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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.**

Summary and evaluation (of the Parts XVI -- XXVIII concerning the promorphology of complex-shaped two-dimensional crystals).

In the present document we will summarize and evaluate the results of the discussions concerning two-dimensional crystals with more complex intrinsic shapes, as we had them in the foregoing documents. These more complex (intrinsic) shapes were **:**

- Amphitect Gyroid Hexagon.
- Amphitect Gyroid Octagon.
- Amphitect Hexagon.
- Amphitect Octagon.
- Bi-isosceles Trapezium.
- Bilateral Octagon.
- Isosceles Trapezium.
- Isosceles Triangle.
- Bilateral Pentagon.
- Regular Gyroid Ditetragon.
- Regular Ditetragon.
- Regular Gyroid Ditrigon.
- Regular Gyroid Dihexagon.

**Amphitect Gyroid Hexagon :**

- Heterogyrostaura hexamera.
- Heterogyrostaura tetramera.
- Heterogyrostaura dimera.

- Heterogyrostaura octomera.
- Heterogyrostaura hexamera.
- Heterogyrostaura tetramera.
- Heterogyrostaura dimera.

- Autopola Oxystaura hexaphragma.
- Autopola Orthostaura Tetraphragma radialia.
- Autopola Orthostaura Tetraphragma interradialia.
- Autopola Orthostaura diphragma.

- Autopola Oxystaura octophragma.
- Autopola Orthostaura Tetraphragma radialia.
- Autopola Orthostaura Tetraphragma interradialia.
- Autopola Orthostaura diphragma.

- Allopola Zygopleura eudipleura.
- Allopola Amphipleura hexamphipleura.
- Allopola Zygopleura Eutetrapleura radialia.
- Allopola Zygopeura Eutetrapleura interradialia.
- Allopola Amphipleura triamphipleura.

- Allopola Zygopleura eudipleura.
- Allopola Amphipleura hexamphipleura.
- Allpola Amphipleura octamphipleura.
- Allopola Zygopleura Eutetrapleura radialia.
- Allopola Zygopleura Eutetrapleura interradialia.
- Allopola Amphipleura triamphipleura.

- Allopola Zygopleura eudipleura.
- Allopola Amphipleura hexamphipleura.
- Allopola Zygopleura Eutetrapleura radialia.
- Allopola Zygopleura Eutetrapleura interradialia.
- Allopola Amphipleura triamphipleura.

- Allopola Zygopleura eudipleura.
- Allopola Amphipleura triamphipleura.
- Allopola Amphipleura hexamphipleura.
- Allopola Zygopleura Eutetrapleura radialia.
- Allopola Zygopleura Eutetrapleura interradialia.
- Allopola Amphipleura pentamphipleura.

- Allopola Zygopleura eudipleura.
- Allopola Amphipleura pentamphipleura.
- Allopola Amphipleura triamphipleura.

- Homogyrostaura tetramera.
- Homogyrostaura ditetramera.

- Homostaura Isopola tetractinota.
- Dihomostaura Isopola tetractinota.

- Homogyrostaura trimera.
- Homogyrostaura ditrimera.

- Homogyrostaura hexamera.
- Homogyrostaura dihexamera.

We will now summarize these same results, but this time from the viewpoint of **promorph**.

Only holomorphic crystals are listed. Recall that a *holomorphic* crystal has its intrinsic symmetry the same as the symmetry of its intrinsic shape. Holomorphic crystals can come in two different varieties, viz. eupromorphic or non-eupromorphic. An *eupromorphic* crystal is such that its promorph is directly evident from its intrinsic shape.

*Promorphs* are geometric figures. And, in particular, two-dimensional promorphs are **polygons**. Of the latter, first the more general promorphological category will be given in our listing below, i.e. we first give the (more) general category of polygons to which the given particular promorph belongs, say **Heterostaura autopola**, which are amphitect polygons (i.e. flattened polygons with two mirror lines perpendicular to each other). Then, if necessary, a lower, but still general subcategory is given, say, **Autopola oxystaura** (amphitect polygons with 6, 8, 10, **. . . , ** antimers). Finally, the ultimate promorph is given, say, **Autopola Oxystaura octophragma**, which are amphitect polygons with eight antimers. The planimetric figure representing this promorph is then given.

After this, one or more examples of two-dimensional *holomorphic crystals* are given that possess this promorph. It is then stated whether this crystal is *eupromorphic* or *non-eupromorphic*, and finally that crystal's *intrinsic shape* is indicated.

Intrinsic shape

Intrinsic shape

General promorphological category

Autopola oxystaura (autopola with 6, 8, 10, ..., antimers, therefore at least three radial cross axes present, that must, consequently intersect at a c u t e angles) :

Intrinsic shape

Intrinsic shape

General promorphological category

Autopola orthostaura (autopola with 4 or 2 antimers, i.e. tetraphragma or diphragma

Orthostaura tetraphragma (orthostaura with four antimers) :

Intrinsic shape

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Intrinsic shape

Intrinsic shape

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Intrinsic shape

Intrinsic shape

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Intrinsic shape

Intrinsic shape

Intrinsic shape

The next two Figures illustrate this basic form still further by giving equivalent geometric figures all representing and expressing the same promorph

Intrinsic shape

Intrinsic shape

Intrinsic shape

Intrinsic shape

Intrinsic shape

Intrinsic shape

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Intrinsic shape

Intrinsic shape

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Intrinsic shape

Intrinsic shape

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Intrinsic shape

Allopola

Intrinsic shape

Intrinsic shape

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Intrinsic shape

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Intrinsic shape

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Intrinsic shape

Intrinsic shape

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Intrinsic shape

Intrinsic shape

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Intrinsic shape

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Intrinsic shape

Allopola

Zygopleura Eutetrapleura (symmetric zygopleura with four antimers) :

or, another, but equivalent representation (with straight interradial cross axes)

As one can see, the geometric figure representing this promorph consists of two isosceles triangles connected base to base.

Intrinsic shape

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Intrinsic shape

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Intrinsic shape

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Intrinsic shape

or, another, but equivalent representation (with straight radial cross axes)

Intrinsic shape

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Intrinsic shape

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Intrinsic shape

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Intrinsic shape

Allopola zygopleura (half rhombic polygons [i.e. isosceles triangles], or isosceles trapezia [i.e. bilateral tetragons] ; bilateral forms with four or two antimers),

Zygopleura

(isosceles triangles, symmetric zygopleura with two antimers, bilateral forms s.str.)

Intrinsic shape

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Intrinsic shape

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Intrinsic shape

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Intrinsic shape

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Intrinsic shape

A two-dimensional crystal with intrinsic shape according to an

(quarter-rhombi, i.e. right-angled triangles, or asymmetric zygopleura with two (unequal) antimers, irregular triangles)

**No antimers** (Anaxonia acentra) **:**

Intrinsic shape

Two non-congruent antimers (green, yellow)

Intrinsic shape

This

While in the foregoing documents promorphs of two-dimensional crystals were studied from the viewpoint of intrinsic

But before we do this, we will dwell for a while in the

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