CS 39R:  Symmetry & Topology
Lecture #2 -- Mon. 2/4, 2013.

PREVIOUS <- - - - > CS 39R  HOME < - - - - > CURRENT < - - - - > NEXT

Preparation:

Bring pictures of hubcaps and company logos to class or send them to me: sequin@cs.berkeley.edu


Warm-up -- Classifying Frieze Symmetry Patterns:

Which friezes in the two panels correspond to one another?

A Key Point:

Exploiting symmetry is a great way to reduce the amount of design work that needs to be done
-- and, possibly, to increase the quality of a resulting design.
Fortunately the numbers of all possible symmetries can be nicely catalogued ...

A Finite Number of Frieze Symmetry Groups!

Symmetry operations form groups.
The key characteristics that make something a "group":
Closure: A,B ==> AB, BA;   --- All combinations of operations are also elements of the group.
Associativity:  (AB)C = A(BC);  --- The order in which elements are combined may matter, but the sequence in which the combinations are calculated does not.
Identity: IA = AI = A;  --- The identity element makes no change.
Inverse:  A ==> A-1:  AA-1 = A-1A = I };  --- for every element there is also an inverse element; an element may be its own inverse.


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 ...

Symmetry in 2D Space:

Symmetry in 3D Space:

Discussion Points:

Approximate Symmetry: Deviation from perfect symmetry;  (what you find in nature).

Forced Symmetry: Making some symmetric that intrinsicly is NOT symmetric;  e.g. Name cards, company logos, ... (Inspiration from Scott Kim).

What is "Antisymmetry" ?


New Homework Assignment:
For the next lecture, think about the following issues:

PREVIOUS <- - - - > CS 39R  HOME < - - - - > CURRENT < - - - - > NEXT

Page Editor: Carlo H. Séquin