# CS 39R:  Symmetry & Topology Lecture #3 -- Mon. 2/11, 2019.

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### Preparation:

Bring along a couple of objects with some higher-order 3D symmetry.
Try to find all its symmetry operations: rotation axes, mirror- and glide-planes.

Determine for each which symmetry group from these two charts it belongs to: chart 1chart 2.

# 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.  Check symmetry classification.(See posters on the wall!)

The symmetries of a pair of gearwheels:  A, B, C, D.

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

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; it has a mixture of pure rotation points and kaleidoscope points;
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 arranged may matter, but the sequence in which they are combined 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!} [DONE last time]

-- The 2D square  or {D4} hubcap -- under rotations and mirroring:  {8 elements};
-- The thick 3D square plate {D4h}
-- under rotations and mirroring:  {16 elements};
-- Permutation of 4 books on a shelf -- while switching pairs of books:
{24 elements};  {mathematicians call this "S4"};
-- All symmetry operations on a tetrahedron
:  {24 elements};  {when "oriented": only 12 elements};
-- The octahedron (==> see plexiglass model);  == same as a cube
:  {48 elements};

A more complicated structure: -- step into the hallway ...

### "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:

Why does an ordinary wall mirror reverse left and right, but not up and down ?

## Visualization of Symmetry Groups Using Shape Generator Programs

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

7 families of rotational groups based on the 7 friezes wrapped around a cylinder: Cn, Dn, S2n, Dnd, Cnh, Cnv. Dnh.

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

7 groups of "really 3D" symmetries based on the Platonic and Archimedean solids.

Jane Yen and C. H. Séquin: "Escher Sphere Construction Kit" Presentation at I3D conference (PPT)

# 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. 25, 2019 (in two weeks).

Design a genus-6 object of high symmetry:   Can you do this with "spherical" symmetry {chart 2} --?
Present your design as a physical model, using: clay, paper, styro-foam, pipe cleaners, . . .,   or as a computer model;   or as a very nice drawing.
Be prepared to tell us what its symmetry is.

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