\magnification = \magstep1
\nopagenumbers
\noindent
{\it Astrid Ludwig}
\bigskip
What you gave here is a specific example of a group.
It illustrates the key properties of groups, but I 
was also looking for a bit more detail about 
what those properties are.  Recall that we said a group is
a collection $G$ of numbers or transformations, or ... with
an operation $*$, and the operation has to satisfy several 
properties:  
\bigskip
\item{1)}  For all $x,y$ in $G$, $x*y$ is in $G$.
\item{2)}  For all $x,y,z$ in $G$, $(x*y)*z = x*(y*z)$ (associativity)
\item{3)}  There is an {\it identity element} $e$ in $G$ that satisfies
$e*x = x*e = x$ for all $x$ in $G$.
\item{4)}  For each $x$ in $G$ there is an inverse element $y$ in $G$  
that satisfies $x*y = y*x = e$.  
\bigskip
\noindent
(See class notes handout.) \bigskip  

The other thing that your
essay did not describe very clearly is the idea that
it is possible to {\it classify} the periodic plane patterns
by saying two patterns have the same type if their groups
of symmetries are ``the same'' (made up of the same
types of transformations).  This is where the 
classification into 17 types comes from.  Two patterns
of the same type just have ``the same'' symmetries.
The motifs or shapes that make up the pattern, the colors, 
and so forth can be different.  
\vfill
\eject
\noindent
{\it Peter Zona}
\bigskip
What you gave here is a specific example of a group.
It illustrates the key properties of groups, but I 
was also looking for a bit more detail about 
what those properties are.  Recall that we said a group is
a collection $G$ of numbers or transformations, or ... with
an operation $*$, and the operation has to satisfy several 
properties:  
\bigskip
\item{1)}  For all $x,y$ in $G$, $x*y$ is in $G$.
\item{2)}  For all $x,y,z$ in $G$, $(x*y)*z = x*(y*z)$ (associativity)
\item{3)}  There is an {\it identity element} $e$ in $G$ that satisfies
$e*x = x*e = x$ for all $x$ in $G$.
\item{4)}  For each $x$ in $G$ there is an inverse element $y$ in $G$  
that satisfies $x*y = y*x = e$.  
\bigskip
\noindent
(See class notes handout.) \bigskip  

The other thing that your
essay did not describe very clearly is the idea that
it is possible to {\it classify} the periodic plane patterns
by saying two patterns have the same type if their groups
of symmetries are ``the same'' (made up of the same
types of transformations).  This is where the 
classification into 17 types comes from.  Two patterns
of the same type just have ``the same'' symmetries.
The motifs or shapes that make up the pattern, the colors, 
and so forth can be different.  
\vfill\eject
\noindent
{\it Kylee Sullivan}
\bigskip
The main thing that your
essay did not describe very clearly is the idea that
it is possible to {\it classify} the periodic plane patterns
by saying two patterns have the same type if their groups
of symmetries are ``the same'' (made up of the same
types of transformations).  This is where the 
classification into 17 types comes from.  Two patterns
of the same type just have ``the same'' symmetries.
The motifs or shapes that make up the pattern, the colors, 
and so forth can be different.  
\vfill\eject
\noindent
{\it Lucia Bostrom}
\bigskip
Most of the essay is very good.  
The only thing that your
essay did not describe very clearly is the idea of
what a group is.  Recall that we said a group is
a collection of numbers or transformations, or ... with
an operation, and the operation has to satisfy several 
properties (see class notes handout).  

Also, as you said,  it is possible to {\it classify} the 
periodic plane patterns by saying two patterns have the 
same type if their groups of symmetries are ``the same'' 
(made up of the same types of transformations).  This is where the 
classification into 17 types comes from.  Two patterns
of the same type just have ``the same'' symmetries.
The motifs or shapes that make up the pattern, the colors, 
and so forth can be different.  
\vfill\eject
\noindent
{\it Catherine Caracciolo}
\bigskip
The first thing that your
essay did not describe very clearly is the idea of
what a group is.  Recall that we said a group is
a collection $G$ of numbers or transformations, or ... with
an operation $*$, and the operation has to satisfy several 
properties:  
\bigskip
\item{1)}  For all $x,y$ in $G$, $x*y$ is in $G$.
\item{2)}  For all $x,y,z$ in $G$, $(x*y)*z = x*(y*z)$ (associativity)
\item{3)}  There is an {\it identity element} $e$ in $G$ that satisfies
$e*x = x*e = x$ for all $x$ in $G$.
\item{4)}  For each $x$ in $G$ there is an inverse element $y$ in $G$  
that satisfies $x*y = y*x = e$.  
\bigskip
\noindent
(See class notes handout.) \bigskip  

Also, your
essay did not describe very clearly is the idea that
it is possible to {\it classify} the periodic plane patterns
by saying two patterns have the same type if their groups
of symmetries are ``the same'' (made up of the same
types of transformations).  This is where the 
classification into 17 types comes from.  Two patterns
of the same type just have ``the same'' symmetries.
The motifs or shapes that make up the pattern, the colors, 
and so forth can be different.  
\vfill\eject
\noindent
{\it Matt Kasuba}
\bigskip
What you gave here is some specific examples of how a
particular collection of rotations with each other.
They illustrate the key properties of groups, but I 
was also looking for a bit more detail about 
what those properties are.  Recall that we said a group is
a collection $G$ of numbers or transformations, or ... with
an operation $*$, and the operation has to satisfy several 
properties:  
\bigskip
\item{1)}  For all $x,y$ in $G$, $x*y$ is in $G$.
\item{2)}  For all $x,y,z$ in $G$, $(x*y)*z = x*(y*z)$ (associativity)
\item{3)}  There is an {\it identity element} $e$ in $G$ that satisfies
$e*x = x*e = x$ for all $x$ in $G$.
\item{4)}  For each $x$ in $G$ there is an inverse element $y$ in $G$  
that satisfies $x*y = y*x = e$.  
\bigskip
\noindent
(See class notes handout.) \bigskip  

The other thing that your
essay did not describe very clearly is the idea that
it is possible to {\it classify} the periodic plane patterns
by saying two patterns have the same type if their groups
of symmetries are ``the same'' (made up of the same
types of transformations).  This is where the 
classification into 17 types comes from.  Two patterns
of the same type just have ``the same'' symmetries.
The motifs or shapes that make up the pattern, the colors, 
and so forth can be different. 
\vfill
\eject
\noindent
{\it Andrew Marzo}
\bigskip
You certainly have almost all of the right elements I was looking for here.
The reasons I took off the two points are that there seems to be
a pair of slight misunderstandings here, if I'm ``reading you'' correctly.

First, it is not really correct to say that the mathematical
concept of a group is the algebraic generalization of ``a symmetry
transformation.''  The group properties can be thought of a 
a generalization of the properties of the collection of
{\it all transformations that leave an object or pattern
invariant}.  It's the properties of a collection of transformations
that are important here, not just the properties of a single
transformation.  

The second thing that your
essay did not describe very clearly is the idea that
it is possible to {\it classify} the periodic plane patterns
by saying two patterns have the same type if their groups
of symmetries are ``the same'' (made up of the same
types of transformations).  This is where the 
classification into 17 types comes from.  Two patterns
of the same type have ``the same'' symmetries.
The motifs or shapes that make up the pattern, the colors, 
and so forth can be different, though.
\vfill\eject
\noindent
{\it Kyle Kadlick}
\bigskip
The last property in the definition of a group is the requirement
that every element has an {\it inverse} -- property 4 below:  
\bigskip
\noindent
A group is
a collection $G$ of numbers or transformations, or ... with
an operation $*$, and the operation has to satisfy several 
properties:  
\bigskip
\item{1)}  For all $x,y$ in $G$, $x*y$ is in $G$.
\item{2)}  For all $x,y,z$ in $G$, $(x*y)*z = x*(y*z)$ (associativity)
\item{3)}  There is an {\it identity element} $e$ in $G$ that satisfies
$e*x = x*e = x$ for all $x$ in $G$.
\item{4)}  For each $x$ in $G$ there is an inverse element $y$ in $G$  
that satisfies $x*y = y*x = e$.  
\bigskip
\noindent
(See class notes handout.) \bigskip  

The other thing was that I was not quite sure what you 
meant by ``degrees of the transformation'' toward the end of your essay.
 Two patterns
of the same type have ``the same'' symmetries in the sense
that their groups of symmetries have the same structure and
are made up of the same types of transformations.  That 
means in particular that every {\it p4} pattern, for instance,
has rotational symmetries {\it only} for $0,90,180,270$-degree rotations.
The motifs or shapes that make up the pattern, the colors, 
and so forth can be different. 
\vfill\eject
\noindent
{\it Xuan Ha Nghiem}
\bigskip
The property in the definition of a group you did not mention
is the requirement that a group must have an {\it identity element} -- property 3 below:  
\bigskip
\noindent
A group is
a collection $G$ of numbers or transformations, or ... with
an operation $*$, and the operation has to satisfy several 
properties:  
\bigskip
\item{1)}  For all $x,y$ in $G$, $x*y$ is in $G$.
\item{2)}  For all $x,y,z$ in $G$, $(x*y)*z = x*(y*z)$ (associativity)
\item{3)}  There is an {\it identity element} $e$ in $G$ that satisfies
$e*x = x*e = x$ for all $x$ in $G$.
\item{4)}  For each $x$ in $G$ there is an inverse element $y$ in $G$  
that satisfies $x*y = y*x = e$.  
\bigskip
\noindent
(See class notes handout.) \bigskip  

The other thing was that I was not quite sure what you 
meant when you said the ``pattern might hinder a transformation
in one but not in the other.''  If two patterns are of the 
same type, then the transformations in their groups of 
symmetries are ``the same.''  If one
pattern has a 90-degree rotation symmetry about a 
point, then there has to be a corresponding point in
the other pattern around which the other also has 
90-degree rotational symmetry. 
\vfill\eject
\noindent
{\it Kaitlin O'Hagan}
\bigskip
 A group is
a collection $G$ of numbers or transformations, or ... with
an operation $*$, and the operation has to satisfy several 
properties:  
\bigskip
\item{1)}  For all $x,y$ in $G$, $x*y$ is in $G$.
\item{2)}  For all $x,y,z$ in $G$, $(x*y)*z = x*(y*z)$ (associativity)
\item{3)}  There is an {\it identity element} $e$ in $G$ that satisfies
$e*x = x*e = x$ for all $x$ in $G$.
\item{4)}  For each $x$ in $G$ there is an inverse element $y$ in $G$  
that satisfies $x*y = y*x = e$.  
\bigskip
\noindent
(See class notes handout.) \bigskip   You had a lot of this, but I was
looking for a bit more detail.  Also, notice that {\it commutativity}
(the statement that $x*y = y*x$ for all $x,y$) is {\it not included}.  In fact, the
groups of symmetries of 16 out of the 17 types of plane period patterns
are {\it not commutative}.  Translations (shifts) do not commute with rotations
and reflections.  
\vfill\eject
\noindent
{\it Max Pettinelli}
\bigskip
This is mostly very good.  However, I was looking for some more
detail in the discussion of the idea of a group.  
\bigskip
 A group is
a collection $G$ of numbers or transformations, or ... with
an operation $*$, and the operation has to satisfy several 
properties:  
\bigskip
\item{1)}  For all $x,y$ in $G$, $x*y$ is in $G$.
\item{2)}  For all $x,y,z$ in $G$, $(x*y)*z = x*(y*z)$ (associativity)
\item{3)}  There is an {\it identity element} $e$ in $G$ that satisfies
$e*x = x*e = x$ for all $x$ in $G$.
\item{4)}  For each $x$ in $G$ there is an inverse element $y$ in $G$  
that satisfies $x*y = y*x = e$.  
\bigskip
\noindent
(See class notes handout.) \bigskip
\bigskip
The other thing is that your
essay did not come out clearly and state the basic idea of
{\it classification} the periodic plane patterns (although you
were certainly saying parts of this, so that's why I only 
took off one point here).  Remember
we said that the  two patterns are put into the same type if their groups
of symmetries are ``the same'' (having the same structure
as groups and made up of the same types of transformations).  
This is where the classification into 17 types comes from.  Two patterns
of the same type have ``the same'' symmetries.
The motifs or shapes that make up the pattern, the colors, 
and so forth can be different, though.
\vfill\eject
\noindent
{\it Kevin Sutman}
\bigskip
What you have in the essay is good, but there are two particular
things that you did not discuss in enough detail.  First, 
part of the question asked about particular types of 
symmetry transformations that we looked at.  I was looking
for the types of transformations we applied to the 
patterns from the Alhambra or the Escher drawings to see
the invariance -- translations (shifts), rotations, reflections,
glide reflections.  Since you brought up the musical analogies,
you might have mentioned transposition, inversion, etc.  there.

The second thing that your
essay did not describe very clearly is the idea of
what a group is.  Recall that we said a group is
a collection $G$ of numbers or transformations, or ... with
an operation $*$.  To give a group, the operation has to satisfy several 
properties:  
\bigskip
\item{1)}  For all $x,y$ in $G$, $x*y$ is in $G$.
\item{2)}  For all $x,y,z$ in $G$, $(x*y)*z = x*(y*z)$ (associativity)
\item{3)}  There is an {\it identity element} $e$ in $G$ that satisfies
$e*x = x*e = x$ for all $x$ in $G$.
\item{4)}  For each $x$ in $G$ there is an inverse element $y$ in $G$  
that satisfies $x*y = y*x = e$.  
\bigskip
\noindent
(See class notes handout.) \bigskip  
\vfill
\eject
\noindent
{\it John Macomber}
\bigskip
This is mostly very good (although your handwriting is very hard to read!).  
However, I was looking for some more
detail in the discussion of the idea of a group.  
\bigskip
 A group is
a collection $G$ of numbers or transformations, or ... with
an operation $*$, and the operation has to satisfy several 
properties:  
\bigskip
\item{1)}  For all $x,y$ in $G$, $x*y$ is in $G$.
\item{2)}  For all $x,y,z$ in $G$, $(x*y)*z = x*(y*z)$ (associativity)
\item{3)}  There is an {\it identity element} $e$ in $G$ that satisfies
$e*x = x*e = x$ for all $x$ in $G$.
\item{4)}  For each $x$ in $G$ there is an inverse element $y$ in $G$  
that satisfies $x*y = y*x = e$.  
\bigskip
\noindent
(See class notes handout.) \bigskip
\vfill\eject
\noindent
{\it Thomas Santa Maria}
\bigskip
This is mostly good.  The deductions were based on the following
other things I was looking for.

First, I was looking for more detail about the 
mathematical definition of what a group is.  Recall we said:
\bigskip
 A group is
a collection $G$ of numbers or transformations, or ... with
an operation $*$, and the operation has to satisfy several 
properties:  
\bigskip
\item{1)}  For all $x,y$ in $G$, $x*y$ is in $G$.
\item{2)}  For all $x,y,z$ in $G$, $(x*y)*z = x*(y*z)$ (associativity)
\item{3)}  There is an {\it identity element} $e$ in $G$ that satisfies
$e*x = x*e = x$ for all $x$ in $G$.
\item{4)}  For each $x$ in $G$ there is an inverse element $y$ in $G$  
that satisfies $x*y = y*x = e$.  
\bigskip
\noindent
(See class notes handout.) \bigskip

The other thing is your
essay did not come out clearly and state the basic idea of
{\it classification} the periodic plane patterns (although you
were certainly saying parts of this, so that's why I only 
took off one point here).  Remember
we said that the  two patterns are put into the same type if their groups
of symmetries are ``the same'' (having the same structure
as groups and made up of the same types of transformations).  So 
there is no ``probably'' about it -- this is actually 
where the classification into 17 types comes from.  Two patterns
of the same type have ``the same'' group of symmetries (same algebraic
structure, same types of symmetry transformations).
The motifs or shapes that make up the pattern, the colors, 
and so forth can be different, though.  
\vfill\eject
{\it Muhammad Javaid}
\bigskip
\noindent
This is not bad.  The deductions were based on the following
other things I was looking for.

First, I was looking for more detail about the 
mathematical definition of exactly what a group is.  Recall we said:
\bigskip
 A group is
a collection $G$ of numbers or transformations, or ... with
an operation $*$, and the operation has to satisfy several 
properties:  
\bigskip
\item{1)}  For all $x,y$ in $G$, $x*y$ is in $G$.
\item{2)}  For all $x,y,z$ in $G$, $(x*y)*z = x*(y*z)$ (associativity)
\item{3)}  There is an {\it identity element} $e$ in $G$ that satisfies
$e*x = x*e = x$ for all $x$ in $G$.
\item{4)}  For each $x$ in $G$ there is an inverse element $y$ in $G$  
that satisfies $x*y = y*x = e$.  
\bigskip
\noindent
(See class notes handout.) \bigskip
\bigskip
The other thing was that I was looking for more of a discussion
of what types of transformations we looked at as symmetries
of the Escher drawings or Alhambra tilings.  The types of
transformations (hence the types of symmetries we found)
were based on translations (shifts), rotations, reflections,
and also glide reflections.  
\bye
