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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Some slides were removed due to outdated links/resources.