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

Let us consider a fully developed two-dimensional crystal having as its intrinsic shape the **equilateral triangle** **:**

A regularly-triangular crystal could, at first sight, be conceived of as being built up by a stacking of microscopic regularly-triangular units, as the next Figure illustrates.

However, the plane group underlying this triangular tiling of the plane is the plane group P6mm

Because the 6-fold rotation axis, and three of the six mirror lines, implied by this plane group (P6mm) and as such belonging to its point symmetry, do not occur among the symmetries of the Equilateral Triangle as such, a triangular tiling of the plane, as given above, cannot support regularly-triangular crystal shape. And moreover, the small triangles, making up the large triangle, are not genuine building blocks of crystals at all, because their stacking is not periodic. So we will not encounter triangular building blocks in crystals at all.

A regularly-triangular crystal can, however, be conceived as being built up by the

But a regularly-triangular crystal can, in appropriate cases (in fact they are equivalent), also be accomplished by the

These rectangles, however, are directly related to the above mentioned rhombi (viz. those with 60 and 120 degrees angles) by their diagonals, as the next Figure shows.

The next Figure analyses this relation of the rectangles to the rhombi formed by their diagonals.

Figure above **:** The rhombi formed by the diagonals of the rectangles are rhombi with 60 and 120 degrees angles, and therefore appropriate building blocks for a regularly-triangular crystal. To see this, consider the angles **a, b, c, d** and **e** . First of all **a** is 60^{0} (in virtue of the special dimensions of the rectangle). **b** is therefore 30^{0}. But then **c** is also 30^{0}. It then follows that **d** is 120^{0} and from this follows that **e** is 60^{0}.

The next two Figures show the

Figure above **:** Two-dimensional intrinsically regularly-triangular crystal built up by the periodic stacking of rhomb-shaped (empty) building blocks. The rhombi possess 60 and 120 degrees angles. As such these rhombi together constitute a **hexagonal lattice.**

When motifs are inserted we will need larger rhombi (but with the same angles) to represent (filled-in) unit meshes. See next Figure.

The true

That a **regularly-triangular crystal** (i.e. a crystal having as its intrinsic *shape* the equilateral triangle) can have these intrinsic *symmetries* can be explained succinctly as follows (and will be further evident in the sequel) **:**

An **Equilateral Triangle** as such has the following (point) symmetries, i.e. it will be superposed upon itself by the following transformations (which are therefore symmetry transformations) **:**

- 0
^{0}or 360^{0}rotation about any axis. - 120
^{0}rotation about the center of the triangle. - 240
^{0}rotation about the center of the triangle. - Reflection in the bisector
**m**(_{1}**Figure below**). - Reflection in the bisector
**m**._{2} - Reflection in the bisector
**m**._{3}

The regularly-triangular crystal originates by development of crystal faces. These faces will appear in accordance with the available symmetry, i.e. according to the set of symmetry transformations available in each case. And this set is dependent on the symmetry of the motif. An initially given face will imply copies of itself according to this set of available symmetry transformations. The configuration of faces so obtained then constitutes a **Form** (in the crystallographic sense). So it is clear that **:**

- When no symmetry is available (
**a**of the above list)**three Forms**are needed to produce a regular triangle (equilateral triangle, regular trigon). - When only a 3-fold rotation axis is available (
**b**and**c**of the above list)**one Form**is needed to produce a regular triangle. - When only one reflection is available (
**d**or**e**or**f**of the above list)**two Forms**are needed to produce a regular triangle. - When one reflection and a 3-fold rotation axis (implying 120
^{0}or 240^{0}rotation) is available (the other two reflections are then implied)**one Form**is needed to produce a regular triangle.

Plane Group |
Point Symmetry |
Number of Forms |

P1 |
C_{1} ( 1 ) |
3 |

PmCm Pg |
D_{1} ( m ) |
2 |

P3 |
C_{3} ( 3 ) |
1 |

P3m1 |
D_{3} ( 3m ) |
1 |

* The plane group **P31m** does not support regularly-triangular crystals, as we will see in the sequel.

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

An equilateral triangle, which has a point symmetry according to the group D

Figure above **:** The symmetries (indicated by symmetry elements) of the Equilateral Triangle **:** Three mirror lines and a three-fold rotation axis at their point of intersection.

Figure above **:** Two-dimensional regularly-triangular crystal, with point symmetry **D _{3}** , and supported by a

For convenient overview of shape we give yet a smaller version of the same crystal type

Figure above **:** Two-dimensional regularly-triangular crystal, with point symmetry **D _{3}** , and supported by a

The next Figures show that the point symmetries of of the plane group, viz. three mirror lines and a 3-fold rotation axis, are also symmetries of the equilateral triangle representing the shape of the crystal. The pattern of motifs s.str. is extended a little bit in order to let the symmetries be evident (Recall that the microscopic structural pattern of motifs s.str. of a (two-dimensional) crystal must be imagined to extend indefinitely over the plane). Also recall that the lattice connection lines do not as such belong to the pattern.

In the above Figures we had

Figure above **:** Two-dimensional regularly-triangular crystal (blue), with point symmetry **D _{3}** , and supported by a

Here there is a snag **:** One would perhaps be tempted to let the plane group P31m support a regularly-triangular crystal shape according to the next two Figures **:**

Figure above **:** Alleged two-dimensional regularly-triangular crystal, with point symmetry **D _{3}** , and supported by a

For convenient overview of shape we give yet a smaller version of the same crystal type

Figure above **:** Alleged two-dimensional regularly-triangular crystal, with point symmetry **D _{3}** , and supported by a

The next Figure emphasizes the alleged possibility to form a regularly-triangular crystal on the basis of our plane group P31m.

But the relevant internal symmetry elements do not align with those of the equilateral triangle allegedly representing the shape of the crystal, and so do not coincide with them. See next Figures.

Figure above **:** The plane group pattern has mirror lines in three directions. No one of them coincides with a mirror line of the equilateral triangle allegedly representing the shape of the crystal.

Figure above **:** In the direction shown there is no mirror line of the plane group pattern that coincides with a mirror line of the equilateral triangle allegedly representing the shape of the crystal.

Figure above **:** Relevant (i.e. relevant with respect to the equilateral triangle allegedly representing the shape of the crystal) mirror lines of all three directions of the plane group pattern (plane group P31m). No one of them coincides with one of the mirror lines of the mentioned equilateral triangle. The center of the latter is highlighted. As one can see there is a 3-fold rotation axis of the periodic P31m pattern that does coincide with the 3-fold rotation axis of the (mentioned) equilateral triangle, but there are no three mirror lines of the periodic pattern such that their point of intersection coincides with the location of this axis. The symmetry of the triangular crystal, i.e. its intrinsic symmetry, is, consequently, according to the Cyclic Group **C _{3}** instead of the Dihedral Group D

The next two Figures depict our regularly-triangular crystal without the lattice connection lines (which do not belong to the internal pattern as such). One can see that the intrinsic symmetry of this crystal is not D

Figure above **:** Two-dimensional crystal with the intrinsic shape of an equilateral triangle as it was discussed above, drawn without lattice connection lines.

Figure above **:** Same as previous Figure. The 3-fold rotation axis as the only symmetry element of the triangular crystal, is indicated. The blue lines are __not__ mirror lines of the crystal.

So the plane group

One could wonder whether this plane group can support crystals at all. But of course it does.

Crystals with intrinsic

Figure above **:** Two-dimensional crystal with intrinsic (regularly) *hexagonal* shape, supported by the plane group P31m. The connection lines of the hexagonal point lattice not drawn. The next Figure shows its intrinsic symmetry.

Figure above **:** Two-dimensional crystal with intrinsic (regularly) *hexagonal* shape, supported by the plane group P31m. The connection lines of the hexagonal point lattice not drawn. While the hexagon -- as being the shape of the crystal -- itself has D_{6} symmetry, the intrinsic symmetry of the crystal is lower, namely according to the group **D _{3}**

Figure above **:** Two-dimensional regularly-triangular crystal, with point symmetry **C _{3}** , and supported by a

The next two Figures show the symmetry -- C

Figure above **:** The two-dimensional regularly-triangular crystal of the previous Figure, with point symmetry **C _{3}** , and supported by a

Figure above **:** Same as previous Figure. Lattice connection lines removed (they do not belong to the pattern), and the 3-fold rotation axis (as the only symmetry element of the crystal) indicated. The intrinsic symmetry of the triangular crystal is clearly according to the group **C _{3}** (which is implied by the supporting plane group P3 ).

The next Figure shows that the periodic stacking of rhombi is equivalent to a periodic stacking of corresponding rectangles (as was discussed above).

A regularly-triangular two-dimensional crystal can (also) be built by the periodic stacking of special rectangles, viz. rectangles where the diagonals involve angles of 60

Figure above **:** Rectangular building blocks of special dimensions, namely such that their diagonals involve angles of 60^{0}, can be periodically stacked in such a way that possible crystal faces form an equilateral triangle. The next Figures provide these rectangular building blocks with appropriate motifs, resulting in the plane groups **Pm** (point group D_{1}) and **Pg** (point group D_{1}), both based on a primitive rectangular lattice. The latter is thus a *special* primitive rectangular lattice in virtue of the angles mentioned, and as such it in fact turns into a **hexagonal lattice.**

Figure above **:** Two-dimensional regularly-triangular crystal, with point symmetry **D _{1}** , and supported by a

For convenient overview of shape we give yet a smaller version of the same crystal type

Figure above **:** Two-dimensional regularly-triangular crystal, with point symmetry **D _{1}** , and supported by a

The next Figure shows the intrinsic symmetry -- D

Figure above **:** The (larger version of the) two-dimensional regularly-triangular crystal considered above, with point symmetry **D _{1}** . Its only symmetry element -- a mirror line (

In the rectangular point lattice of the above crystals we can inscribe a rhombic net (by drawing connection lines coinciding with the diagonals of the rectangles). But this rhombic net as drawn cannot be a lattice that underlies the mode of repetition of the motifs of the crystal, because it turns out that we then have

Figure above **:** Two types of filled-in rhombic meshes of the above discussed triangular crystal, meaning that the rhombus as drawn does not contain all the information of the pattern, i.e. it is too small for it to be a unit mesh .

In order to let a unit rhombus to contain all the (morphological) information of the motif we must choose a larger rhombus, as the next Figure shows.

Figure above **:** A larger rhombus must be taken to represent a unit mesh (a filled-in building block) of the above triangular crystal. See also next Figure.

Figure above **:** The larger rhombus, as established in the previous Figure, is indicated to represent a unit mesh (a filled-in building block) of the above triangular crystal. See also next Figure.

Figure above **:** Two-dimensional regularly-triangular crystal with D_{1} intrinsic symmetry (plane group Pm) based on the periodic stacking of filled-in rhombi (with 60 and 120 degrees angles). As such these rhombi directly represent the hexagonal point lattice of the crystal.

In order to get crystals with plane group symmetry **Pg** (and, consequently, with point group symmetry D_{1}), and having their intrinsic shape to be that of the equilateral triangle, we start from the (special) rectangular building block of the previous Figures provided with a D_{1} motif consisting of two commas. By pulling the commas apart along the former mirror line ( **m** ) by an appropriate distance, we get a unit mesh still with two commas, but which is such that when repeated vertically a sequence of commas is obtained that relate to each other by a glide line ( **g** ). See next Figure.

Figure above **:** Construction of a **Pg** pattern from a **Pm** pattern involving rectangular building blocks that can support a regularly-triangular shape. Repetition of this building block in the horizontal direction as well (i.e. in addition to a vertical repetition), results in a **Pg** pattern supporting a regularly-triangular configuration of possible crystal faces. See next Figure.

Figure above **:** Two-dimensional regularly-triangular crystal, with point symmetry **D _{1}** , and supported by a

For convenient overview of shape we give yet a smaller version of the same crystal type

Figure above **:** Two-dimensional regularly-triangular crystal, with point symmetry **D _{1}** , and supported by a

The next Figure shows the intrinsic symmetry -- D

Figure above **:** The (larger version of the) two-dimensional regularly-triangular crystal considered above, with point symmetry **D _{1}** . Its only symmetry element -- a mirror line (

Of course, in the crystal just depicted, there is, strictly speaking, no such mirror line present. The triangular crystal is based on the plane group

As we already know, the rectangular lattice of the above Figures (triangular D

Figure above **:** Hexagonal net. As drawn it contains not one but two types of filled-in rhombic meshes of the above discussed triangular crystal, meaning that the rhombus is not exactly repeated **:** One rhombus as drawn does not contain all the information of the pattern, meaning that it is too small for it to be a (filled-in) unit mesh.
.

In order to let a unit rhombus to contain all the (morphological) information of the motif we must choose a larger rhombus, as the next Figure shows.

Figure above **:** A larger rhombus must be taken to represent a unit mesh (a filled-in building block) of the above triangular crystal. See also next Figure.

Figure above **:** Two-dimensional regularly-triangular crystal with D_{1} intrinsic symmetry (plane group Pg) based on the periodic stacking of filled-in rhombi (with 60 and 120 degrees angles). As such these rhombi directly represent the hexagonal point lattice of the crystal.

A (two-dimensional) crystal, having a total symmetry according to the plane group

Figure above **:** Two-dimensional regularly-triangular crystal, with point symmetry **D _{1}** , and supported by a

For convenient overview of shape we give yet a smaller version of the same crystal type

Figure above **:** Two-dimensional regularly-triangular crystal, with point symmetry **D _{1}** , and supported by a

The next Figure shows the

Figure above **:** Mirror line ( **m** ) of the two-dimensional regularly-triangular crystal, with point symmetry **D _{1}** , and supported by a

The above crystal can also be seen as being built up by a periodic stacking of

Figure above **:** Two-dimensional triangular D_{1} crystal (as discussed above). Instead of rhombic building blocks, also rectangular building blocks, of which the diagonals involve 60^{0} angles, can be considered to build up the crystal by periodic stacking.

A regularly-triangular shape can, finally, be supported by the plane group **P1** if the *oblique lattice* happens to be such that its two (independent) translations are perpendicular to each other and imply rectangular meshes with their diagonals making angles of 60 and 30^{0} with the sides. As we know, such a (special) net is in fact a *hexagonal point lattice*. See next Figures.

Figure above **:** Two-dimensional regularly-triangular crystal, with point symmetry **C _{1}** , and supported by an

For convenient overview of shape we give yet a smaller version of the same crystal type

Figure above **:** Two-dimensional regularly-triangular crystal, with point symmetry **C _{1}** , and supported by a

As we already know, the rectangular lattice of the above Figures (triangular C

Figure above **:** Hexagonal net. As drawn it contains not one but two types of filled-in rhombic meshes of the above discussed triangular C_{1} crystal, meaning that the rhombus is not exactly repeated **:** One rhombus as drawn does not contain all the information of the pattern, meaning that it is too small for it to be a (filled-in) unit mesh.

In order to let a unit rhombus to contain all the (morphological) information of the motif we must choose a larger rhombus, as the next Figure shows.

Figure above **:** A larger rhombus must be taken to represent a unit mesh (a filled-in building block) of the above triangular crystal. See also next Figure.

Figure above **:** Two-dimensional regularly-triangular crystal with C_{1} intrinsic symmetry (plane group P1) based on the periodic stacking of filled-in rhombi (with 60 and 120 degrees angles). As such these rhombi directly represent the hexagonal point lattice of the crystal.

As soon as we change the mode of repetition of the motifs, resulting in a different lattice, i.e. a lattice different from the hexagonal lattice, an intrinsically triangular shape (equilateral triangle) cannot be supported anymore. See next Figures.

Figure above **:** By displacing the rectangles (with their motifs) of the above Figures, a different lattice results. In the case here depicted, an *oblique lattice*. This lattice consists of parallelograms. Periodic stacking of these parallelograms cannot result in a regularly-triangular crystal shape. See next Figure.

Figure above **:** Periodic repetition of asymmetric motifs according to an *oblique point lattice* resulting in a P1 pattern (point symmetry C_{1}). A *regularly*-triangular shape cannot be accomplished by the periodic stacking of this pattern's unit parallelograms.

The plane groups

This concludes a summary, showing when and in what way, regularly-triangular two-dimensional crystals are supported.

The

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