CS 285: SOLID MODELING

Lecture #5 -- Tue, Sep. 11, 2007.

Warm-up Exercise: Match the patterns: letters to numbers

What is symmetry ?

Symmetry operations form groups {Closure: A,B ==> AB, BA;  Associativity:  (AB)C = A(BC);  Identity: IA = AI = A;  Inverse:  A ==> A^-1:  AA^-1 = A^-1A = I }.

Understanding the symmetries of 1D friezes is crucial to the understanding of the symmetries of 2D figures, 3D objects, and 2D and 3D tilings.

Examples of Friezes on the web ...

• More Friezes to Match
• Woven Rope Friezes
• Frieze Patterns in Cast Iron
• Brownian Friezes
• Impossible Frieze...
• Others? -- If you find any cool references, please let me know.

Some fine books on symmetry and tilings:

• Hermann Weyl: "Symmetry", Birkhaeuser, 1955.
• Alan Holden: "Shapes, Space, and Symmetry", Dover, 1971.
• Alan Holden: "Orderly Tangles", Columbia, 1983.
• Robert Williams: "The Geometrical Foundation of Nature and Structure", Dover, 1972.
• Gruenbaum / Shephard: "Tilings and Patterns", Freeman, 1986.

Symmetry in 2D Space:

• 7 frieze symmetries.
• 2 families of rotational groups Cn, Dn.
• 17 lattice groups.

Symmetry in 3D Space:

• 7 families of rotational groups based on the 7 friezes wrapped around a cylinder: Cn, Dn, S2n, Dnd, Cnh, Cnv. Dnh.
• 7 groups of "really 3D" symmetries based on the Platonic and Archimedean solids.
  (pictures of the double-tetrahedron and the oriented double-tetrahedron)
  
• 230 3D lattice groups !

Open your eyes to the symmetry of everyday objects...

• What are the symmetries of the objects you encounter in your daily life ?
• How to determine the symmetry group of an object:
  Find a maximal-valence rotation axis, make it the z-axis, go to chart 1,  look for C₂ axes perpendicular to it, also for mirror planes, ...
  if you find more than one rotation axis with valence >= 3, go to chart 2;  5-fold axes ==> icosa/dodeca;  4-fold axes at right angles ==> cube/octa, ...
  
  
• Brain teaser: -- Why does an ordinary wall mirror reverse left and right, but not up and down ?

Current Homework Assignment#2: (Due 9/13/07:  08 am, electronically)

Install SLIDE,  and  create a 5-variable Venn diagram with 5-fold symmetry.