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.**

Square (Promorphs)

The **Vector Rosette of Actual Growth** of a quadratic crystal, i.e. of any two-dimensional crystal having (when fully developed) its intrinsic shape to be that of a square (which, all by itself has D_{4} symmetry), and (such a crystal) of whatever (possible) intrinsic symmetry, is then as follows (blue lines) **:**

This vector rosette, viz. a vector rosette of a square 2-D crystal, in fact consists of four lines,

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

In other cases, different numbers, i.e. different from 4, antimers can occur in square (2-D) crystals (dependent on the chemico-geometric features of the crystal's motifs, expressed as the equivalence or non-equivalence of the vectors

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

The D_{4} symmetry of the square crystal (as a first case of possible intrinsic point symmetry of a *square* two-dimensional crystal) is indicated in the next Figure by the pattern of **symmetry elements** [rotation axis (green), reflection lines (red)].

The square shape of the D

A promorph with D

Figure above **:** A two-dimensional intrinsically quadratic crystal with intrinsic D_{4} symmetry with its four antimers (green, yellow).

And with the Vector Rosette of Actual Growth added

As before, in the drawings the

In order to illustrate __what__, in the present case, makes the four vectors of the vector rosette of our square crystal equivalent, we insert the following motif compatible with the intrinsic symmetry of the crystal **:**

The next Figure is the same as was depicted

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

The next Figure depicts and assesses the **promorph** of our square crystal, which (crystal) has, in the present case, a D_{4} intrinsic point symmetry. This promorph turns out to be the 2-D (i.e. two-dimensional) analogue of a quadratic pyramid (i.e. a regular 4-fold pyramid, or pyramid with a square as its base), which has four antimers (green, yellow) [ The group describing the symmetry of our 2-D promorph (a square) is isomorphic with the group describing the symmetry of the quadratic pyramid (which is a 3-D promorph)].

The intrinsic point symmetry of an intrinsically square (quadratic) crystal can also be according to the group

Also for a square crystal with point symmetry C

The next Figure depicts an example of the

Figure above **:** A two-dimensional intrinsically quadratic crystal with intrinsic C_{4} symmetry and its four antimers (green, yellow).

And with the Vector Rosette of Actual Growth added

Again, in order to (better) illustrate

The next Figure is the same as was depicted

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

The next three Figures are about the **promorph** (and its name) of our square crystal with its intrinsic C_{4} point symmetry.

Figure above **:** Base of a regular gyroid pyramid with four antimers. The 4-fold rotation axis (main axis) is perpendicular to the plane of the drawing and goes through the center of the image. The two s i g m o i d radial cross axes are shown in red. Their shape, as drawn, only symbolizes their sigmoid character.

The following Figure gives, for clarity, the

Figure above **:** Slightly oblique top view of a regular gyroid pyramid with four antimers ( *Homogyrostaura tetramera*). The group, representing the symmetry (C_{4}) of this regular four-fold gyroid pyramid is isomorphic to that representing the symmetry of its base alone, i.e. of (any) regular four-fold gyroid polygon.

Alternatively, the promorph of our square crystal with intrinsic C

Figure above **:** Alternative expression of the promorph representing the (2-D analogue of the) *Homogyrostaura tetramera* (C_{4} promorph). The geometric image, expressing the promorph has no receding angles (as they also do not occur in single crystals).

We now consider crystals, again possessing the square as their intrinsic shape, but now with intrinsic symmetry according to the point group

The next Figure depicts the

Figure above **:** Symmetry elements of a square (quadratic) crystal having D_{2} intrinsic point symmetry **:** One 2-fold rotation axis (small yellow solid ellipse) and two diagonal mirror lines ( **m** ).

To generate crystal faces, together forming a square, i.e. to generate a square D

Figure above **:** To generate a square two-dimensional crystal with point symmetry D_{2} , only one crystallographic Form (dark blue) is needed.

Natural objects with either D

See for some more explanation

In the present case we have to do with q u a d r a t i c crystals, which here means that their intrinsic s h a p e is a square. In such a square we can indicate the crystal's antimers (in the present case, four). While a square as such has not one but two sets of perpendicular cross axes, and so does not possess definite directional axes, the intrinsic symmetry is -- in the present case -- such that the promorph (of the square crystal) is expected to be an amphitect polygon (as such the two-dimensional analogue of an amphitect pyramid), and the two perpendicular mirror lines (implied by the intrinsic symmetry of the crystal), that we have drawn in the square (See

The next Figure depicts the

Figure above **:** The four antimers (two-by-two unequal) and the Vector Rosette of Actual Growth (blue lines, **a, b, c, d** ) of a two-dimensional square crystal with intrinsic D_{2} point symmetry. The two-by-two unequality of the antimers demonstrates the two-by-two non-equivalence of the (four) vectors ( **a, b, c, d** ) of the rosette. The configuration of the four antimers is r a d i a l because the mirror lines, which are perpendicular to each other and as such point to the directional axes of the promorph, run right through the antimers.

The next Figure is the same, but without the vector rosette.

Figure above **:** The four antimers (two-by-two unequal) of a two-dimensional square crystal with intrinsic D_{2} point symmetry.

The next Figure again depicts our crystal under investigation (square, D

Another possible compatible motif, showing perhaps more clearly the number and configuration of the four antimers, could be as follows

While in the above Figures we had a

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

The next two Figures, finally, depict the **promorph** of our quadratic crystal with intrinsic D_{2} point symmmetry and four antimers. It can either be the 2-D analogue of a *rhombic* pyramid (in fact it is its base) with four antimers (yellow, green) **:**

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

For an explanation of

The delineation of the four antimers can be indicated in several equivalent ways. The next Figure shows this for the radial configuration.

The next Figure illustrates the same as the last-but-one, but now for the case of the two mirror lines being non-diagonally oriented (a case, considered further below).

Also here the delineation of the four antimers can be indicated in several equivalent ways. The next Figure shows this for the radial configuration.

A two-dimensional, intrinsically square crystal, with intrinsic D_{2} symmetry, can, in some cases, possess only **two antimers** instead of four. See next Figure.

Figure above **:** A two-dimensional square crystal, having D_{2} symmetry and (only) two antimers (yellow, green).

To emphasize the crystal's D

Figure above **:** Same as previous Figure, but now with a D_{2} motif inserted, representing the crystal's point symmetry and indicating the two antimers.

The next Figure gives the

A two-dimensional, intrinsically square crystal, with intrinsic D_{2} symmetry, can, in some cases, possess **six antimers** . See next Figure.

Figure above **:** Two-dimensional square crystal with intrinsic D_{2} symmetry and six antimers (green, yellow), indicated by the numerals 1, 2, 3, 4, 5 and 6.

The next Figure is the same as the previous one, but with the

In the next Figure the

Figure above **:** Two-dimensional square crystal with intrinsic D_{2} symmetry and six antimers. The Vector Rosette of Actual growth is indicated (blue lines).

The next Figure, finally, gives

Figure above **:** Promorph of the two-dimensional square crystal considered above (D_{2} , six antimers). It is a six-fold amphitect polygon and as such the two-dimensional analogue of a six-fold amphitect pyramid.

In the foregoing we considered

In a much earlier case (Part V and VI), where we considered

The next Figure indicates the pattern of **symmetry elements** of such a crystal.

Figure above **:** Symmetry elements of a two-dimensional intrinsically square crystal with intrinsic D_{2} point symmetry **:** One 2-fold rotation axis and two mirror lines ( **m** ) intersecting at the piercing point of the rotation axis. The two mirror lines do __not__ coincide with the diagonals of the square.

Two crystallographic

Figure above **:** The two Forms (blue, red) of a two-dimensional square crystal with intrinsic D_{2} symmetry and non-diagonal mirror lines.

With such a crystal, i.e. with a square crystal having

Figure above **:** The four antimers (green, yellow) of a square D_{2} crystal with non-diagonal mirror lines. Symmetry elements indicated. All figures with D_{2} symmetry have two (and no more than two) mirror lines perpendicular to each other. And because they are perpendicular, they must correspond with the *directional axes* of the corresponding promorphs of such D_{2} figures. In the crystal of the present Figure we can see that these (directional) axes are *radial* (because they coincide with the median lines of the antimers [geen, yellow]), and so the corresponding promorph must also be *radial*.

A second possible configuration of the four antimers is the

Figure above **:** The four antimers (green, yellow) of a square D_{2} crystal with non-diagonal mirror lines. Symmetry elements indicated. The antimers correspond to the corner areas of the square, and their configuration is interradial, because the mirror lines, necessarily representing to the promorphological directional axes, are interradial ( They run between antimers).

Figure above **:** Same as previous Figure **:** The four antimers (green, yellow) of a square D_{2} crystal with non-diagonal mirror lines. Symmetry elements not drawn. Radial (R) and interradial (IR) directions indicated. Configuration of the four antimers interradial.

The next Figure indicates the

Figure above **:** Same as previous Figure **:** The four antimers (green, yellow) of a square D_{2} crystal with non-diagonal mirror lines. Vector Rosette of Actual Growth with its four vectors **a, b, c, d** indicated (dark blue lines).

The above three Figures are, however only partially adequate, because they suggest that each antimer is in itself mirror symmetric, and that they relate to each other by a 4-fold rotation axis. All this cannot be, because it is not compatible with the pattern of symmetry elements in these three Figures. In order to express the true nature of the four antimers we insert a motif

Figure above **:** Same as previous Figure. Motif (black) inserted in order to highlight the D_{2} intrinsic symmetry of the crystal (i.e. the intrinsic point symmetry of the crystal is __not__ that of its quadratic shape, which is D_{4} ). Compare with the quadratic crystal with intrinsic D_{4} symmetry, considered **above** . Compare also with the quadratic D_{2} crystal with its two mirror lines coinciding with the diagonals of the square, also considered **above** . The configuration of the four antimers is interradial.

The next Figure shows the second configuration, mentioned earlier, viz. the

Figure above **:** A two-dimensional intrinsically square crystal with intrinsic point symmetry according to the group D_{2} , and having four antimers (green, yellow). Radial configuration of antimers, indicated by an inserted motif (black), which also expresses the D_{2} intrinsic symmetry of the crystal.

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

The **promorph** of our quadratic D_{2} crystal with non-diagonal mirror lines and four antimers is the same (viz. *(Autopola Orthostaura) Tetraphragma radialia* or *interradialia* ) as that of the quadratic D_{2} crystal with diagonal mirror lines (i.e. with its only two mirror lines being diagonal) and four antimers, as considered **above** .

A quadratic D

Figure above **:** The (case of having only) two antimers (green, yellow) of a quadratic D_{2} crystal with its two mirror lines non-diagonally positioned.

And with the Vector Rosette of Actual Growth added

Figure above **:** The (case of having only) two antimers (green, yellow) of a quadratic D_{2} crystal with its two mirror lines non-diagonally positioned. Vector Rosette of Actual Growth added. All vectors, **a, b, c, d** are equivalent.

The

We now consider the case of a quadratic D

Figure above **:** The six antimers (green, yellow) of a quadratic D_{2} crystal with its two mirror lines non-diagonally positioned. The antimers are (also) indicated by numerals.

The next Figure adds the

Figure above **:** The six antimers (green, yellow) of a quadratic D_{2} crystal with its two mirror lines non-diagonally positioned. The antimers are (also) indicated by numerals. The Vector Rosette of Actual Growth with its four vectors **a, b, c, d** is indicated (blue lines). All four vectors are equivalent. Compare with the quadratic D_{2} crystal, also with six antimers, but with its mirror lines in a diagonal position, considered **above** .

The

We now consider crystals, again possessing the square as their intrinsic shape, but now with intrinsic symmetry according to the point group

Again, we have two possibilities. The position of the mirror line could either be diagonal or non-diagonal with respect to the quadratic crystal. We begin with the first case (mirror line diagonal).

The next Figure depicts the

Figure above **:** The only symmetry element (with respect to point symmetry) -- a mirror line ( **m** ) -- of a square crystal with intrinsic **D _{1}** point symmetry.

To generate the whole crystal conceptually, two crystallographic

A crystal with intrinsic D

Figure above **:** Two-dimensional, and intrinsically square crystal, with point symmetry D_{1} and two antimers (green, yellow). A motif (black) is added to indicate that there is only one mirror line present.

The next Figure is the same as the previous one, but now provided with the crystal's

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

The next Figure gives and names the **promorph** of our intrinsically quadratic D_{1} two-dimensional crystal with its mirror line (as an element of its point group) coinciding with a diagonal of the square, and with two antimers.

We will now consider the case of a quadratic D

Figure above **:** Two-dimensional quadratic D_{1} crystal, with four antimers. The inserted motif (black) clearly shows which lines are going to represent the *directional axes* of the promorph **:** One of them is the mirror line (coincident with the NE diagonal, see **Figure above** ), the other is perpendicular to it (and is not a mirror line). And we see that these lines run *between* antimers, so the configuration of the antimers is *interradial*.

Figure above **:** Same as previous Figure. Vector Rosette of Actual growth with its four vectors **a, b, c, d** added.

The four antimers allow for two types of configuration. One type we have just depicted, viz. the interradial configuration. It promorphologically corresponds to the

Figure above **:** Two-dimensional quadratic D_{1} crystal, with four antimers. Alternative configuration (viz. radial) of the four antimers. Also here the inserted motif expresses the intrinsic symmetry and shows where the lines that are going to represent the directional axes of the promorph are to be found **:** NE diagonal mirror line and the line perpendicular to it, and they are radial.

Figure above **:** Same as previous Figure. Vector Rosette of Actual growth with its four vectors **a, b, c, d** added.

As has been noted earlier, the motif (black) in the above figures does not as such occur in a crystal. It here only serves to indicate the intrinsic D

The next Figures give and name the **promorphs** respectively possessed by the two types (radial, interradial) of quadratic D_{1} crystal with diagonal mirror line (One can see that whether the mirror line is diagonal or non-diagonal does not make a difference in promorph) **:**

Figure above **:** Promorph of the quadratic D_{1} crystal with four antimers. It is an isosceles trapezium, and as such a two-dimensional analogue of the trapezoid pyramid (i.e. it is its base). The four antimers (green, yellow) and the radial and interradial directions are indicated. The configuration of the antimers is interradial.

Figure above **:** Promorph of the quadratic D_{1} crystal with four antimers. Alternative configuration (viz. radial) of the four antimers. This promorph is a bi-isosceles triangle, and as such the two-dimensional analogue of the bi-isosceles pyramid (i.e. it is its base). The four antimers (green, yellow) and the radial and interradial directions are indicated.

As we did already in Part VI (concerning rectangular crystals), 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 ordinary 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.

A two-dimensional quadratic crystal (i.e. a crystal of which the intrinsic shape is a square), with intrinsic D

Figure above **:** A two-dimensional quadratic D_{1} crystal with six antimers (green, yellow). The six antimers are indicated by numerals.

Figure above **:** Same as previous Figure **:** a two-dimensional quadratic D_{1} crystal with six antimers (green, yellow). The Vector Rosette of Actual Growth, with its four vectors **a, b, c, d** indicated (blue lines).

The next Figure gives (with two images) the

The next Figure is the same, but with the median lines of the antimers omitted, in order to let the latter come out more clearly.

A two-dimensional quadratic crystal, with intrinsic D

Figure above **:** A two-dimensional quadratic D_{1} crystal with three antimers (green, yellow). The three antimers are indicated by numerals.

Figure above **:** Same as previous Figure **:** a two-dimensional quadratic D_{1} crystal with three antimers (green, yellow). The Vector Rosette of Actual Growth, with its four vectors **a, b, c, d** indicated (blue lines).

The next Figure gives the

The next Figure is the same, but with the median lines of the antimers omitted, in order to let the latter come out more clearly.

We now consider the

Figure above **:** The only symmetry element of a two-dimensional D_{1} quadratic crystal **:** a mirror line ( **m** ). The latter is, in the present case, non-diagonal. It connects the centers of two opposite sides of the square.

Three crystallographic

Figure above **:** Three Forms (blue, red, green) are needed to construct a quadratic D_{1} two-dimensional crystal, if the mirror line is non-diagonal **:**

An initial crystal face (blue) implies a second face (blue) in virtue of the mirror line. No further faces will be implied. The result is one Form consisting of two parallel faces (blue). It is an open Form, and cannot therefore represent a crystal. A second initial face (red), perpendicular to the mirror line does not imply new faces. So this is a second Form consisting of one face only (red). These two Forms combined are still an open Form and cannot represent a crystal. A third initial face (green) parallel to the second and (also) perpendicular to the mirror line also does not imply new faces. So it is a Form -- a third Form -- consisting of one face only (green). The three Forms together constitute a closed structure, and the latter is a square and as such represents our crystal.

A two-dimensional quadratic crystal, with intrinsic D

Figure above **:** A two-dimensional quadratic D_{1} crystal with its mirror line in a non-diagonal position and possessing two antimers (green, yellow) indicated by numerals.

Figure above **:** Same as previous Figure **:** a two-dimensional quadratic D_{1} crystal with its mirror line in a non-diagonal position and possessing two antimers (green, yellow) indicated by numerals. Included is the crystal's Vector Rosette of Actual Growth, with its four vectors **a, b, c, d** .

In order to highlight the asymmetry of the antimers, i.e. in order to highlight the absence of a horizontal mirror line in addition to the vertical one, and at the same time to demonstrate the equivalence and non-equivalence of the vectors of the Rosette, we insert a motif (black), first without Vector Rosette

And with the Vector Rosette of Actual Growth added

Figure above **:** Same as earlier Figure (crystal with vector rosette, but without inserted motif) **:** a two-dimensional quadratic D_{1} crystal with its mirror line in a non-diagonal position and possessing two antimers (green, yellow). A motif (black) is inserted. It demonstrates the equivalence of the vectors **a** and **b** , and of **c** and **d** , and the non-equivalence of **a** and **c** , of **b** and **d** , of **a** and **d** and, finally, of **b** and **c** .

The

A two-dimensional quadratic crystal, with intrinsic D

Figure above **:** A two-dimensional quadratic D_{1} crystal with its mirror line in a non-diagonal position and possessing four antimers (green, yellow) indicated by numerals.

Figure above **:** Same as previous Figure **:** a two-dimensional quadratic D_{1} crystal with its mirror line in a non-diagonal position and possessing four antimers (green, yellow) indicated by numerals. Included is the crystal's Vector Rosette of Actual Growth, with its four vectors **a, b, c, d** , where **a** and **b** , as well as **c** and **d** are equivalent, while **a** and **c** , as well as **b** and **d** and also **a** and **d** as well as **b** and **c** are non-equivalent.

The four antimers allow for two types of configuration. One of them is just depicted, the other is depicted in the next Figure

Figure above **:** A two-dimensional quadratic D_{1} crystal with its mirror line in a non-diagonal position and possessing four antimers (green, yellow) indicated by numerals. Alternative configuration of the four antimers.

The

A two-dimensional quadratic crystal, with intrinsic D

Figure above **:** A two-dimensional quadratic D_{1} crystal with its mirror line in a non-diagonal position and possessing five antimers (green, yellow) indicated by numerals. Only the members of the left-right pairs of vectors of the Vector Rosette of Actual Growth (not drawn) are equivalent.

The next Figures give and name the

Figure above **:** Promorph of the two-dimensional quadratic D_{1} crystal with five antimers. It is half a 10-fold amphitect polygon, and as such a two-dimensional analogue of half a 10-fold amphitect pyramid (i.e. it is its base). In the right image the five antimers (green, yellow) and the radial and interradial directions are indicated.

Figure above **:** Same as previous Figure **:** promorph of the two-dimensional quadratic D_{1} crystal with five antimers. Median lines of the antimers omitted in order for the latter to come out more clearly.

A two-dimensional quadratic crystal, with intrinsic D

Figure above **:** A two-dimensional quadratic D_{1} crystal with its mirror line in a non-diagonal position and possessing six antimers (green, yellow) indicated by numerals.

Figure above **:** Same as previous Figure **:** a two-dimensional quadratic D_{1} crystal with its mirror line in a non-diagonal position and possessing six antimers (green, yellow) indicated by numerals. The Vector Rosette of Actual Growth, with its four vectors **a, b, c, d** is indicated (blue lines), where **a** and **b** , as well as **c** and **d** are equivalent, while **a** and **c** , as well as **b** and **d** and also **a** and **d** as well as **b** and **c** are non-equivalent.

The

A two-dimensional quadratic crystal, with intrinsic D

Figure above **:** A two-dimensional quadratic D_{1} crystal with its mirror line in a non-diagonal position and possessing three antimers (green, yellow) indicated by numerals.

Figure above **:** Same as previous Figure **:** a two-dimensional quadratic D_{1} crystal with its mirror line in a non-diagonal position and possessing three antimers (green, yellow) indicated by numerals. The Vector Rosette of Actual Growth, with its four vectors **a, b, c, d** is indicated (blue lines), where **a** and **b** , as well as **c** and **d** are equivalent, while **a** and **c** , as well as **b** and **d** and also **a** and **d** as well as **b** and **c** are non-equivalent.

The

A two-dimensional crystal having the square as its intrinsic shape (like in the previous cases), can have an intrinsic point symmetry according to the cyclic group

The next Figure indicates the only **symmetry element** (with respect to point symmetry) of such a crystal.

Figure above **:** A two-dimensional quadratic C_{2} crystal. Its only symmetry element is a two-fold rotation axis (indicated as a small solid green ellipse) going through the crystal's center (and being, of course, perpendicular to the plane of the drawing).

Figure above **:** A two-dimensional quadratic C_{2} crystal. Two crystallographic Forms are needed to construct the square **:** An intial crystal face (blue) implies another face in virtue of the two-fold rotation axis, resulting in a Form consisting of two parallel faces (blue). Another initial face (red), perpendicular to the first one, implies a second face, also resulting in a Form consisting of two parallel faces (red). Combining these two Forms yields the quadratic crystal consisting of four crystal faces.

Quadratic crystals can occur if the (two-fold) motif s.str. plus its corresponding surroundings (together forming the motif s.l.) are stacked according to a quadratic net, which, in the present case, is in fact an oblique net where the angles happen to be 90^{0} and where the two translations happen to be equal in length. See for such a stacking the **relevant Figure in Part VII** .

The next three Figures depict a two-dimensional intrinsically quadratic crystal with intrinsic symmetry according to the cyclic group C

Figure above **:** A two-dimensional quadratic C_{2} crystal with two antimers (green, yellow). A motif (black) is inserted in order to indicate the exclusive 2-fold rotational (point) symmetry of the crystal, and therefore the absence of mirror lines. Radial (R) and interradial (IR) directions are indicated.

Figure above **:** Same as previous Figure **:** a two-dimensional quadratic C_{2} crystal with two antimers (green, yellow). The Vector Rosette of Actual Growth, with its four vectors **a, b, c, d** is indicated (blue lines), where **a** is equivalent to **d** , and **b** to **c** .

The next Figure shows an alternative orientation of the two antimers within the quadratic crystal.

Figure above **:** Same as previous Figure. Alternative orientation of the two antimers.

It is possible to draw the above quadratic C

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

The next Figure gives and names the

Figure above **:** Promorph of the quadratic C_{2} crystal with two antimers. It is a two-fold amphitect gyroid polygon, and as such a two-dimensional analogue of the corresponding three-dimensional two-fold amphitect gyroid pyramid, which is depicted in the next Figure.

Figure above **:** Three-dimensional analogue of the promorph of the quadratic two-dimensional C_{2} crystal with two antimers **:** slightly oblique top-view of a two-fold amphitect gyroid pyramid.

As has been said, a crystal with C

The next two Figures depict a two-dimensional intrinsically quadratic crystal with intrinsic symmetry according to the cyclic group C

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

Figure above **:** Same as previous Figure **:** a two-dimensional quadratic C_{2} crystal with eight antimers (green, yellow). The Vector Rosette of Actual Growth, with its four vectors **a, b, c, d** is indicated (blue lines). Only opposite vectors are equivalent.

The next Figure gives and names the

Figure above **:** Promorph of the above square C_{2} crystal with eight antimers. It is an 8-fold amphitect gyroid polygon and as such the two-dimensional analogue of an 8-fold amphitect gyroid pyramid (namely its base). The eight antimers (green, yellow) are indicated.

The promorphs of C_{2} crystals were represented here by polygons (pyramids) with *receding* angles. They can equally be represented by polygons (pyramids) not involving such angles, as is the case in all single crystals. But because in organisms receding angles are allowed, and because they more clearly indicate the gyroid nature of those promorphs, we prefer such angles to figure in the geometric structures representing such promorphs.

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

Figure above **:** Crystals with point symmetry C_{1} do not have symmetry elements, apart from the trivial 1-fold rotation axis.

Figure above **:** For a two-dimensional quadratic C_{1} crystal to be conceptually constructed, four crystallographic Forms are needed (blue, red, green, black).

In the next Figure an asymmetric motif (black) is added to indicate the asymmetry of the crystal, i.e. to indicate that the crystal's point symmetry is according to the group C

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

The next Figure (two images) gives the possible **promorphs** (and their names) of such a quadratic 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).

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