1. The Nobel Prize in Chemistry 2011 Dan Shechtman Technion – Israel Institute of Technology, Haifa, Israel Prize motivation: "for the discovery of quasicrystals"

2. Matter Solid LiquidGas Plasma Liquid Crystal Amorphous Crystalline QUASICRYSTALLINE 1984

3. L+  T Stable liquid Under Cooled liquid log t TmTm TTT Diagram for liquid-to-solid transformation Coarse grained crystals Fine grained crystals glass

4. Crystal Amorphous SiO 2

5. Crystal 3D Periodic arrangement of atoms Amorphous Random arrangement of atoms Long-range translational order Short-range order No Long-range order Short-range order Sharp diffraction pattern Diffuse diffraction pattern

6. Sharp Crystalline Amorphous Diffuse Diffraction Patterns

8. Electron Diffraction and symmetry Beam :

9. 9/87 7 crystal Systems Cubic Defining Crystal system Conventional symmunit cell 4 1 3 1 1 1 none Tetragonal Orthorhombic Hexagonal Rhombohedral Triclinic Monoclinic a=b=c,  =  =  =90  a=b  c,  =  =  =90  a  b  c,  =  =  =90  a=b  c,  =  = 90 ,  =120  a=b=c,  =  =  90  a  b  c,  =  =90  a  b  c, 

11. Rotational Symmetries Z 180  120  90  72  60  2345 6 45  8 Angles: Fold: Graphic symbols

12. Crsytallographic Restriction 5-fold symmetry or Pentagonal symmetry is not possible for Periodic Tilings Symmetries higher than 6-fold also not possible Only possible rotational symmetries for periodic tilings 23456789…

14. A crystal with 10-Fold symmetry???

16. Five-fold Two-fold Five-fold Two-fold Three-fold Icosahedral Symmetry Icosahedron

17. Regular Polygons: All sides equal all angles equal How many regular polygons are possible? 3 4 5 6 There are infinitely many regular polygons Triangle square pentagon hexagon…

18. 3D: Regular Polyhedra or Platonic Solids Cube How many regular solids? Tetrahedron All faces regular congruent polygons, all corners identical.

19. There are 5 and only 5 Platonic or regular solids ! Icosahedron Octahedron Tetrahedron Cube Dodecahedron

20. What is the structure of Quasicrystal?

21. One Dimensional Quasicrystal Fibbonacci Chain

22. Two-dimensional Quasicrystal Penrose Pattern

23. Hexagons always tile periodically Square can tile periodically or aperiodically. Is there a tile or a set of tile that will tile only aperiodically?

26. Thank you