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 Hexagon

**REMARK :** In this document (and also the next) the terms "*hexagonal*" and "*hexagon*" always refer to respectively "*regularly-hexagonal*" and "*regular hexagon*" .

Let us consider a fully developed two-dimensional crystal having an intrinsically

Any intrinsically hexagonal crystal can be conceived as being built up by a periodic stacking of microscopic rhombic units, where the rhombi possess angles of 60 and 120 degrees. See next Figure.

Figure above **:** Two-dimensional crystal consisting of the periodic stacking of rhombi with 60 and 120 degrees angles. These rhombi are thus unit meshes. In the drawing these rhombi are provided with lines dividing each one of them into two equilateral triangles. These triangles are repeated, but not periodically, because they show two different orientations. So they do not represent unit meshes. They represent a triangular non-periodic *tiling* of the two-dimensional plane. Six such triangles form a hexagon, which is, like the rhombus, a genuine unit mesh but not the smallest possible (which is the rhombus as drawn). The next Figures illustrate all this.

In the previous Figure a few half-rhombi are involved to smooth out some edges of the hexagon which should otherwise be jagged. The next Figure shows how things are if we strictly adhere to the crystal as consisting of whole rhombic building blocks only. (One should realize that these features, like the mentioned jaggednes, or the involvement of half-rhombi, are macroscopically not visible at all

Figure above **:** Two-dimensional crystal consisting of the periodic stacking of rhombi with 60 and 120 degrees angles. These rhombi are thus unit meshes. As has been said, the triangles are not building blocks of the crystal in the sense of unit meshes, because their repetition is not periodic. Two edges are jagged, because of the strict adherence to *periodic* building blocks, in the present case rhombi with 60 and 120 degrees angles.

Figure above **:** This Figure clearly shows, by means of coloration, that the stacking of the rhombi (with 60 and 120 degrees angles) is *periodic* **:** their orientation (and size and shape) is the same throughout the pattern.

The next Figure shows the

Figure above **:** This Figure indicates the hexagonal point symmetry (D_{6}) of the two-dimensional crystal of the above Figures.

As has been said, a larger building block (larger than the above discussed rhombi) for our hexagonal crystal is possible

Figure above **:** Two-dimensional crystal of the above Figures. Hexagonal building blocks.

The next Figure indicates the

Figure above **:** Symmetry elements of the two-dimensional crystal of the above Figures. Rhombic (60/120^{0}) building blocks. Six mirror lines and one 6-fold rotation axis at their point of intersection.

As was the case in the previous two documents, the hexagonal lattice, i.e. a lattice consisting of the periodic stacking of 60/120

Figure above **:** Two-dimensional hexagonal crystal consisting of the stacking of special rectangles (red lines), of which the diagonals involve 60^{0} angles. Recall that the lines delineating small triangles or rhombi do not, strictly speaking, belong to the pattern presently under discussion. The 'motifs' of this pattern are the (empty) hexagonal contents of unit hexagons (each divisible into six equilateral triangles, see **Figure above** ). Such a motif has D_{6} symmetry, letting the pattern have the maximal plane group symmetry that a hexagonal net can support. As has been said, this symmetry is according to the group P6mm. Other motifs, but of the same point group symmetry (D_{6}), do not change this maximal plane group symmetry. If, however, instead of such motifs, lower symmetric motifs are inserted in the hexagonal net, then the plane group symmetry (and by definition also the point group symmetry) will be lower (with "*motif*" here is meant **:** *translation-free residue*).

In the Figures to come, concerning hexagonal crystals based on one or another plane group, we will use the 60/120

We will now give some general preliminaries concerning the ensuing discussion and exemplifications of the several possible types of two-dimensional crystals which have an intrinsically hexagonal shape.

The true, i.e. intrinsic, point symmetry of such a crystal is __either__ the same as the symmetry of its intrinsic shape, the regular hexagon, and thus according to the Dihedral Group **D _{6}** (crystallographically denoted as

That a **hexagonal crystal** can have these intrinsic symmetries can be explained succinctly as follows (and will be further evident in the sequel) **:**

A **regular hexagon** as such has the following symmetries, i.e. it will be superposed upon itself by the following transformations (which are then for that reason symmetry transformations) (See also **Figure below** ) **:**

- 0
^{0}or 360^{0}rotation about any axis. - 60
^{0}rotation about the center of the hexagon. - 120
^{0}rotation about the center of the hexagon. - 180
^{0}rotation about the center of the hexagon. - 240
^{0}rotation about the center of the hexagon. - 300
^{0}rotation about the center of the hexagon. - Reflection in the line
**m**through the center of the hexagon (_{1}**Figure below**). - Reflection in the line
**m**through the center of the hexagon (_{2}**Figure below**). - Reflection in the line
**m**through the center of the hexagon (_{3}**Figure below**). - Reflection in the line
**m**through the center of the hexagon (_{4}**Figure below**). - Reflection in the line
**m**through the center of the hexagon (_{5}**Figure below**). - Reflection in the line
**m**through the center of the hexagon (_{6}**Figure below**).

- When no symmetry is available (
**a**of the above list)**six Forms**are needed to produce a regular hexagon. - When only a 6-fold rotation axis is available (
**b, c, d, e**and**f**of the above list) only**one Form**is needed to produce a regular hexagon. - When only one reflection is available (
**g**or**h**or**i**or**j**or**k**or**l**of the above list) ,**three Forms**are needed to produce a regular hexagon. - When one reflection and a 6-fold rotation axis (implying 60
^{0}, 120^{0}, 180^{0}, 240^{0}, and 300^{0}rotations) are available (the other five reflections are then implied),**one Form**is needed to produce a regular hexagon. - When only a 3-fold rotation axis (implying 120 and 240
^{0}rotations) is available (**c**and**e**of the above list),**two Forms**are needed to produce a regular hexagon. - When one reflection and a 3-fold rotation axis (implying 120 and 240
^{0}rotations) are available (**g, h, i, j, k**or**l**, and**c**and**e**of the above list), only**one Form**is needed to produce a regular hexagon. - When only a 2-fold rotation axis is available (
**d**of the above list),**three Forms**are needed to produce a regular hexagon. - When a reflection and a 2-fold rotation axis are available (
**g, h, i, j, k**or**l**, and**d**of the above list),**two Forms**are needed to produce a regular hexagon.

Plane Group |
Point Symmetry |
Number of Forms |

P1 |
C_{1} ( 1 ) |
6 |

P2 |
C_{2} ( 2 ) |
3 |

PmCm Pg |
D_{1} ( m ) |
3 |

P2mmC2mm P2mg P2gg |
D_{2} ( 2mm ) |
2 |

P3 |
C_{3} ( 3 ) |
2 |

P3m1P31m |
D_{3} ( 3m ) |
1 |

P6 |
C_{6} ( 6 ) |
1 |

P6mm |
D_{6} ( 6mm ) |
1 |

For each of these plane groups we will show how it supports a

A regular hexagon, which has a point symmetry according to the group D

Figure above **:** The symmetries (indicated by symmetry elements) of the Regular Hexagon **:** Six mirror lines and a six-fold rotation axis at their point of intersection.

Figure above **:** A pattern according to the plane group **P6mm** .

Each rhombic lattice mesh contains two **D _{3}** motifs s.str. that are rotated 60

On the periodic pattern of the above Figure a

Figure above **:** A two-dimensional hexagonal crystal with point symmetry **D _{6}** based on a hexagonal lattice, i.e. on a periodic stacking of 60/120

Figure above **:** Same as previous Figure. Lattice connection lines (which do not belong to the pattern) erased.

The next Figure shows the D

Figure above **:** Symmetry elements of the hexagonal crystal of the previous Figures **:** Six mirror lines and one 6-fold rotation axis at their point of intersection. And this pattern of symmetry elements is the fingerprint of D_{6} symmetry. In the present case the intrinsic point symmetry of the crystal is the same as that of its intrinsic shape (regular hexagon).

Figure above **:** A pattern representing the plane group **P6** . It has its three-fold motifs s.str. at the mid-points of the equilateral triangles that make up the rhombic unit meshes of the hexagonal point lattice. A lattice mesh is indicated (green).

Hexagonal crystals can be supported by this (P6) pattern, as the next Figure shows.

Figure above **:** A two dimensional hexagonal crystal, supported by a P6 pattern. It can be conceived as consisting of a periodic stacking of 60/120^{0} rhombic building blocks. The intrinsic point symmetry of the crystal is according to the group **C _{6}** . This is shown in the next two Figures, where the first one depicts, to begin with, the pattern proper, i.e. without the lattice connection lines.

Figure above **:** The above two-dimensional C_{6} crystal with hexagonal shape. Lattice connection lines omitted (they do not belong to the pattern).

Figure above **:** A 6-fold rotation axis as the only symmetry element of the above two-dimensional C_{6} crystal with hexagonal shape. Its point symmetry is therefore C_{6} . As also all the crystals below, the present crystal (thus) has a lower intrinsic symmetry than that of its (intrinsic) shape (which is a regular hexagon and has D_{6} symmetry).

Figure above **:** A two-dimensional hexagonal crystal supported by a P3m1 pattern. The point symmetry of this crystal is **D _{3}** . The crystal results from a periodic stacking of 60/120

Figure above **:** Same as previous Figure. Lattice connection lines removed.

Figure above **:** Symmetry elements (three mirror lines and one 3-fold rotation axis at their point of intersection) of the above depicted two-dimensional hexagonal crystal based on a P3m1 pattern. The point symmetry of the crystal is accordingly D_{3} .

Figure above **:** A two-dimensional hexagonal crystal supported by a P31m pattern. The point symmetry of this crystal is **D _{3}** . The crystal results from a periodic stacking of 60/120

Figure above **:** Same as previous Figure. Lattice connection lines removed.

Figure above **:** Symmetry elements (three mirror lines and one 3-fold rotation axis at their point of intersection) of the above depicted two-dimensional hexagonal crystal based on a P31m pattern. The point symmetry of the crystal is accordingly D_{3} .

Figure above **:** A two-dimensional hexagonal crystal supported by a P3 pattern. The point symmetry of this crystal is **C _{3}** . The crystal results from a periodic stacking of 60/120

Figure above **:** Same as previous Figure. Lattice connection lines removed.

Figure above **:** Symmetry elements (one 3-fold rotation axis only) of the above depicted two-dimensional hexagonal crystal based on a P3 pattern. The point symmetry of the crystal is accordingly C_{3} .

Figure above **:** Slightly smaller version of the above crystal (hexagonal, C_{3}), for convenient overview of its hexagonal shape.

Figure above **:** Two-dimensional hexagonal crystal, i.e. a crystal having as its intrinsic shape a regular hexagon, consisting of a stacking of special rectangles, viz. rectangles such that their diagonals involve angles of 60^{0}. The lattice plus D_{2} motifs is a periodic pattern with a symmetry according to the plane group P2mm. The crystal therefore, despite its regularly-hexagonal intrinsic shape, has **D _{2}** (point) symmetry (and not D

Figure above **:** The crystal's six-sided shape obtained by periodic stacking of special rectangles (having diagonals involving 60^{0} angles) has corners of 120^{0}. The next Figure shows that also the six sides are equal in length.

Figure above **:** Because of the fact that the relevant angles are all 60^{0}, the polygon (with corners of 120^{0}) consists of six congruent equilateral triangles, proving it to be a regular hexagon.

Figure above **:** As in the three previous Figures. A two-dimensional crystal, having a regular hexagon as its intrinsic shape, obtained by periodic stacking of special rectangles (having diagonals involving 60^{0} angles). Larger version.

Figure above **:** Same as previous Figure. Also here the corners are 120^{0} each.

Also this larger version (with corners of 120

Figure above **:** Same as previous Figure. Also here the six-sided polygon is a regular hexagon.

The next Figures show the

Figure above **:** Construction of a **P2mg** pattern based on special rectangles, namely rectangles such that their diagonals involve angles of 60^{0} . Some glide lines ( **g** ) indicated. See also next Figure.

Figure above **:** Result of the construction of a **P2mg** pattern based on special rectangles, namely rectangles such that their diagonals involve angles of 60^{0} (As always, the lattice connection lines -- here the lines delineating the rectangles -- do not belong to the pattern).

The next Figure demonstrates the P2mg nature of the pattern. Some glide lines (

From the just constructed P2mg pattern we derive a two-dimensional crystal with point symmetry D

Figure above **:** A two-dimensional intrinsically hexagonal crystal based on a P2mg pattern. The lattice consists of special rectangles, viz. rectangles such that their diagonals involve angles of 60^{0}.

The next two Figures indicate the

The two Figures above **:** Symmetry elements -- two mirror lines ( **m _{1}, m_{2}** ) and a two-fold rotation axis -- of the just discussed two-dimensional intrinsically hexagonal crystal, based on a P2mg pattern. The set of these symmetry elements determines the point symmetry of the crystal to be according to the group

The next two Figures show the construction of a P2gg pattern based on a stacking of special rectangles, viz. rectangles such that their diagonals involve angles of 60^{0} .

The two Figures above **:** Construction and its result of a **P2gg** pattern consisting of a stacking of (filled-in) special rectangles, viz. rectangles of which the diagonals involve angles of 60^{0} . Some glide lines ( **g** ) indicated. See also next Figure.

Figure above **:** The pattern is clearly a P2gg pattern **:** Some glide lines and some 2-fold rotation axes are indicated (Recall that a periodic pattern, as depicted here, must always be imagined to extend indefinitely over the plane).

From the just constructed P2gg pattern we will derive hexagonal crystals, i.e. crystals based on the periodic stacking of special rectangles filled in with motifs (as described above), resulting in P2gg plane group symmetry and D

Figure above **:** Intrinsically hexagonal two-dimensional crystal based on a P2gg pattern. Its point symmetry is according to the group **D _{2}** .

The next Figures are about the

Figure above **:** Intrinsically hexagonal two-dimensional crystal based on a P2gg pattern. Its symmetry elements are indicated **:** Two mirror lines (red) perpendicular to each other, and a two-fold rotation axis (indicated by a small yellow circle) at the intersection point of the mirror lines. The point symmetry of the crystal is therefore according to the group D_{2} . In fact the solid lines fail to express mirror lines of the crystal. But this is only seemingly so **:** First of all they must be seen as the translation-free residues of glide reflections, because the translations (shifts) involved in the glide reflections are macroscopically undetectable, changing in this way the glide line into a mirror line at the macroscopical level. Second, the relevant glide lines (indicated by dashed lines) are somewhat displaced with respect to the relevant crystal corners, but this displacement also is not detectable macroscopically [Realize that -- as in real (3-dimensional) crystals -- the number of building blocks making up our crystal must conceived to be huge].

Figure above **:** Same as previous Figure. Smaller version of crystal. Also here the displacements of the glide lines (dashed lines) -- which macroscopically will present themselves as mirror lines -- are macroscopically undetectable.

Figure above **:** Construction of a 60/120^{0} rhombic unit mesh (with compatible motif) from the rectangle used in the Figures above, viz. a rectangle the diagonals of which involve angles of 60^{0}. This unit mesh can be the building block for an intrinsically hexagon-shaped crystal based on the plane group C2mm. See next Figures.

Figure above **:** C2mm pattern based on 60/120^{0} rhombic building blocks as constructed in the previous Figure.

Figure above **:** Construction of a hexagon-shaped crystal based on the C2mm pattern of the previous Figure.

The next three Figures show the result of this construction.

Figure above **:** Result of the construction of a hexagon-shaped crystal based on the C2mm pattern as given above. In fact, two edges are not smooth. See next two Figures.

Figure above **:** A hexagon-shaped crystal based on the C2mm pattern as given above, and formed by the periodic stacking of (whole) 60/120^{0} rhombic building blocks (as constructed above). Two edges are not smooth.

Figure above **:** A hexagon-shaped crystal based on the C2mm pattern as given above. Two edges are not smooth.

Including half building blocks, in order to let the hexagonal shape stand out clearly, as in the

Figure above **:** A hexagon-shaped crystal based on the C2mm pattern as given above. Lattice connection lines removed (they do not belong to the pattern).

Figure above **:** Symmetry elements (with respect to point symmetry **:** two perpendicular mirror lines and a 2-fold rotation axis -- yellow circle -- at their point of intersection) of a hexagon-shaped crystal based on the C2mm pattern as given above. The point symmetry of the crystal is clearly according to the group D_{2} .

On the basis of the plane group Pm a hexagon-shaped crystal can be formed by the periodic stacking of special rectangular building blocks (provided with a compatible motif) **:** The rectangles are such that their diagonals involve angles of 60^{0}. See next Figure.

Figure above **:** Intrinsically hexagon-shaped two-dimensional crystal, based on a **Pm** pattern with special rectangular building blocks, and D_{1} motifs.

Figure above **:** Symmetry elements (with respect to point symmetry **:** a mirror line (red) only) of the above given intrinsically hexagon-shaped two-dimensional crystal. The point symmetry of the latter is indeed clearly according to the group D_{1} .

On the basis of the plane group Pg a hexagon-shaped crystal can be formed by the periodic stacking of special rectangular building blocks (provided with a compatible motif) **:** The rectangles are such that their diagonals involve angles of 60^{0}. See next Figure.

Figure above **:** Intrinsically hexagon-shaped two-dimensional crystal, based on a **Pg** pattern with special rectangular building blocks, and D_{1} motifs as translation-free residues (i.e. after elimination of the translational component of the glide reflection -- which is equivalent to looking at the crystal from a macroscopic perspective -- we end up with a mirror line).

Figure above **:** Symmetry elements (with respect to point symmetry **:** a mirror line (red) only) of the above given intrinsically hexagon-shaped two-dimensional crystal. In fact, the glide line, which macroscopically expresses itself as a mirror line, and in this way representing the mentioned symmetry element of the crystal, is a little displaced (dashed line) with respect to the corners of the crystal, but this displacement is macroscopically undetectable. So the point symmetry of the latter is indeed clearly according to the group D_{1} .

On the basis of the plane group Cm a hexagon-shaped crystal can be formed by the periodic stacking of special rhombic building blocks (provided with a compatible motif) **:** The rhombi have angles of 60 and 120^{0}. The stacking of these rhombi results in a hexagonal lattice. See next Figure.

Figure above **:** Intrinsically hexagon-shaped two-dimensional crystal, based on a **Cm** pattern with 60/120^{0} rhombic building blocks, and D_{1} motifs.

Figure above **:** Symmetry elements (with respect to point symmetry **:** a mirror line (red) only) of the above given intrinsically hexagon-shaped two-dimensional crystal. The point symmetry of the latter is indeed clearly according to the group D_{1} .

An intrinsic hexagon-shaped crystal can be derived from a **P2** pattern, based on an oblique lattice, where the two independent translations happen to be perpendicular to each other, with lengths such that the diagonals of the resulting rectangular meshes involve angles of 60^{0} . See next Figure.

Figure above **:** Intrinsically hexagon-shaped two-dimensional crystal based on a **P2** pattern with oblique lattice, where the meshes happen to be rectangles with diagonals involving angles of 60^{0} .

Figure above **:** Symmetry elements (with respect to point symmetry **:** a 2-fold rotation axis [yellow ellipse] only) of the hexagonal crystal of the previous Figure. The point symmetry of this crystal is clearly according to the group **C _{2}** .

An intrinsic hexagon-shaped crystal can be derived from a **P1** pattern, based on an oblique lattice, where the two independent translations happen to be perpendicular to each other, with lengths such that the diagonals of the resulting rectangular meshes involve angles of 60^{0} . See next Figure.

Figure above **:** Intrinsically hexagon-shaped two-dimensional crystal based on a **P1** pattern with oblique lattice, where the meshes happen to be rectangles with diagonals involving angles of 60^{0} . The crystal has no symmetry elements (with respect to point symmetry), so its point symmetry is according to the Asymmetric Group **C _{1}** .

This concludes an overview of how and when intrinsically hexagon-shaped two-dimensional crystals are supported.

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