CS 39R:  Symmetry & Topology
Lecture #3 -- Mon. 2/8, 2016.

Preparation:

Warm-up:

Identify the symmetry classes for the following depicted objects:

Also:  Discuss with your neighbors the symmetry of the objects that you brought along.
(See posters on the wall!)

A Key Point: Any finite physical object falls into one of the 14 symmetry classes described in: chart 1 and  chart 2.

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, ...
the difficult one (for me) is the oriented double tetrahedron;
      the 3 mirror planes transform one (right-handed) tetrahedron into the other (left-handed) one.

Let's review (and count) the elements of some symmetry groups:

REVIEW:  the required properties to make this 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.

==> Now let's apply this to some of the symmetry classes we have encountered: 

-- The simplest possible frieze with an asymmetrical fundamental domain (repetitive element); translatory symmetry only:
    This is equivalent to the group formed by all integer numbers!  (e.g., right shift = "+1")  {infinitely many elements!}

-- The 2D square  or {D4} hubcap:  {8 elements};
-- The thick 3D square plate {D4h}:  {16 elements};
-- Permutation of 4 books on a shelf  {mathematicians call this "S4"}:  {24 elements};
-- All symmetry operations on a tetrahedron:  {24 elements};  {when "oriented": only 12 elements}

-- The octahedron (==> see plexiglass model);  == same as a cube:  {48 elements};

"Rainbow-Bits" by George Hart -- a Propellerized Icosahedron

It has oriented icosidodecaheral symmetry; no mirror planes.

The Platonic Solids:

Tetrahedron, cube (hexahedron), octahedron, dodecahedron, icosahedron.
(Could you explain to a high-school student why there are exactly (and only) five Platonic Solids ?

Discussion:

Visualization of Symmetry Groups Using Shape Generator Programs

Understanding Chart I:  with "Sculpture Generator I"  {The program for you to experiment with}

Understanding Chart II:  with "Escher Sphere Editor" {The program for you to experiment with}

A First Glimpse of Topology:

The  genus  of simple "handle-bodies"  (g=1;  g=2;  g=3;  g=7).

These are all equivalent definitions of "genus" (in this particular context):
How many tunnels are there through this (volumetric) body?
How many handles are attached to a simple spherical blob?
How many closed boundary loops can be drawn on this surface that do not yet partition it into separate regions?
         (You can still go from any point to any other point without crossing any of those boundary loops).

New Homework Assignments:

Due: Feb. 22, 2016