CS 39R:  Symmetry & Topology
Lecture #2 -- Mon. 9/11, 2017.


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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 (the transformations that map an artifact back onto itself) 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.

DEMONSTRATE: Symmetry operations on a cardboard square ...


NEW: Translatory Symmetry

Where might you find artifacts with (finite) translatory symmetry ? . . .
An example of  translatory symmetry.


Examples of Friezes on the web ...
Translatory Symmetry in Musical Scales:
To what degree do keyboards have translatory symmetry?

Symmetry in 2D Space:

A 2D pattern with a rotational center is either Cn-symmetric, if a rotation through 360/n degrees places it back onto itself;
or it is Dn-symmetric if in addition it also has reflective mirror symmetry on n axes going through the center.
A pattern with only one mirror-symmetry axis would be D1.

DISCUSS: Homework 1.

Symmetry in 3D Space:

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



New Homework Assignment:

For the next lecture, think about the following issues:



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