1. MATERIALS SCIENCE &ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK Part of http://home.iitk.ac.in/~anandh/E-book.htm A Learner’s Guide

2. Symmetry of Solids  We consider the symmetry of some basic geometric solids (convex polyhedra).  Important amongst these are the 5 Platonic solids (the only possible regular solids* in 3D):  Tetrahedron  Cube  Octahedron (identical symmetry)  Dodecahedron  Icosahedron (identical symmetry) The symbol  implies the “dual of”.  Only simple rotational symmetries are considered (roto-inversion axes are not shown).  These symmetries are best understood by taking actual models in hand and looking at these symmetries.  Certain semi-regular solids are also frequently encountered in the structure of materials (e.g. rhombic dodecahedron). Some of these can be obtained by the truncation (cutting the edges in a systematic manner) of the regular solids (e.g. Tetrakaidecahedron, cuboctahedron) * Regular solids are those with one type of vertex, one type of edge and one type of face (i.e. ever vertex is identical to every other vertex, every edge is identical to every other edge and every face is identical to every other face)

3. Symmetry of the Cube {4,3}* 4- fold axes pass through the opposite set of face centres 3 numbers * The schläfli symbol for the cube is {4,3}  4-sided squares are put together in 3 numbers at each vertex Yellow 4-fold Blue 3-fold Pink 2-fold

4. The body diagonals are 3-fold axes (actually a  3 axis) 4 numbers

5. 2-fold axes pass through the centres of opposite edges 6 numbers

6. 3 mirrors Centre of inversion at the body centre of the cube

7. Important Note:  These are the symmetries of the cube (which are identical to those present in the cubic lattice)  A crystal based on the cubic unit cell could have lower symmetry as well  A crystal would be called a cubic crystal if the 3-folds are NOT destroyed

8. Symmetry of the Octahedron {3,4}  Octahedron has symmetry identical to that of the cube  Octahedron is the dual of the cube (made by joining the faces of the cube as below) 3-fold is along centre of opposite faces 2-fold is along centre of opposite edges 4-fold is along centre of opposite vertices Centre of inversion at the body centre

9. Yellow 4-fold Blue 3-fold Pink 2-fold

10. Symmetry of the Tetrahedron {3,3} No centre of inversion No 4-fold axis 3-fold connects vertex to opposite face 2-fold connects opposite edge centres

11. Yellow 3-fold Blue 3-fold Pink 2-fold

12. Yellow 5-fold Blue 3-fold Pink 2-fold Symmetry of the Dodecahedron {5,3}

13. Symmetry of the Icosahedron {3,5} Yellow 5-fold Blue 3-fold Pink 2-fold

14.  Certain semi-regular solids can be obtained by the truncation of the regular solids.  Usually truncation implies cutting of all vertices in a systematic manner (identically)  E.g. Tetrakaidecahedron, cuboctahedron can be obtained by the truncation of the cube. In these polyhedra the rotational symmetry axes are identical to that in the cube or octahedron.  Tetrakaidecahedron {4,6,6}:  Two types of faces: square and hexagonal faces  Two types of edges: between square and hexagon & between hexagon and hexagon  Cuboctahedron {3,4,3,4}={3,4} 2 :  Two types of faces: square and triangular faces Truncated solids Tetrakaidecahedron Cuboctahedron Cuboctahedron formed by truncating a CCP crystal

15. Cuboctahedron Yellow 4-fold Blue 3-fold Pink 2-fold

16.  Space filling solids are those which can ‘monohedrally’ tile 3D space (i.e. can be put together to fill 3D space such that there is no overlaps or no gaps).  In 2D the regular shapes which can monohedrally tile the plane are: triangle {3}, Square {4} and the hexagon {5}.  The non regular pentagon can tile the 2D plane monohedrally in many ways.  The cube is an obvious space filling solid. None of the other platonic solids are space filling.  The Tetrakaidecahedron and the Rhombic Dodecahedron are examples of semi-regular space filling solids. Space filling solids Tetrahedral configuration formed out the space filling units Video: Space filling in 3D Video: Space filling in 2D