz. P1 and P2 support crystals with the parallelogram as their intrinsic shape. See the **table given in the previous document** , where also the corresponding point groups are given (with the crystallographic notation between brackets), and the number of crystallographic Forms needed to construct a parallelogram in each case (of point group).

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

A parallelogrammatic crystal possessing this (C

Two crystallographic

Figure above **:** Two Forms are needed to construct a parallelogram on the basis of intrinsic C_{2} symmetry **:** An initially given crystal face (blue) implies a second face (blue) parallel to it, in virtue of the 2-fold rotation axis. The result is an open Form consisting of two parallel faces (blue). As such this Form cannot represent a crystal, so a second Form is needed **:** Another initially given crystal face (red), not parallel to the first one, implies a second face (red) parallel to it, in virtue of that same 2-fold rotation axis. The two Forms together constitute a parallelogram.

Crystals with intrinsic C

The geometry of the motif (translation-free residue of our C

Figure above **:** A parallelogrammatic two-dimensional crystal with intrinsic C_{2} symmetry and four antimers (green, yellow).

And with the Vector Rosette of Actual growth added

Figure above **:** A parallelogrammatic two-dimensional crystal with intrinsic C_{2} symmetry and four antimers (green, yellow). Vector Rosette of Actual Growth (blue lines) added. The vectors are two by two equivalent **:** **a** with **d** and **b** with **c** .

The next Figure gives the

Its three-dimensional counterpart (i.e. the corresponding pyramid) is given in the next Figure (slightly oblique top-view)

Chemico-morphological features of motifs could be such that the true symmetry of the parallelogrammatic two-dimensional crystal is again

And with the Vector Rosette of Actual Growth added

Figure above **:** Vector Rosette of Actual Growth of the above crystal (C_{2} , two antimers). The vectors are two by two equivalent **:** **a** with **d** and **b** with **c** .

The next Figure gives the

Its three-dimensional analogue -- the pyramid -- is depicted in the next Figure (slightly oblique top-view)

There can in principle be any even number of antimers, dependent on the geometry of the translation-free chemical motif. The next Figures show the case of eight antimers (green, yellow).

And with the Vector Rosette of Actual Growth added

The next Figure gives the

The next Figure gives this same promorph, but now with the median lines of the antimers omitted, in order to let the latter stand out more clearly.

Chemico-morphological features of motifs could be such that the true symmetry of the parallelogrammatic two-dimensional crystal is

Figure above **:** A parallelogrammatic two-dimensional crystal with intrinsic C_{1} symmetry. There are no symmetry elements.

Figure above **:** A parallelogrammatic two-dimensional crystal with intrinsic C_{1} symmetry. There are no symmetry elements. The inserted asymmetric motif (black) is meant to express the intrinsic C_{1} symmetry of the crystal (despite its intrinsic parallelogrammatic shape), and shows that the four vectors **a, b, c, d,** are all non-equivalent, because there is no symmetry transformation that transforms one (vector) into another. There are no genuine antimers.

Figure above **:** Any initially given crystal face will not imply other faces, because there are no symmetry transformations that generate new faces. Consequently any such initially given crystal face is a Form. So to construct a parallelogrammatic (two-dimensional) crystal, four Forms are needed (blue, red, green, purple).

The next Figure (two images) gives the possible

This

Next we consider the Rhombus

The **Vector Rosette of Actual Growth** of a rhombus-shaped crystal, i.e. of any two-dimensional crystal having as its intrinsic shape (when fully developed) the **Rhombus** (which itself has D_{2} symmetry) and possessing whatever (possible) intrinsic (point) symmetry, is as follows (blue lines) **:**

This Vector Rosette, viz. a vector rosette of a rhombic two-dimensional crystal, in fact consists of four vectors,

The following **plane groups** support a rhombus as the intrinsic shape of a two-dimensional crystal **:**

P1, P2, Pm, Cm, Pg, P2mm, C2mm, P2mg and P2gg.

See also the **table in the previous document** indicating these plane groups, the implied point groups (with the crystallographic notation between brackets) and the number of crystallographic Forms needed to construct the rhombus-shaped crystal in each case of point symmetry. In the table we can see that a rhomb-shaped crystal can have either an intrinsic symmetry equal to that of a rhombus, or a lower symmetry, down to no symmetry at all.

We will now consider several possible cases of the given example of a rhombus-shaped two-dimensional crystal as depicted just above.

This is the case of a rhombus-shaped two-dimensional crystal having as its intrinsic point symmetry the full symmetry of a Rhombus, i.e. D

The pattern of

Figure above **:** Symmetry elements of an intrinsically rhomb-shape two-dimensional crystal with intrinsic D_{2} symmetry **:** Two mirror lines (m_{1}, m_{2}), perpendicular to each other, and connecting opposite corners. One two-fold rotation axis at the intersection point of the mirror lines.

If we take an initially given crystal face to be such that it is not perpendicular to one of the mirror lines, only

Figure above **:** One Form (blue) is needed to conceptually construct a rhomb-shaped D_{2} crystal **:** An initially given crystal face not parallel to one of the mirror lines implies three more faces, together making up a closed Form having the shape of a rhombus **:** The initial face is reflected in one of the mirror lines resulting in two faces. This set of two faces is then reflected in the other mirror line resulting in a rhombus.

Crystals having intrinsic D

The geometry of the 'chemical' motif of a 2-dimensional intrinsically rhomb-shaped D

Figure above **:** A two-dimensional rhomb-shaped D_{2} crystal with four antimers (green, yellow). Radial (R) and interradial (IR) directions indicated.

And with the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional rhomb-shaped D_{2} crystal with four antimers (green, yellow). Vector Rosette of Actual Growth added (blue lines). The vectors of the rosette are two by two equivalent **:** **a** with **d** and **b** with **c** .

Depending on the motif (translation-free residue) the four antimers can be configured differently, as the next Figures illustrate.

Figure above **:** Alternative configuration of four antimers (green, yellow) in a two-dimensional rhomb-shaped D_{2} crystal.

And with the Vector Rosette of Actual Growth added

Figure above **:** The two-dimensional rhomb-shaped D_{2} crystal with four antimers (green, yellow) of the previous Figure. Vector Rosette of Actual Growth added (blue lines).

The two types of configuration of the four antimers as shown above imply two slightly different

The **first configuration** corresponds to the following promorph **:**

Figure above **:** Promorph of a two-dimensional (rhomb-shaped) D_{2} crystal with four antimers arranged according to the **first configuration**. It is a rhombus, and as such the two-dimensional analogue of the rhombic pyramid. The directional axes (see next Figure), i.e. those cross axes that are perpendicular to each other, run through the antimers, and are therefore *radial* directional axes.

The next Figure indicates the

The

Figure above **:** Promorph of a two-dimensional (rhomb-shaped) D_{2} crystal with four antimers arranged according to the **second configuration**. It is a rectangle, and as such the two-dimensional analogue of the rectangular pyramid. The directional axes (red lines), i.e. those cross axes that are perpendicular to each other, run *between* the antimers, and are therefore *interradial* directional axes.

The motif of a two-dimensional intrinsically rhomb-shaped D

Figure above **:** A two-dimensional rhomb-shaped crystal with intrinsic D_{2} symmetry and two antimers (green, yellow).

And with the Vector Rosette of Actual growth added

Figure above **:** A two-dimensional rhomb-shaped crystal with intrinsic D_{2} symmetry and two antimers (green, yellow). Vector Rosette of Actual growth added. The vectors are two by two equivalent **:** **a** with **d** and **b** with **c** .

The next Figure gives the

The geometry of the motif of a rhomb-shape two-dimensional crystal with intrinsic D

Figure above **:** A two-dimensional rhomb-shaped crystal with intrinsic D_{2} symmetry and six antimers (green, yellow, blue).

And with the Vector Rosette of Actual growth added

Figure above **:** A two-dimensional rhomb-shaped crystal with intrinsic D_{2} symmetry and six antimers (green, yellow, blue). Vector Rosette of Actual growth added. The vectors are two by two equivalent **:** **a** with **d** and **b** with **c** .

The next Figure gives the

The geometry of the motif of a rhomb-shape two-dimensional crystal with intrinsic D

Figure above **:** A two-dimensional rhomb-shaped crystal with intrinsic D_{2} symmetry and eight antimers (green, yellow).

And with the Vector Rosette of Actual growth added

Figure above **:** A two-dimensional rhomb-shaped crystal with intrinsic D_{2} symmetry and eight antimers (green, yellow). Vector Rosette of Actual growth added. The vectors are two by two equivalent **:** **a** with **d** and **b** with **c** .

The next Figure gives the

Intrinsically rhomb-shaped two-dimensional crystals can, dependent on the symmetry of the translation-free residue, which represents the chemical motif, have a lower symmetry than that of a rhombus. Here we consider such crystals with D

The next Figure gives the pattern of **symmetry elements** of such a crystal, a pattern which consists of just one mirror line.

Figure above **:** A mirror line (m) as the only symmetry element of a two-dimensional rhomb-shaped crystal with intrinsic D_{1} symmetry.

Figure above **:** Two crystallographic Forms (red, dark blue) are needed to construct a rhomb-shaped D_{1} two-dimensional crystal.

D

As is by now clear, these four antimers can be arranged in two ways, which we have called the

Figure above **:** A two-dimensional intrinsically rhomb-shaped D_{1} crystal with four antimers (green, yellow, indicated by numerals). **Interradial configuration** of antimers.

And with the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically rhomb-shaped D_{1} crystal with four antimers (green, yellow, indicated by numerals). Interradial configuration of antimers. Vector Rosette of Actual Growth added. The vector **b** is equivalent with the vector **c** .

We have just depicted the

Figure above **:** One would be tempted to interpret the purple axes (which are perpendicular to each other) as *directional axes,* which would then imply that one of them runs between antimers while the other runs through two antimers, so that we cannot decide whether the directional axes taken together (as a pair) are *radial* or *interradial*. The directional axes of which we assess whether together they are radial or interradial -- and which assessment in turn determines whether the promorph is radial or interradial -- must be the directional axes of the **promorph.** The purple lines in the present Figure are the directional axes of the *rhombus,* which here is the (intrinsic) shape of the crystal. The geometric figure representing the promorph of this crystal (i.e. the D_{1} crystal with four antimers, arranged as in the above illustration **:** two at one side of the mirror line and two at the other) is not a rhombus, but an isosceles trapezium (See the promorph **below** ). And in such a trapezium both directional axes run between antimers. They are therefore definitely interradial.

The next Figures illustrate the

Figure above **:** A two-dimensional intrinsically rhomb-shaped D_{1} crystal with four antimers (green, yellow, indicated by numerals). **Radial configuration** of antimers.

And with the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically rhomb-shaped D_{1} crystal with four antimers (green, yellow). Radial configuration of antimers. Vector Rosette of Actual Growth added. The vector **b** is equivalent with the vector **c** .

The

The

The geometry of the motif (as translation-free residue) of a rhomb-shaped D

Figure above **:** A two-dimensional intrinsically rhomb-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, i.e. to express the fact that there is only one mirror line present.

And with the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically rhomb-shaped D_{1} crystal with two antimers (green, yellow). Vector Rosette of Actual Growth added. The vector **b** is equivalent with the vector **c** .

The

An intrinsically rhomb-shaped crystal can contain a motif (as translation-free residue) with D

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

And with the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically rhomb-shaped D_{1} crystal with three antimers (green, yellow). Vector Rosette of Actual Growth added. The vector **b** is equivalent with the vector **c** .

The

An intrinsically rhomb-shaped crystal can contain a motif (as translation-free residue) with D

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

And with the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically rhomb-shaped D_{1} crystal with five antimers (green, yellow). Vector Rosette of Actual Growth added. The vector **b** is equivalent with the vector **c** .

The

A two-dimensional rhomb-shaped crystal with intrinsic D

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

And with the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional intrinsically rhomb-shaped D_{1} crystal with six antimers (green, yellow). Vector Rosette of Actual Growth added. The vector **b** is equivalent with the vector **c** .

The

An intrinsically rhomb-shaped two-dimensional crystal can have an intrinsic symmetry according to the Cyclic Group C

The only

Figure above **:** The only symmetry element of a two-dimensional (intrinsically rhomb-shaped) crystal with intrinsic C_{2} symmetry is a two-fold rotation axis (green).

Figure above **:** Two crystallographic Forms are needed to construct a rhomb-shaped C_{2} two-dimensional crystal **:** An initially given crystal face (red) implies (in virtue of the 2-fold rotation axis) one more face parallel to it, resulting in an open Form consisting of two faces (red). A second initially given face (blue), not parallel to the first, implies a second face, in virtue of the same rotation axis, also resulting in an open Form consisting of two faces (blue). The two Forms together (red, blue) constitute a rhombus, and therefore constitute the whole rhomb-shaped crystal.

The geometry of the motif (as translation-free residue) of a rhomb-shaped two-dimensional crystal with intrinsic C

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

And with the Vector Rosette of Actual Growth added

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

**a** is equivalent to **d** , and **b** to **c** .

The

The next Figure gives the three-dimensional analogue of the just given (two-dimensional) promorph. It is a two-fold amphitect gyroid pyramid.

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

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

And with the Vector Rosette of Actual Growth added

Figure above **:** Two-dimensional rhomb-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 rhomb-shaped C_{2} crystal. The four antimers are indicated (green, yellow).

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

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 rhomb-shaped crystal with intrinsic C

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

And with the Vector Rosette of Actual Growth added

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

**a** is equivalent to **d** , and **b** to **c** .

The

The promorphs of C

The symmetry of the motif (as translation-free residue) of an intrinsically rhomb-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 **:** Four crystallographic Forms are needed to construct a rhomb-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 four such Forms are needed to construct a rhombus, and thus the whole rhomb-shaped C_{1} crystal.

C

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

The next Figure (two images) gives the possible

This

In the

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