The Fascination of Crystals and Symmetry

Unit 3.8

by Frank Hoffmann & Michael Sartor

Summary and Outlook

Space Symmetry (3D)Plane Symmetry (2D)

TranslationInversion (point mirroring)

Rotation-

ReflectionGlide (reflect, then translate, 2D)

5 Bravais lattices

TranslationInversion (point mirroring)

RotationRoto-Inversion (rotate, then invert)

ReflectionGlide (reflect, then translate, 3D)

Screw (rotate, then translate)

230 space groups

32 crystal classes

14 Bravais lattices

17 plane groups

Point Symmetry (3D)

-Inversion (point mirroring)

RotationRoto-Inversion (rotate, then invert)

Reflection--

32 crystal classes

Algorithm to determine the plane symmetry group

Brian Sanderson's Pattern Recognition Algorithm:

http://www.math.toronto.edu/~drorbn/Gallery/Symmetry/Tilings/Sanderson/index.html

Is the maximum rotation order 1,2,3,4 or 6?

Is there a mirror (m)?

Is there an indecomposable glide reflection (g)?

Is there a rotation axis on a mirror?

Is there a rotation axis not on a mirror?

Web and other Tools regarding plane groups

http://www.scienceu.com/geometry/handson/kali/ http://escher.epfl.ch/escher/ http://www.mathsisfun.com/geometry/symmetry-artist.html

Web and other Tools regarding plane groups

http://imaginary.org/program/morenaments/applet http://weavesilk.com/

Some slides were removed due to outdated links/resources.