What are the 32 Point Groups?The interaction of symmetry elements around a common

center, as found in all natural minerals and in their 3-D lattices and external forms

BariteBaSO4

orthorhombic lattice

USC Mineralogy Geol 315 (Anderson)Acknowlegement: some images from John Winter and Joe Smythe

Six Crystal Systems

Name axes angles

Triclinic a b c 90o

Monoclinic a b c = 90o 90o

Orthorhombic a b c = 90o

Tetragonal a1 = a2 c = 90o

Hexagonal

Hexagonal (4 axes) a1 = a2 = a3 c = 90o 120o

Rhombohedral a1 = a2 = a3 90o

Isometric a1 = a2 = a3 = 90o

3-D Lattice Types

a

b

c

PMonoclinic

=a g=90o ¹ ba ¹ b ¹ c

a

b

c

I = Ca

b

PTriclinica ¹ b ¹ g

a ¹ b ¹ c

c

c

aP

Orthorhombic=a b=g=90o a ¹ b ¹ c

C F Ib

a1

c

PTetragonal

=a b=g=90o a1 = a2 ¹ c

Ia2

a1

a3

PIsometric

=a b=g=90o a1 = a2= a3

a2

F I

a1

c

P or C

a2

RHexagonal Rhombohedral

900

a1a2

c90

a1 = a2 = a3

Triclinic and Monoclinic

Orthorhombic and Tetragonal

Hexagonal

Isometric

Stereographic Projections

Stereographic Projections

Illustrated above are the stereographic projections for Triclinic point groups 1 and -1

The 32 Point Groups

Triclinic System

Point Groups 1, 1 bar

The Triclinic Systemmicrocline

Monoclinic System

One symmetry axis = b

Point Groups 2, m, 2/m

The Monoclinic SystemOrthoclase

Orthorhombic System

Point Groups: 3 characters on a, b, c

The Orthorhombic System

anhydrite

barite

Triclinic, Monoclinic, and Orthorhombic Point Groups

Trigonal Subsystem

Point Groups; 1 or 2 characters1st: Symmetry along c-axis2nd: Symmetry along a’s

Trigonal Point Groups

The Trigonal Subsystem

The Trigonal Subsystem

Tourmaline on quartz

Tetragonal System

Point Groups: 1 or 3 characters1st: symmetry on the c-axis2nd: symmetry on a1 and a23rd: symmetry between a’s

Tetragonal Point Groups

The Tetragonal System

scapolitevesuvianite

The Tetragonal System

rutile

The Tetragonal Systemtopaz

Hexagonal Subsystem

Point Groups: 1 or 3 characters1st: Symmetry along c-axis2nd: Symmetry along a’s3rd: Symmetry between a’s

Hexagonal Point Groups

The Hexagonal Systemcinnabar

The Hexagonal Systemberyl

The Hexagonal System

zincite

Isometric System

Point Groups: 2 or 3 characters1st: symmetry along a’s2nd: symmetry on [111]3rd: symmetry between a’s

The Isometric System

garnet

The Isometric Systempyrite