Lecture #3 -- Mon. 2/11, 2019.

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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 1, chart 2.

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

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 C_{2} 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.

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

Identity

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 ...

It has **oriented icosidodecaheral
symmetry**; no mirror planes.

Tetrahedron, cube (hexahedron), octahedron, dodecahedron,
icosahedron.

(Could you explain to a high-school student why there are
exactly (and only) **five
Platonic Solids **?

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).

**Due: Feb. 25, 2019 (in two
weeks).**

Present your design as a physical model, using: clay, paper, styro-foam, pipe cleaners, . . ., or as a computer model; or as a

Be prepared to tell us what its symmetry is.

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