2. WHY CRYSTALLORAPHY WHY CRYSTALLORAPHY? perfect order … the perfect order of particles in the space is amazing… crystals ground"as is" To think that such crystals come from the ground "as is" is surprising to many… Is maths in beauty of world or is maths the beauty of world is maths the beauty of world?

3. Geometrical structures give to minerals specific physical and chemical properties … beauty and charm… as well beauty and charm… solid geometry In particular we must study solid geometry in crystallography. crystal geometrical If we talk about crystal we must talk about crystal’s straordinary geometrical structure and properties. The universe [...] is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. ( Il Saggiatore, Galileo Galilei)

4. WHAT IS A CRYSTAL? A crystal is regular polyhedral form a regular polyhedral form bounded by crystal faces) smooth faces (called crystal faces), chemical compound which is assumed by a chemical compound, interatomic forces due to the action of its interatomic forces, when pass from liquid or gasous state to solid one orderly when compound crystallize atoms, molecules, or ions are arranged in an orderly, repeating repeating pattern unit cell three spatial dimensions (the unit cell) extending in all three spatial dimensions

5. three dimensions CRYSTALLOGRAPHIC AXES If we talk about three dimensions we must use three axes called CRYSTALLOGRAPHIC AXES The axes pass through the center of the crystal and, by using them, we can describe the intersection of any given face and identify it. cvertical The first axis, the c axis, is vertical bhorizontal The second axis, the b axis, is horizontal and passes through the center of the c axis atird dimension The third axis is the a axis, that of tird dimension, passes through the join of the b and c axes and is normal to plane made by this two axes. Opposite ends of these axes are indicated by plus and minus signes. AXIAL CROSS The combination of the three axes is called AXIAL CROSS.

6. grow During the process of crystallization, crystals assume various geometric shapes dependent on the physical and chemical conditions under which they grow. predominant a direction If, for exemple, is predominant a direction in which the mineral forms or change the angles beetwen two crystallogrphic axes, we’ll have different unit cell and different crystal form. crystal systemsgroups We have 7 crystal systems divided in to 3 groups: 1) Isometric a=b=c 2) Dimetric a=b≠c Cubic α=β=γ=90° Tetragonal α=β=γ=90°

7. a=b≠c 3)Trimetric a≠b≠c Rhombohedral α ≠ β≠γ≠90° Hexagonal α=γ=90°, β=120° Orthorhombic α=β=γ=90°

8. a≠b≠c Monoclinic α=γ=90° β ≠90° Triclinic α ≠ γ ≠ β ≠ 90°

9. THE CONSTANCY OF INTERFACIAL ANGLES law of constancy of interfacial angles To classificate crystalline substances is also important the law of constancy of interfacial angles stated in 1669 by Nicholas Steno, a Danish physician and natural scientist. same mineral By examination of numerous specimens of the same mineral, he found that in all crystals of a given substance, measuring at the same temperature and pressure, anglessimilar crystal faces (dihedral angles) constant, the angles between similar crystal faces (dihedral angles) remain constant, regardless of the size or the shape of the crystal

10. ELEMENTS OF SYMMETRY degreesymmetry But if I know only seven crystal system and this law I can’t know all possible polyhedral forms and all possible shapes of crystals, infact each crystals have a different degree of symmetry. Due to different combination of elements of symmetry We can divided elements of simmetry in: Real Real such as Ideal Ideal such as Planes of symmetry Axes of symmetry Center of symmetry Faces Edges Vertices

11. FACES, EDGES AND VERTICES elationshipEuler’s law Exist a relationship beetwen these elements stated by Euler’s law: V + F = E + 2 Ex: In the case of the cube, we've that V = 8, E = 12 and F = 6. V + F = 8 + 6 = 14 E + 2=12+2=14

12. PLANES OF SYMMETRY mirror immage plane of symmetry Any two dimensional surface (we can call it flat) that, when passed through the center of the crystal, divides it into two symmetrical parts that are mirror immage is a plane of symmetry Is a plane of symmetry the plane which pass from GDE F 1 C = F 2 C The plane which pass from ABED is a plane of symmetry

13. Ex: The solid of tetragonal system ( a=b≠c α=β=γ=90°) had 5 planes of symmetry A cube has 9 planes of symmetry, 3 of one set and 6 of another

14. AXES OF SYMMETRY rotated Any line through the center of the crystal, around which the crystal may be rotated, angular revolution same and so that after a definite angular revolution the crystal form appears the same as before, axis of symmetry is termed an axis of symmetry.

15. degrees of rotation necessary Depending on the amount or degrees of rotation necessary, four four types of axes of symmetry are possible when you are considering crystallography: repeats 60 sixfold When rotation repeats form every 60 degrees, then we have sixfold or HEXAGONAL SYMMETRY. 90fourfold When rotation repeats form every 90 degrees, then we have fourfold or TETRAGONAL SYMMETRY

16. 180 twofold When rotation repeats form every 180 degrees, then we have twofold or BINARY SYMMETRY 120 threefold When rotation repeats form every 120 degrees, then we have threefold or TRIGONAL SYMMETRY. 360 When rotation repeats form every 360 degrees, we have NO SYMMETRY.

17. CENTER OF SYMMETRY Most crystals have a center of symmetry, even though they may not possess either planes of symmetry or axes of symmetry. imaginary line through the center of the crystal equidistance center of symmetry If you can pass an imaginary line from an real element through the center of the crystal (the axial cross), and it intersects a similar point on that element similar to the first, equidistance from the center, then the crystal has a center of symmetry that coincide with the center of crystal.

18. Depending upon what elements of symmetry are present in a crystal we can clasificate it in classes of symmetry. CLASSES OF SYMMETRY Each crystal system includes some particular classes of symmetry for different characteristics depending faces length and angles beetwen crystallographic axes In particular we can define a FORM is a group of crystal faces, all having the same relationship between elements of symmetry and crystal system.

19. We have general forms… In each crystal system, there is a form in which the faces intersect each crytallographic axes at different lengths. We talk about dimetric and trimetric system and so to non-isometric (noncubic) system. Divided in to 32 classes of symmetry (31+1) …And special forms including all other form interesting isometric or cubic system. Divided in to 15 classes of symmetry

20. Crystal forms in the isometric system have the highest degree of SYMMETRY. ISOMETRIC SYSTEM Only sphere in geometrical universe had infinite planes of symmetry pass through its center, infinite rotational axes are present, and no matter how little or much you rotate it on any of its infinite number of axes, it appears the same! A sphere is the HOLY GRAIL of symmetry!

21. No crystal system even approaches a sphere's degree of symmetry, but the isometric system is often quickly recognizable because some of the forms and combinations of forms somewhat approach sphericity. For these forms - There are 4 diagonal axes of 3-fold rotation(120°) (4A 3 ) - The 3 crystallographic axes are axes of 4-fold rotation (90°) (3A 4 ) - We have 6 directions of 2-fold symmetry (180°) (6A 2 ) - We find 9 mirror planes (9P) - There is also a center of symmetry (C) This combination of symmetry elements defines the highest possible symmetry of crystals.

22. To this group belongs: Cube: The cube is composed of 6 square faces at 90 degree angles to each other. Each face intersects one of the crystallographic axes and is parallel to the other two. An example of crystal that have this form is galena or pyrite. galena pyrite NaCl