CS 39R:  Symmetry & Topology
Lecture #2 -- Mon. 2/04, 2019.

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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 ...

Only a Finite Number of Frieze Symmetry Groups!

How many are there ... ?   ---  Why ... ?

Fill in Worksheet_7-Friezes ...

A little bit of Math:
Remember the definition of "symmetry" ...   ==> Transformations
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.
:  (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.

As an example: consider all integer numbers under addition.  Check the four rules above ...

DEMONSTRATE: Symmetry operations (rotation and mirroring) on a cardboard square ...

Friezes have Translatory Symmetry

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

Examples of Friezes on the web ...
An example of purely translatory symmetry:
Translatory Symmetry in Musical Scales (keyboards):

To what degree do keyboards have translatory symmetry?

Symmetry in 2D Space:

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.

Conway Notation for Symmetries:

     * (star)      ==>  mirror plane
     *3             ==>  kaleidoscope with 3 mirror planes
     • (dot)       ==>  exactly one point remains in place under all symmetry operations
     7•              ==>  gyration point with 7 angle positions in 360 degrees
     X "miracles"        ==>  a glide-symmetry axis
     O "wonder-ring"  ==>  two pure translation vectors

New Homework Assignment:

For the next lecture, think about the following issues:

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