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06/02/13 Tetragonal Crystal System V

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The Tetragonal Crystal System (PartFive)

(Alternative name : Quadratic System)

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The Tetragonal-pyramidal Class

(= Hemimorphy of the Pyramidal Hemihedric) 4

The Forms of the Tetragonal-bipyramidal Class (See Part Two) can, like the Forms of the holohedric Class (= the Ditetragonal-bipyramidal Class), be, in their turn, subjected to a hemihedric, in this case a

hemimorphy. So like we have a hemimorphy of holohedric Forms, we also can have a hemimorphy of

Forms of the Tetragonal-bipyramidal Class, thus a hemimorphy of the pyramidal hemihedric Forms. Recall

that hemimorphy is a special type of hemihedric, namely the suppression of the main mirror plane, in our case

the equatorial mirror plane. What is, in our case, left is just the 4-fold rotation axis, coincident with the main

crystallographic axis, which now has become polar, which means that the two ends of the axis are not

equivalent. At those ends different states of affairs are to be expected.

The pyramids and the basic pinacoid each decay into two independent upper and lower halves, while the

prisms remain unchanged with respect to their outer shape.

The symmetry content of this Class is :

One 4-fold rotation axis, coincident with the main crystallographic axis.
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Derivation of the Forms of the Tetragonal-pyramidal Class

MEROHEDRIC APPROACH.

We will now derive all the Forms of this Class by applying hemimorphy to all the pyramidal hemihedric

Forms. These latter Forms were obtained from the holohedric Forms in the following way :

1. The holohedric Protopyramid gave rise to the

pyramidal hemihedric Type I Tetragonal Bipyramid.

2. The holohedric Deuteropyramid gave rise to the

pyramidal hemihedric Type II Tetragonal Bipyramid.

3. The holohedric Ditetragonal Bipyramid gave rise to the

tetragonal tritopyramid, i.e. the pyramidal hemihedric Type III Tetragonal Bipyramid.

4. The holohedric Protoprism gave rise to the

pyramidal hemihedric Type I Tetragonal Prism.

5. The holohedric Deuteroprism gave rise to the

pyramidal hemihedric Type II Tetragonal Prism.

6. The holohedric Ditetragonal Prism gave rise to the

Tetragonal Tritoprism, i.e. a pyramidal hemihedric Type III Tetragonal Prism

7. The holohedric Basic Pinacoid gave rise to the

pyramidal hemihedric Basic Pinacoid

From these pyramidal hemihedric Forms (marked above in bold face) we will derive the Forms of the presentClass, i.e. the Tetragonal-pyramidal Class (hemimorphy of pyramidal hemihedric).

In doing so we first show (again) the relevant pyramidal hemihedric Form, and then show the derivation of the

hemimorphous Form from it.

1. Derivation from the pyramidal hemihedric Type I Tetragonal Bipyramid.


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Figure 1.

Left image.

The pyramidal hemihedric Type I Tetragonal Bipyramid, derived from the holohedric Protopyramid.

Right image.

Applying hemimorphy to the pyramidal hemihedric Type I Tetragonal Bipyramid.

The Naumann symbol clearly shows the repeated hemihedric (pyramidal hemihedric and

hemimorphy), i.e. the successive application of two hemihedrics to a holohedric Form.

The "h"in 2 h signifies the second hemihedric (being applied), namely the hemimorphy.

The hemimorphy demands the suppression of the equatorial mirror plane which was present in the pyramidal

hemihedric Form, which implies that in the derivation we must mark the bottom faces in such a way that they

are distinguished from any upper face, garanteeing the absence of the equatorial mirror plane. See Figure 1,

right image.

The result is a Type I tetragonal monopyramid possessing no mirror planes whatsoever.Of course two are possible, the upper half of the initial bipyramid and the lower half of that pyramid. They are

independent of each other and can occur together as a combination. But also the one or the other can

combine with other Forms of this Class. The monopyramids are open Forms, so a single monopyramid

cannot occur all by itself.

Figure 2 shows them.


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Figure 2. The two possible Tetragonal Monopyramids. Forms of the Tetragonal-pyramidal Crystal

Class.

2. Derivation from the pyramidal hemihedric Type II Tetragonal Bipyramid.

The hemimorphy suppresses the equatorial mirror plane, resulting in the suppression of either the lower facesor the upper ones. So we obtain two hemimorphous type II tetragonal monopyramids. See Figure 3.

Figure 3. The derivation of the Monopyramids by subjecting a Type II Tetragonal Bipyramid to

hemimorphy. It will result in two monopyramids, as depicted in Figure 4.


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Figure 4. Two independent Tetragonal Monopyramids, originated by hemimorphy applied to a

pyramidal hemihedric Type II Tetragonal Bipyramid.

3. Derivation from the pyramidal hemihedric Type III Tetragonal Bipyramid, the Tetragonal

Tritopyramid.

Also here the suppression of the equatorial mirror plane ( = application of hemimorphy) results in twoindependent type III monopyramids, one of which is shown in Figure 5.

Figure 5.Derivation of the Type III monopyramid from the pyramidal hemihedric Type III Bipyramid

by applying hemimorphy. This monopyramid is exactly identical to the above monopyramids of type I and

type II, except its orientation with respect to the crystallographic axes. Also here we can indicate the absence

of (vertical) mirror planes as done in the next Figure.


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Figure 6. The hemimorphic Type III Tetragonal Pyramid. Absence of vertical mirror planes is

indicated.

4. Derivation from the pyramidal hemihedric Type I Tetragonal Prism.

The suppression of the equatorial mirror plane ( = applying hemimorphy) does not have any effect on the

external shape of the prism, but of course it looses some symmetry. So we obtain again a type I tetragonal

prism. See Figure 7.

Figure 7.Derivation of the hemimorphous Type I Tetragonal Prism from the pyramidal hemihedric

Type I Tetragonal Prism. The absence of vertical as well as of horizontal mirror planes is indicated.

5. Derivation from the pyramidal hemihedric Type II Tetragonal Prism.

Applying hemimorphy to the pyramidal hemihedric type II tetragonal prism gives again rise to a type II

tetragonal prism but with lower symmetry. See Figure 8.


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Figure 8. Getting a hemimorphous Type II Tetragonal Prism from a pyramidal hemihedric Type II

Tetragonal Prism. The absence of all the mirror planes is indicated.

6. Derivation from the pyramidal hemihedric Type III Tetragonal Prism, the Tetragonal Tritoprism.

Applying hemimorphy to the tritoprism does not change its external shape, but lowers its (crystallographic)

symmetry accordingly. See Figure 9.


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Figure 9. The pyramidal hemihedric Tritoprism has become a hemimorphous (pyramidal hemihedric)

Type III Tetragonal Prism, having no mirror planes whatsoever, despite the fact that the Figure

suggests some vertical mirror planes.

7. Derivation from the pyramidal hemihedric Basic Pinacoid.

The basic pinacoid consists of two parallel faces. When hemimorphy is applied to this Form we obtain two

independent halves of it, two pedions. So this new Form, the pedion, is just one face. The derivation of onesuch pedion is depicted in Figure 10 and 10a.

Figure 10. The pyramidal hemihedric Basic Pinacoid, from which the Pedion is derived by

hemimorphy.

Figure 10a.A Pedion (monohedron), derived from the pyramidal hemihedric Basic Pinacoid.


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All these Forms can engage in combinations.

As an example could serve the combination of Forms in crystals of the mineral Wulfenite (BRUHNS, 1912,

p. 63). See Figure 11. On top we see the primary protopyramid (here a monopyramid) (p in

the Figure), of which the corners are cut off

in an oblique fashion by the tritoprism (r in the Figure). At the bottom the

polar corner of the other monopyramid (p' in the Figure) is cut off by a pedion.

Figure 11. Combination of two Tetragonal Type I Pyramids (p and p') and a Tritoprism (r).

FACIAL APPROACH

We will now derive those same Forms by subjecting the basic faces (compatible with the Tetragonal Crystal

System) one by one to the symmetry operations of the present Class (the Tetragonal-pyramidal Crystal

Class).

Recall that the basic faces were the following :

a : a : c

a : ~a : c


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a : na : mc

a : a : ~c

a : ~a : ~c

a : na : ~c

~a : ~a : c

The stereogram of the symmetry elements of the present Class is :

Figure 12. Stereogram of the symmetry elements of the Tetragonal-pyramidal Crystal Class and of all

the face poles of the most general Form.

The face a : a : c is the unit face of the Tetragonal Crystal System. Its position in the stereographic projection

of the symmetry elements of the present Class is shown in Figure 13.

Figure 13. Stereogram of the symmetry elements of the Tetragonal-pyramidal Crystal Class and the

position of the facea : a : c.

This face a : a : c generates, when subjected to the symmetry elements of the present Class, consisting of a

4-fold rotation axis only, a type I tetragonal monopyramid (Figure 2). Its stereographic projection is

depicted in Figure 13a.


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Figure 13a. Stereogram of the hemimorphous Type I Tetragonal Pyramid.

The face a : ~a : c is parallel to the east-west crystallographic axis and cuts off a unit piece from the maincrystallographic axis. Its position is depicted in Figure 14.

Figure 14.Position if the facea : ~a : c in the stereographic projection of the symmetry elements ofthe Tetragonal-pyramidal Crystal Class.

The 4-fold rotation axis generates from this face a type II tetragonal monopyramid (Figure 4). The

stereographic projection of it is shown in Figure 14a.

Figure 14a. Stereogram of the hemimorphous Type II Tetragonal Pyramid.

The face a : na : mc is the general face. Its position in the stereogaphic projection is shown in Figure 15.


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Figure 15.Position of the facea : na : mcin the stereographic projection of the symmetry elements of

the Tetragonal-pyramidal Crystal Class.

The 4-fold rotation axis generates a type III tetragonal monopyramid (Figure 6). Its stereographic

projection is shown in Figure 15a.

Figure 15a. Stereogram of the hemimorphous Type III Tetragonal Pyramid.

The face a : a : ~c is vertical. Its position is indicated in Figure 16.

Figure 16. The position of the facea : a : ~c in the stereographic projection of the symmetry elementsof the Tetragonal-pyramidal Crystal Class.

The 4-fold rotation axis generates from this face a type I tetragonal prism (Figure 7). Its stereographic

projection is indicated in Figure 16a.


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Figure 16a. Stereogram of the hemimorphous Type I Tetragonal Prism.

The face a : ~a : ~c is also vertical. It is moreover parallel to the east-west crystallographic axis. Its positionin the stereographic projection is indicated in Figure 17.

Figure 17. Stereogram of the position of the facea : ~a : ~cin the stereographic projection of thesymmetry elements of the Tetragonal-pyramidal Crystal Class.

The 4-fold rotation axis generates from this face the type II tetragonal prism (Figure 8). Its stereographic

projection is shown in Figure 17a.

Figure 17a. Stereogram of the hemimorphous Type II Tetragonal Prism.

The face a : na : ~c is also vertical but cuts off a non-unit piece of the east-west crystallographic axis. Itsposition in the stereographic projection is indicated in Figure 18.


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Figure 18.Position of the facea : na : ~cin the stereographic projection of the symmetry elements ofthe Tetragonal-pyramidal Crystal Class. The 4-fold rotation axis generates from this face a type III

tetragonal prism (Figure 9). Its stereographic projection is shown in Figure 18a.

Figure 18a. Stereogram of a hemimorphous Type III Tetragonal Prism.

Recall that the second coefficient (relating to the east-west crystallographic axis) is variable. Its value ranges

between n = 1 (type I prism) and n = infinity (type II prism). So the next stereogram is also that of a type III

tetragonal prism :

Figure 18b. Stereogram of another Type III hemimorphous Tetragonal Prism.

Finally we have the face ~a : ~a : c. It is horizontally oriented. Its position in the stereographic projection inindicated in Figure 19.


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Figure 19.Position of the face ~a : ~a : cin the stereographic projection of the symmetry elements ofthe Tetragonal-pyramidal Crystal Class.

The only symmetry element of the Class, the 4-fold rotation axis, has no effect on this face, i.e. the face will

not be multiplied by this axis. Said differently, when we consider this face to be subjected to the symmetry

elements of the present Class, it will be transformed onto itself, by the actions of the 4-fold rotation axis,which actions are : rotations (of the Form) about the main crystallographic axis by 900, 1800, 2700 (and of

course, 00 [= 3600] ). So the face ~a : ~a : c turns out to be already a Form all by itself of the presentClass. It consists of that one face and is called a pedion (Figure 10a).

Its stereographic projection is of course identical with the one depicted in Figure 19.

This concludes our exposition of the derivation of all the Forms of the Tetragonal-pyramidal Crystal Class,

by means of two approaches.

To continue, klickHERE for Part Six.

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