This document continues the investigation of special categories (If / Then constants), and compares crystals with organisms. It, and the ensuing documents, brings crystals closer to organisms by demonstrating that

In fact we have demonstrated this already in **Second Part of Website** , viz. in the Series
**"Basic Forms of Crystals"**
(which can be found by scrolling down the left frame associated with this Second Part of Website). There we could establish that crystals do in fact have a promorph (on the basis of the geometry of the translation-free residue), but, as thus established, this promorph was not expressed *macroscopically* in the crystal. In the present series of document we let it be so expressed, by showing the macroscopic regions of the crystal that represent genuine antimers. In this way we succeed in bringing crystals even more close to organisms (in which the promorph and its corresponding antimers are always macroscopically expressed).

Crystals and Organisms, Shape, Symmetry and Promorph.

Rectangle (Promorphs)

The **Vector Rosette of Actual Growth** of any intrinsically **rectangular** two-dimensional (fully developed) crystal of whatever (possible) intrinsic point symmetry (while the symmetry of a rectangle itself is always that of D_{2}) is as follows (blue lines) **:**

This vector rosette in fact consists of four lines (vectors), **a, b, c, d,** each originating in the center of the crystal and ending up at a corner. As such it encodes the crystal's shape. And the lines of this rosette *generally* point to the presence of four or two antimers (dependent on the chemico-geometric features of the crystal's motifs, expressed as the equivalence or non-equivalence of the vectors **a, b, c, d** in the rosette) where, of course, in addition to these four lines, also the boundaries between the antimers indicate special directions, as well as the mirror lines, if present, of the crystal (which can coincide with the boundaries between antimers, or coincide with the median lines of antimers).

In some cases the vectors of actual growth do not point to antimers at all, viz. in all (2-D) crystals with their symmetry group according to the plane group **P1** (and therefore having **C _{1}** point symmetry). Here an asymmetric motif is repeated along two independent translations, that, in the case of a rectangular crystal (See the

In other cases, different numbers, i.e. different from 2 and 4, antimers can occur in rectangular (2-D) crystals.

We will now consider several possible cases of the given example of a rectangular two-dimensional crystal as depicted

The intrinsic point symmetry of crystals with intrinsic rectangular shape is often identical to the point symmetry of a (geometric) rectangle, which is according to the dihedral group D

The shape of the rectangular D

If a rectangularly shaped crystal has intrinsic D

Intrinsic D

The

which choice is arbitrary in the absence of further 'chemical' information about the crystal. In such a case we prefer the first way of depicting. If, on the other hand, such information is available as to the geometry and orientation of the motifs, then we should assess which of these two possiblities is actually realized. They constitute two different types of configuration of the four antimers, and they correspondingly represent two slightly different promorphs.

In the drawings the

The next Figure is the same as was depicted

And the next Figure depicts the same, but with another, also compatible motif, i.e. we still have the case of a rectangular crystal having D

Both Figures also represent the

Figure above **:** Interradial configuration of the four antimers (green, yellow) of the above described rectangular two-dimensional crystal with intrinsic D_{2} symmetry.

The

Figure above **:** Radial configuration of the four antimers (green, yellow) of the above described rectangular two-dimensional crystal with intrinsic D_{2} symmetry.

To symbolize this second type of configuration, we insert a motif (black)

To see in what way the motif (shown by a comparable motif, also having D

The next two Figures depict and name the **promorph** of the above considered two-dimensional crystals. It is either the 2-dimensional analogue of a *rectangular* pyramid (in fact it is its base) with four antimers (yellow, green) (**first mentioned type of configuration of the four antimers** ) **:**

The next Figure explains these two promorphological possibilities. They depend on how the motif lets itself be interpreted in terms of antimers.

Figure above **:** The chemical and morphological features of a two-dimensional ('chemical') D_{2} motif determine which parts of it to be interpreted as antimers. In the case of a motif consisting of four antimers the promorph can then be either that of the *(Autopola Orthostaura) Tetraphragma interradialia,* or that of the

The delineation of the four antimers can be expressed in several equivalent ways, as is shown in the next Figure for the radial configuration

The next five Figures are about

Figure above **:** System of cross axes of a rectangle.

A rectangle (but also all other amphitect [ = laterally compressed] polygons) possesses a *special* system of **cross axes.** Generally, cross axes are lines that go through the center, where each such line connects two definite parts (for example corners or centers of sides) of a polygon. When, as in amphitect polygons, there is one (and only one) pair of such axes of which the members are *perpendicular* to each other, we call them **directional axes** ( In the Figure the lines **dd** and **d'd'** ). And if the directional axes are *radial,* then we call the corresponding promorph *radial,* as in "*(Autopola Orthostaura) Tetraphragma radialia*". If, on the other hand, the directional axes are *interradial,* then we call the corresponding promorph *interradial,* as in "*(Autopola Orthostaura) Tetraphragma interradialia*".

The next Figure shows the system of

The next Figure, finally, shows the system of

Figure above **:** System of cross axes (three pairs, red, blue, blue) of a 6-fold amphitect polygon. The perpendicular ones (red) are the *directional axes* of the polygon.

In the above drawing there is another pair of cross axes of which the members seem to be perpendicular to each other, in addition to the pair (red) mentioned earlier. But the perpendicularity of that second pair is only of an accidental nature, implied by the special demensions of this particular figure. Generally, in any amphitect polygon all cross axes involve angles different from 90

Chemico-morphological features of motifs could be such that there are only

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

And with the Vector Rosette of Actual Growth added

The next Figure is the same ( D

The next Figure gives the

There can in principle be any even number of antimers, dependent on the geometry of the translation-free chemical motif. The Figures below show the cases of six and of eight antimers.

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

And with the Vector Rosette of Actual Growth added

Figure above **:** A two-dimensional rectangular D_{2} crystal with six antimers (green, yellow, blue, and indicated by numerals). Vector Rosette of Actual Growth added.

The corresponding

Figure above **:** Promorph of a D_{2} two-dimensional crystal with six antimers (green, yellow, blue).

and with the Vector Rosette of Actual Growth added

The next Figures give the

Figure above **:** Same as previous Figure. Median lines of antimers omitted, in order to let the latter stand out more clearly.

The degree of flattening of the amphitect polygon that depicts this promorph is arbitrary (as long as it remains flattened), so the next will do also

Chemico-morphological features of motifs could be such that the true symmetry of the rectangular two-dimensional crystal is not D

Its only symmetry element is then a 2-fold rotation axis (Its symmetry group therefore consists of two group elements, viz. the identity element and a half-turn

The shape of our rectangular C

Crystals with intrinsic C

The

where the C

A somewhat better representation of an intrinsically rectangular two-dimensional crystal with intrinsic C

and with the Vector Rosette of Actual Growth added

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 rectangular 2-D crystal is again

where the C

A slightly better representation of our two-dimensional rectangular C

and with the Vector Rosette of Actual Growth added

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.

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 rectangular two-dimensional crystal is

The rectangular shape of the crystal consists of a combination of three crystallographic

Such a D

Such a case (two antimers, green, yellow) is depicted in the next Figure, where the D

Figure above **:** A two-dimensional intrinsically rectangular crystal with intrinsic D_{1} symmetry and two antimers (green, yellow). The D_{1} symmetry of the crystal is expressed by the insertion of a motif (black).

With another way of expressing the D

Figure above **:** Same crystal as in previous Figure. Vector Rosette of Actual Growth added. The two antimers (green, yellow) indicated by numerals.

The next Figure is the same ( D

The next Figure gives the

Here (as comparable with the D

**Interradial :**

and with the Vector Rosette of Actual Growth added

or, alternatively (but promorphologically equivalently)

and with the Vector Rosette of Actual Growth added

The next two Figures give the corresponding

**Interradial case (perpendicular directional axes interradially positioned) :**

The next three Figures elaborate a little more on the

Figure above **:** Cross axes (two pairs, blue, red) of the isosceles trapezium, the basic form of the *Eutetrapleura interradialia*. The perpendicular cross axes (red) are the *directional axes*, while the other axes (blue) are just cross axes. The latter are bent, but this is neither typical, nor necessary as the next Figure shows.

Figure above **:** Cross axes of the isosceles trapezium, the basic form of the *Eutetrapleura interradialia*. Two sets of (straight) cross axes (red, blue). The perpendicular ones (red) are the *directional axes* of the isosceles trapezium. The horizontal directional axis does not connect the centers of opposite sides, but this is to be expected because of the absence in the promorph of a horizontal mirror line.

Also the basic form of the

Figure above **:** Cross axes of a bi-isosceles triangle, the basic form of the *Eutetrapleura radialia*. All cross axes are straight. The perpendicular ones (red) are the *directional axes* of the bi-isosceles triangle.

Here we have six antimers, three at each side of the mirror line.

and with the Vector Rosette of Actual Growth added

Indicating the arrangement and number of antimers, to express a certain promorph, often allows for some freedom. The next two Figures are wholly equivalent to the two Figures directly above. The six antimers are indicated by the colors green, yellow and blue (See also next Figure).

Figure above **:** Same as previous Figure. Vector Rosette omitted. Coloration of the six antimers (green, yellow) expressing the intrinsic reflectional symmetry of the crystal.

The next Figure gives (with two images) the

Figure above **:** Same as previous Figure. Median lines of antimers omitted, in order to let the latter stand out more clearly.

The next three Figures illustrate the case of three antimers (green, yellow). One median antimer (yellow) is mirror symmetric. The two others (green) are (generally) asymmetric and lie at either side of the mirror line

and with the Vector Rosette of Actual Growth added

Also here there are alternative ways to indicate the three antimers in a rectangular D

The next Figure gives the

Figure above **:** Same as previous Figure. Median lines of antimers omitted, in order to let the latter stand out more clearly.

In the case of five antimers we again have one symmetric median antimer, and (now) four other antimers each of which is (generally) asymmetric, and which are symmetrically arranged at either side of the mirror line

and with the Vector Rosette of Actual Growth added

As before, there are several alternative ways to express five antimers in a rectangular D

Figure above **:** Alternative representation of five antimers (green, yellow) in a rectangular D_{1} crystal.

The next Figure gives (with two images) the

**:** Same as previous Figure. Median lines of antimers omitted, in order to let the latter stand out more clearly.

In the case of seven antimers we again have one symmetric median antimer, and (now) six others each of which (generally) is asymmetric, and which are symmetrically arranged at either side of the mirror line

Figure above **:** A two-dimensional rectangular crystal with intrinsic D_{1} symmetry and seven antimers (green, yellow).

and with the Vector Rosette of Actual Growth added

There are, of course, several different ways to express the presence of seven antimers in a rectangular D

The next Figure gives (with two images) the

Figure above **:** Same as previous Figure. Median lines of antimers omitted, in order to let the latter more clearly stand out.

Rectangular crystals can have a true symmetry according to the point group C

Figure above **:** A two-dimensional intrinsically rectangular crystal with intrinsic C_{1} symmetry. There are no genuine antimers. A motif (black) is inserted to clearly express the C_{1} symmetry of the crystal.

And with the Vector Rosette of Actual Growth added

The inserted asymmetric motif (black) clearly shows that the vectors

The next Figure (two images) gives the possible **promorphs** (and their names) of this rectangular 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

In the

**e-mail :**

To continue click HERE

**e-mail :**