Bring pictures of hubcaps and company logos to class or send
them to me: sequin@cs.berkeley.edu
Which friezes in the two
panels correspond to one another?
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.
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.
Understanding the symmetries of 1D friezes is crucial to the understanding of the symmetries of 2D figures, 3D objects, and 2D and 3D tilings.