It is possible for an object to possess more

Than one type of symmetry.

Elements of symmetry - Symmetry centre

Identical features are

encountered, along a line passing

through a central point. Object and

image are at equal distance of i.

Symmetry centre. The symmetry centre terms such point in a figure, at holding through

which one any straight line will meet the identical and reversibly laid out parts of a figure

on spacing interval equal from it. The symmetry centre is designated by the letter (i).

If each crystal face has to itself equal, though and reversibly laid out facet the

yielded crystal possesses symmetry centre.

Some crystals can not have symmetry centre.

Elements of symmetry - Plane of symmetry Bilateral symmetryAny plane of symmetry

divides the crystal

form into two mirror

images.

Identical features are encountered, at equal but opposite

distances along any line perpendicular to the mirror plane.

Plane of symmetry. It is an imagined plane which one divides a figure on two equal parts so

that, one of parts is a reflexion of the other. The plane of symmetry is designated by (P).

a little before a plain label the number of planes is put, for example - 3Р, three planes of symmetry.

In crystals there can be one, two, three, four, five, six, seven and nine planes of symmetry.

Many crystals have no any plane of symmetry.

Line or bilateral symmetry

Shape has line symmetry when one half of it is the mirror image of the other half.

You get two halves,

each one is the mirror

image of the other.

So a symmetry plane

Is called a mirror

Plane with the symbol

(m)

The yellow line is the trace of a symmetry plane or mirror plane (m).

(m) letter is used in the Hermann and Mauguin international symbols

Elements of symmetry - Symmetry axes

Rotational symmetry

2-fold axis Rotation

Symmetry axes. The « symmetry axis » terms an imagined straight line,

at rotational displacement around which one, always on the same angle,

there is a combination of equal parts of a figure.

The Symmetry axis is designated by letter L

Elements of symmetry - Symmetry axes Rotation

3-fold axis

4-fold axis

6-fold axis

At rotational displacement on 360 ° may show a combination of facets

which may appear two, three, four or six times

(i.e. at each rotational displacement on 180, 120, 90 and 60 °).

The four points a₁, a₂, a₃, and a₄, consitute a set of symmetrically

equivalent points or equipoints.

1-fold axeExample: an astronaut on a fixed station looking at the earth.

He will see the same landscape after a total rotation of the

Earth, that is after a 360° rotation.

So, there is no repetitive image and thus, no symmetry at all.

The hermann-Mauguin symbol 1 signifies this total

lack of symmetry,

That is the existence of 1-fold axe.

Symmetry operations

The geometric mouvement to bring a given point a₁ into coincidence with a symmetrically equivalent point a₂ is called

a symmetry operation.

2 Major kinds of symmetry operations

Centre of symmetry

(inversion)-

Symbols: i or minus 1

Plane of symmetry (reflexion):

mOperations of the first kind :

The symmetry operation is generating the same hand as shown on the figure.

Operations of the second kind :

The symmetry operation is promoting a change of hand.

Symmetry through a point or a mirror generates a change of hands.

Proper and improper axesThe 1-, 2-, 3-, 4- and 6-fold axes represent operations of

the first kind. They are called proper axes.

Crystallographers recognize five other axes:

Called (bar 1, bar 2, etc) resulting from the operation of

1-, 2-, 3-, 4- and 6-fold axes combined with the operation

of inversion through a central point on the axis.

They are called inversion axes or improper axes as they

Represent the operations of the second kind.

External symmetry of crystalsThe external symmetry of any crystal must necessarily

corresponds to :

1. One of the five proper axes

2. One of the five improper axes

3. One of the 22 combinaisons of these axes