MATERIALS SCIENCEMATERIALS SCIENCE&&

ENGINEERING 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-BOOKAN INTRODUCTORY E-BOOK

Part of

http://home.iitk.ac.in/~anandh/E-book.htmhttp://home.iitk.ac.in/~anandh/E-book.htm

A Learner’s GuideA Learner’s GuideA Learner’s GuideA Learner’s Guide

Symmetry of SolidsSymmetry 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)

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

Yellow4-fold

Blue3-fold

Pink2-fold

The body diagonals are 3-fold axes (actually a 3 axis)

4 numbers

2-fold axes pass through the centres of opposite edges

6 numbers

3 mirrors

Centre of inversion at the body centre of the cube

The 3-fold axis implies that the 1/3rd of the cube is the ‘asymmetric unit’ (the part which when repeated by the 3-fold creates the whole cube).

This set of figures explain the asymmetric unit for 3-fold rotation and –3 (3 bar roto-inversion) symmetry.

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

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 faces2-fold is along centre of opposite edges4-fold is along centre of opposite vertices

Centre of inversion at the body centre

Yellow4-fold

Blue3-fold

Pink2-fold

Symmetry of the Tetrahedron {3,3}

No centre of inversionNo 4-fold axis

3-fold connects vertex to opposite face2-fold connects opposite edge centres

Yellow3-fold

Blue3-fold

Pink2-fold

Yellow5-fold

Blue3-fold

Pink2-fold

Symmetry of the Dodecahedron {5,3}

Symmetry of the Icosahedron {3,5}

Yellow5-fold

Blue3-fold

Pink2-fold

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

Cuboctahedron

Yellow4-fold

Blue3-fold

Pink2-fold

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 3DVideo: Space filling in 3D

Video: Space filling in 2DVideo: Space filling in 2D

Cluster of 4 Tetrakaidecahedra Cluster of 4 Rhombic Dodecahedra

These can monohedrally tile

3D space

These can monohedrally tile

3D space