INTRODUCING PLANE SYMMETRY TYPES (PART I)
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Today we took a detailed look at 10 of the 17 symmetry types a plane 
pattern may possess.  These 10 have the following names and features:

*2222    Reflections lines break the plane into rectangles, 
         and form four different types of order 2 corner points

*333     Reflections lines break the plane into equilateral triangles
         and form three different types of order 3 corner points

*632     Reflection lines break the plane into 30-60-90 triangles and
         form corner points of orders 6,3 and 2.

*442     Reflection lines break the plane into 45-45-90 triangles
         and form two types of order 4 corner points and also order 2 corner points.

 2222    No reflection lines.  Four different types of gyration points of order 2.

 333     No reflection lines.  Three different types of gyration points of order 3.

 632     No reflection lines.  Gyration points of orders 6,3 and 2.

 442     No reflection lines.  Two types of order 4 gyration points and also order 2 
         gyration points.

 O       No reflection lines, no gyration points, no glide reflections.  The most
         primitive of all the symmetry types.

 **      Reflections lines of two types, all mutually parallel.  No gyrations.


We saw how each such pattern could be interpreted as a flat map of a surface
of some appropriate shape:

*2222    Rectangular ``card''  (pattern prints right through, appears mirror reversed on back)
*333     Equilateral triangular card
*632     30-60-90 triangular card
*442     45-45-90 triangular card
 2222    Rectangular pillowcase (has inside outside, map shows only outside)
 333     Equilateral triangular pillowcase (template may be a rhombus formed from adjacent triangles)
 632     30-60-90 triangular pillowcase (template may be equilateral triangle formed from adjacent 30-60-90's) 
 442     45-45-90 triangular pillowcase (template may be a square formed from adjacent 45-45-90's)
 O       Donut shape (technical term: torus)
 **      Side of a cylinder (alternatively, a rectangular pillow case opened on opposite sides.)

Notes:   A pattern determines the shape of the surface that goes with it, but not the shape of
         the template.  

         By template, I mean a smallest possible region of the pattern that
         generates the whole pattern by means of symmetric repetitions.  Put another way,
         a polygon in which the artist may do whatever he or she pleases afterwhich the whole
         pattern is determined.

         One's eye easily detects the gyrations of order 6 and 3 in a 632, but the order 2
         gyrations tend to be hard to see.  But the fall midway between nearby gyrations of
         order 6, and also midway between nearby gyrations of order 3.

         Your immediate goals:  
         1) Given a pattern, to identify its features and name its symmetry type
         2) Given a symmetry type and a template, to draw the pattern that extends it

         Though I hope the naming conventions seem reasonably natural, I haven't yet
         described the whole naming scheme.  Friday's lecture will address this.

         Your homework for the weekend will be to identify the patterns found in the textbook.
         (Due to a copy shop error, there may be other text illustrations mixed in with these.)
         You can get a start on this now, at least for the 10 patterns we've covered.

         Mathematical terminology I shall define as precisely as our need warrent.  For any 
         difficulties with nontechnical vocabulary, consider consulting a dictionary (as you would 
         in any other university course.)