CS 285: SOLID MODELING

Lecture #5 -- Wednesday, Sept. 14, 2011

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Warm-up Exercise: Match the Patterns: Numbers to Letters

In addition to using some properly tailored procedural utilities,
exploiting symmetry is another great way to reduce the amount of design work that needs to be done
-- and, possibly, to increase the quality of the resulting design.
Fortunately the numbers of all possible symmetries can be nicely catalogued ...

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
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.
The Symmetries of Things by John H. Conway, Heidi Burgiel and Chaim Goodman-Strauss, 2008.

(You could of course spend many hours studying symmetry from these books.
But it is far more efficient on your time to think along in class;
in this way you can absorb the essentials in a couple of hours.)

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 !

(In 4D, there are 4783 space 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 ?

Discussion of Assignment #2 -- Generating Symmetrical Venn Digrams

How do know whether your solution is satisfactory ?
How many regions to you expect in your diagram ?
Pascal's Triangle
Can you make completely symmetrical Venn diagrams with n variables, for n=4, 6, 7, ... ?
Some solutions: with ellipses, isoscelese triangles, equilateral triangles, rectangles
7-fold symmetric solution
Higher order solutions
A nice 2011 solution (by Andrew Lee)
SLIDE issues: weaknesses and how to get around them ...
-- "circle" exists only as a cross section ... may not serve as afull-blown object that can be instantiated.
-- spline curves don't seem to take on any color :-(
-- but you can use anything that looks like a circle, e.g., a skinny torus, a squashed cylinder or cone, a hand-crafted n-gon ...
Most CAD tools have some limitations.
"Learning to use a CAD tool" means finding out what works well, and finding ways around missing or defective primitives.