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 Ditetragon

We will now investigate two-dimensional crystals with an intrinsic shape according to a

Figure above **:** Two-dimensional crystal (microscopic view) built up from a stacking of quadratic building blocks, resulting in a **regular ditetragon** (and not in a regular octagon). If the motifs (not drawn) in the building blocks have D_{4} symmetry (or are such that their translation-free residue has this symmetry), then the intrinsic point symmetry of the crystal is according to the group D_{4} . As one can see, the depicted crystal shape is an octagon, but not a *regular* octagon (it lacks the equality of all its sides). According to the four-fold symmetry the number of antimers must be four, or a multiple of it.

__Either__ the crystal as drawn here, i.e. with empty building blocks, which by themselves already have D_{4} symmetry, implying the intrinsic point symmetry of the crystal to be that of D_{4} ,

__or__ it having those building blocks provided with explicit D_{4} motifs, __is__, although being holomorphic (symmetry of shape is equal to intrinsic symmetry of crystal -- both D_{4} ), not eupromorphic __if__ it has four equal antimers, (because the intrinsic shape suggests eight similar antimers). If it has eight (2x4) similar antimers, then it is eupromorphic.

The next Figure indicates the 'faces' of this crystal.

Figure above **:** The regular ditetragonal D_{4} crystal of the previous Figure. Crystal 'faces' indicated (dark blue lines).

In order to emphasize the D

Figure above **:** A regular ditetragonal D_{4} crystal. Its D_{4} motifs (black) consist of four equal antimers. Moreover each antimer is mirror symmetric. One can see that the shape of the crystal is __not__ that of a regular octagon, but indeed of a regular ditetragon **:** If we set the length of a side of the square building block (the stacking of which results in the crystal) as being **1** , then its diagonal has a length of **square root(2)**. The length of an oblique side of the octagon is then **4{square root(2)}**. The length of a horizontal or vertical side of the octagon is, however, equal to **5** . So the octagon has no equal sides and is therefore irregular. In fact the figure's sides of lenght 5 alternate with those of length 4{square root(2)}, so its shape is that of a regular ditetragon.

Removing the lattice connection lines and the motifs, results in a

Figure above **:** Macroscopic view of the regular ditetragonal D_{4} two-dimensional crystal of the previous Figure.

The next Figure gives the pattern of

Figure above **:** Pattern of symmetry elements of the regular ditetragonal crystal of the above Figures. It consists of a *4-fold rotation axis* at the intersection point of *four mirror lines.* The latter comprise two sets **:** One set consists of a horizontal and vertical mirror line, while the second set consists of two diagonal mirror lines. So in all we have four mirror lines and one 4-fold rotation axis as the symmetry elements of the crystal.

Figure above **:** Two crystallographic Forms are needed to construct the (eight) faces of a regular ditetragonal two-dimensional crystal. An initially given face (dark blue) implies three more faces in virtue of the 4-fold rotation axis. The result is a closed Form -- a square -- consisting of four faces. A second initially given face (red), not parallel nor perpendicular to the first one, also implies three more faces, also resulting in a quadratic closed Form consisting of four faces. Combining these two Forms yields our regular ditetragonal crystal.

Figure above **:** A regular ditetragonal D_{4} two-dimensional crystal. The case of __ f o u r__ equal (and in themselves mirror symmetric) antimers (green, yellow). Note the correspondence between the morphology of the (microscopic) motif (as translation-free residue) and the arrangement of the (macroscopic) antimers of the crystal. In this way the promorph, and in particular the number of antimers is based on the morphology of the translation-free residue of the crystal. This residue is explicitly given in the form of a D

Another possible configuration of four equal antimers is the following

Figure above **:** Alternative configuration of four equal antimers in a regular ditetragonal D_{4} crystal .

Removing the lattice connection lines and the motifs in the above Figure, results in a

Figure above **:** Macroscopic view of the regular ditetragonal D_{4} two-dimensional crystal of the previous figure, with four equal antimers (green, yellow).

The

Figure above **:** The promorph of the regular ditetragonal D_{4} crystal with four antimers. It is a regular 4-fold polygon -- a square -- and as such the two-dimensional analogue of the quadratic pyramid, which represents the promorph of corresponding three-dimensional crystals or other objects. Note the difference in shape of the promorph (square, regular tetragon) and the crystal (regular ditetragon). Radial (R) and interradial (IR) directions are indicated.

We have just found out that the promorph of our regular ditetragonal D

Figure above **:** The regular ditetragonal D_{4} crystal under investigation. Its four mirror lines are indicated (red). The dark blue lines do not possess mirror symmetric poles. In fact these lines do not represent cross axes of the ditetragon.

If we want to find the

Figure above **:** The promorph of the regular ditetragonal D_{4} crystal under investigation. It has four cross axes (dark blue lines). Two of them are radial, and the other two are interradial. For each cross axis its two poles are mirror symmetric, which means that these axes are equipolar.

So now we know the

Figure above **:** The cross axes of the regular ditetragon **:** one horizontal, one vertical, two diagonal. All of them are equipolar (expressed in the term "*Isopola*" figuring in the name of the promorph).

The next Figure shows yet another configuration of four antimers of the regular ditetragonal crystal under investigation. This configuration, however, is

Figure above **:** Incorrect delineation of the four antimers of the regular ditetragonal crystal under investigation. Although the antimers are equal, they are not (each for themselves) mirror symmetric.

Figure above **:** Microscopic view of a regular ditetragonal D_{4} crystal. Its D_{4} motifs have eight similar antimers. Two types of antimer alternate with each other.

Figure above **:** The regular ditetragonal D_{4} two-dimensional crystal of the previous Figure. The case of __ e i g h t__ similar (and in themselves mirror symmetric) antimers (green, yellow). Note the correspondence between the morphology of the (microscopic) motif (as translation-free residue) and the arrangement of the (macroscopic) antimers of the crystal. In this way the promorph, and in particular the number of antimers is based on the morphology of the translation-free residue of the crystal. This residue is explicitly given in the form of a D

Removing the lattice connection lines and the motifs in the above Figure, results in a

Figure above **:** Macroscopic view of the regular ditetragonal D_{4} two-dimensional crystal of the previous figure, with eight similar (and in themselves mirror symmetric) antimers (green, yellow).

The

Figure above **:** The promorph of the regular ditetragonal D_{4} crystal with eight similar antimers. It is a regular ditetragon and as such the two-dimensional analogue of the ditetragonal pyramid, which represents the promorph of corresponding three-dimensional crystals or other objects. Note the identical shapes of the promorph (regular ditetragon) and the crystal (regular ditetragon). Radial (R) and interradial (IR) directions are indicated.

It is important to realize that not only ditetragonal, but also

Figure above **:** A regular tetragonal (i.e. quadratic) D_{4} two-dimensional crystal. The case of __ e i g h t__ similar (and in themselves mirror symmetric) antimers (green, yellow). Note the correspondence between the morphology of the (microscopic) motif (as translation-free residue) and the arrangement of the (macroscopic) antimers of the crystal. In this way the promorph, and in particular the number of antimers is based on the morphology of the translation-free residue of the crystal. This residue is explicitly given in the form of a D

Removing the lattice connection lines and the motifs in the above Figure, results in a

Figure above **:** Macroscopic view of the regular tetragonal (i.e. quadratic) D_{4} two-dimensional crystal of the previous figure, with eight similar (and in themselves mirror symmetric) antimers (green, yellow). The promorph of this crystal (like the ditetragonal crystal discussed earlier) belongs to the *Dihomostaura Isopola tetractinota* and was depicted **above** . Note, with respect to the present case, the difference in shape obtaining between that of the promorph (ditetragon) and that of the present crystal (square) of which it is the promorph.

Let's return to the d i t e t r a g o n a l D

The eight antimers of the translation-free residue are each for themselves mirror symmetric (with respect to a mirror line passing through the residue's center) (See Figure

Figure above **:** Macroscopic view of the regular ditetragonal D_{4} two-dimensional crystal (again) under investigation, with eight supposedly similar antimers (green, yellow). The delineation of the eight antimers is **incorrect** because -- as depicted -- they are not each for themselves mirror symmetric (with respect to a mirror line passing through the crystal's center) **:** The length of the line segment AB is to that of BC as **4{square root(2)}** to **5,** implying that DB is not equal to BE. Moreover, the antimers as depicted here (green, yellow) are congruent (not just similar), which they shouldn't be according to the morphology of the translation-free residue (motif) (See Figure **above** ). In fact the 'antimers' as depicted here are not really antimers. Only when we consider *pairs consisting of two 'antimers' (the latter) related to each other by a reflection,* we have genuine antimers. And then we get the regular ditetragonal crystal with *four* antimers that was discussed **earlier** . And the promorph of that crystal is, consequently, belonging to the *Homostaura Isopola tetractinota* which promorph was depicted **above** .

Above we have established that the

"

The geometric figure, as promorph, representing the

Figure above **:** A **d i t e t r a g o n** is formed when two unequal squares, with their centers coinciding, but differing in orientation by 45^{0}, cut off each other's corners. In two-dimensional crystals these two squares represent two crystallographic Forms (See **Figure above** ).

So the

Of the

Figure above **:** A square ( = regular tetragon) has four cross axes (red), two radial (R-R) and two interradial (IR-IR). Each cross axis is equipolar (which feature is expressed by the term "*isopola*" in the name of the promorph that the square can represent (*Homostaura Isopola tetractinota*)).

Wheras a

Figure above **:** Superimposing two equal squares results in a regular octagon.

A

Figure above **:** Regular octagon. One type of mirror line connects opposite sides.

Figure above **:** Regular octagon. A second type of mirror line connects opposite corners. One example shown.

Figure above **:** Regular octagon. The second type of mirror line connects opposite corners.

Figure above **:** Regular octagon. The total set of its mirror lines, consisting of the two types considered above.

A

Figure above **:** A regular octagon with eight equal antimers (green, yellow).

The

Figure above **:** The planimetric figure (regular octagon) representing the promorph of two-dimensional objects having eight equal, and in themselves mirror symmetric, antimers. If these objects are three-dimensional, then the figure representing their promorph is a stereometric figure, in the present case a regular octagonal pyramid. Radial (R) and interradial (IR) directions indicated. The antimers of the object are represented in the promorph as green and yellow areas.

The next Figure gives an object having the above promorph, a promorph belonging to the

Figure above **:** An object having eight equal, and in themselves mirror symmetric, antimers.

The next object also has eight equal and symmetric antimers.

Figure above **:** An object having eight equal, and in themselves mirror symmetric, antimers.

Of course both objects just depicted have the same promorph (

Without destroying the initial D_{8} symmetry, the number of antimers can be doubled.

If this doubling were such that the new antimers were exact copies of the original ones, and were inserted such that they bisect the angles between the original antimers (into two equal angles), then the symmetry of this object would have changed to D_{16} (in which D_{8} symmetry is still comntained as a subgroup), and the object's promorph would then have been transformed from the *Homostaura Isopola octactinota* to the *Homostaura Isopola hexakaidecactinota* ( The latter is a regular 16-gon, or sixteen-fold regular polygon).

If, on the other hand, the extra 'antimers' are totally different from the original antimers, we cannot consider them as additional antimers, but as a morphological extention of the original antimers, meaning that the number of antimers does not change, it remains eight.

If, finally, the extra antimers resemble the original antimers very closely (but are not identical to them), then the number of antimers is doubled **:** we end up with 16 similar antimers (eight by eight equal) that alternate between two types. An object that satisfies this condition is depicted in the next Figure.

Figure above **:** An object having 16 similar (in themselves mirror symmetric) antimers. The symmetry of this object is still according to the group D_{8} .

The

Figure above **:** Promorph of the just described and depicted object (not a crystal) having 16 similar (in themselves mirror symmetric) antimers. The symmetry of this object is still according to the group D_{8} . The promorph is a *regular dioctagon* (not a regular 16-gon). It is the two-dimensional analogue of the regular dioctogonal pyramid, which represents the promorph of corresponding three-dimensional objects (not crystals). This (two-dimensional) promorph possesses eight mirror lines and an 8-fold rotation axis through their point of intersection. So the symmetry is according to the group D_{8} (and not D_{16} ). Radial (R) and interradial (IR) directions indicated. The 16 similar antimers of the object are represented in the promorph by non-congruent (and thus just similar) areas (green, yellow). The antimers are eight by eight equal.

The next Figure again depicts the dioctagon of the just given promorph, but with an

Figure above **:** Incorrect delineation of the sixteen antimers in the figure supposed to represent the promorph of the object depicted **above** . The antimers of this object are not congruent, and this feature should be expressed in the figure representing its promorph. However, in the present Figure the areas representing antimers are congruent. Moreover, the antimers, as depicted here, are not in themselves mirror symmetric, what they should be (because *Homostaura,* and by implication also *Dihomostaura,* have symmetric antimers). In fact the figure represents a different promorph, and the antimers as presented here (green, yellow) are not antimers. See next Figure.

Figure above **:** The previous Figure actually does not represent the *Dihomostaura Isopola octactinota,* but the **Homostaura Isopola octactinota,** because it expresses in fact eight equal antimers (which are in themselves mirror symmetric). These eight equal and symmetric antimers are shown in the present Figure, and derived from the previous one. Each such an antimer itself consists of two congruent, namely mirror symmetric, 'antimers' (second order antimers), which we could call "*paramers*". Because, in the present case, every two paramers are mirror symmetric, they can be considered as belonging to each other, constituting an antimer ( This in contrast to considering the object, represented by this promorph, as possessing 16 congruent asymmetric antimers).

In the

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