FRIEZE SYMMETRY (Part I)
                         ------------------------

In architecture, the term frieze refers to an ornately decorated band
running along the the top of a wall or other structure.  When the
decoration has a repetitive nature, that raise the mathematical question of
classifying its symmetry type.  It turns out that a frieze may possess one
of exactly 7 different symmetry types.

One may relate frieze symmetry to wallpaper symmetry in several ways:

1) One may view frieze symmetry as a ``toy'' version of wallpaper symmetry,
 a somewhat simpler context where we encounter similar issues.  Thus we can
 study friezes to reinforce our understanding of the plane symmetry
 classification;

2) One may extend a frieze pattern to a plane pattern in two distinct ways
 (see below), so one may use the classification of plane symmetry types to
 derive the classification for frieze symmetry types;

3) One may relate frieze symmetry types to symmetric patterns on the sphere.


         THE SEVEN FRIEZE TYPES AND A CATCHY WAY TO REMEMBER THEM
         --------------------------------------------------------

A trail of footprints counts as a frieze decoration.  The symmetry type
depends on the nature of the stride, as follows:



HOP
-----------------------------------------------------------------------------------------------

  .....  o             .....  o             .....  o             .....  o             .....  o
 .     . o            .     . o            .     . o            .     . o            .     . o
  ...  . o             ...  . o             ...  . o             ...  . o             ...  . o
     ..  o                ..  o                ..  o                ..  o                ..  o

-----------------------------------------------------------------------------------------------

JUMP
-------------------------------------------------------------------------------------------
  .....  o  .....  o  .....  o  .....  o  .....  o  .....  o  .....  o  .....  o  .....  o
 .     . o .     . o .     . o .     . o .     . o .     . o .     . o .     . o .     . o
  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o
     ..  o     ..  o     ..  o     ..  o     ..  o     ..  o     ..  o     ..  o     ..  o

     ..  o     ..  o     ..  o     ..  o     ..  o     ..  o     ..  o     ..  o     ..  o
  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o
 .     . o .     . o .     . o .     . o .     . o .     . o .     . o .     . o .     . o
  .....  o  .....  o  .....  o  .....  o  .....  o  .....  o  .....  o  .....  o  .....  o
-------------------------------------------------------------------------------------------

SIDE-STEP or SIDLE
-------------------------------------------------------------------------------------------
  oooo  oooo   oooo  oooo   oooo  oooo   oooo  oooo   oooo  oooo   oooo  oooo   oooo  oooo
   ..    ..     ..    ..     ..    ..     ..    ..     ..    ..     ..    ..     ..    ..
  .  .  .  .   .  .  .  .   .  .  .  .   .  .  .  .   .  .  .  .   .  .  .  .   .  .  .  .
  .  .  .  .   .  .  .  .   .  .  .  .   .  .  .  .   .  .  .  .   .  .  .  .   .  .  .  .
  . .    . .   . .    . .   . .    . .   . .    . .   . .    . .   . .    . .   . .    . .
  . .    . .   . .    . .   . .    . .   . .    . .   . .    . .   . .    . .   . .    . .
  . .    . .   . .    . .   . .    . .   . .    . .   . .    . .   . .    . .   . .    . .
   .      .     .      .     .      .     .      .     .      .     .      .     .      .
-------------------------------------------------------------------------------------------

WALK
-----------------------------------------------------------------------------------------------
  .....  o             .....  o             .....  o             .....  o             .....  o
 .     . o            .     . o            .     . o            .     . o            .     . o
  ...  . o             ...  . o             ...  . o             ...  . o             ...  . o
     ..  o                ..  o                ..  o                ..  o                ..  o
               ..  o                ..  o                ..  o                ..  o
            ...  . o             ...  . o             ...  . o             ...  . o
           .     . o            .     . o            .     . o            .     . o
            .....  o             .....  o             .....  o             .....  o
-----------------------------------------------------------------------------------------------

SPIN-JUMP
-----------------------------------------------------------------------------------
  .....  o   o  .....     .....  o   o  .....     .....  o   o  .....     .....  o
 .     . o   o .     .   .     . o   o .     .   .     . o   o .     .   .     . o
  ...  . o   o .  ...     ...  . o   o .  ...     ...  . o   o .  ...     ...  . o
     ..  o   o  ..           ..  o   o  ..           ..  o   o  ..           ..  o

     ..  o   o  ..           ..  o   o  ..           ..  o   o  ..           ..  o
  ...  . o   o .  ...     ...  . o   o .  ...     ...  . o   o .  ...     ...  . o
 .     . o   o .     .   .     . o   o .     .   .     . o   o .     .   .     . o
  .....  o   o  .....     .....  o   o  .....     .....  o   o  .....     .....  o
-----------------------------------------------------------------------------------

SPIN-HOP
-----------------------------------------------------------------------------------
  .....  o                .....  o                .....  o                .....  o
 .     . o               .     . o               .     . o               .     . o
  ...  . o                ...  . o                ...  . o                ...  . o
     ..  o                   ..  o                   ..  o                   ..  o
             o  ..                   o  ..                   o  ..
             o .  ...                o .  ...                o .  ...
             o .     .               o .     .               o .     .
             o  .....                o  .....                o  .....
-----------------------------------------------------------------------------------

SPIN-SIDLE
----------------------------------------------------------------------------------------
  oooo  oooo    .      .     oooo  oooo    .      .     oooo  oooo    .      .     oooo
   ..    ..    . .    . .     ..    ..    . .    . .     ..    ..    . .    . .     ..
  .  .  .  .   . .    . .    .  .  .  .   . .    . .    .  .  .  .   . .    . .    .  .
  .  .  .  .   . .    . .    .  .  .  .   . .    . .    .  .  .  .   . .    . .    .  .
  . .    . .   .  .  .  .    . .    . .   .  .  .  .    . .    . .   .  .  .  .    . .
  . .    . .   .  .  .  .    . .    . .   .  .  .  .    . .    . .   .  .  .  .    . .
  . .    . .    ..    ..     . .    . .    ..    ..     . .    . .    ..    ..     . .
   .      .    oooo  oooo     .      .    oooo  oooo     .      .    oooo  oooo     .
----------------------------------------------------------------------------------------


At this point, you should hardly find it obvious that no other symmetry type occurs!

Exercise:  Determine the symmetry type (one of the above)of the following
footprint frieze:

-----------------------------------------------------------------------------------
     ..  o   o  ..           ..  o   o  ..           ..  o   o  ..           ..  o
  ...  . o   o .  ...     ...  . o   o .  ...     ...  . o   o .  ...     ...  . o
 .     . o   o .     .   .     . o   o .     .   .     . o   o .     .   .     . o
  .....  o   o  .....     .....  o   o  .....     .....  o   o  .....     .....  o
-----------------------------------------------------------------------------------


(Beside helping us to remember the 7 types, this nomenclature suggests that frieze
symmetry can model the symmetries of dance steps, or equally well, repetitive
musical motifs.)

Now that we've met the 7 types, let's review some of their features.

HOP possesses only translation symmetry to the left and right.

JUMP possesses  a line of reflection running along the middle of the strip.

SIDLE possesses two sorts of reflection lines running perpendicular to the direction
of the strip.  Reflection occur between consecutive footprint.  One sort lies
close to the big toes, the other close to the little toes.

WALK possesses a line of  glide reflection running along the middle of the strip.

SPIN-JUMP possesses a line of reflection running along the middle of the strip.
as well as two types of perpendicular reflection lines, some near toes, some
near heels.

SPIN-HOP has no reflections or glide reflections, but does have two types of
order 2 gyration points.  Some lie midway between nearby big toes, some
midway between nearby heels.

SPIN-SIDLE has perpendicular reflection lines running midway between
consecutive footprints that point the same way, and gyration points of
order 2 that lie midway between oppositely oriented consecutive footprints.
(A line of glide reflection does run down the middle of the strip, but
since that line contain gyration points, it will not play any further role
in our analysis).


                                 NEW NAMES
                                 ---------

Just as will plane symmetric patterns, extracting a template and folding it
into a polyhedron form a crucial step in our understand of frieze symmetry.

For reason that shall only become clear as we go on, we now choose to think of
each strip as sitting inside a plane and work with the entire plane.  Thus
the templates will have infinite area; this marks frieze symmetry as distinct
from the plane symmetries we've already studied.

In the case of our standard examples, each template should contain one footprint.


The following diagram shows a template for HOP:


              |                    |
              |  Typical template  |
              |                    |
              |                    |
  .....  o    |        .....  o    |        .....  o             .....  o
 .     . o    |       .     . o    |       .     . o            .     . o
  ...  . o    |        ...  . o    |        ...  . o             ...  . o
     ..  o    |           ..  o    |           ..  o                ..  o
              |                    |
              | (extends up and    |
              | down infinitely)   |
              |                    |
              |                    |

To form a(n infinite) polyhedron from a template we just tape together the
edges to form an infinite cylinder.  Let us regard the two ends of the
cylinder as ``cone points'' of zero angle and infinite order.  For now you
should just take this as a suggestive convention.  (Of course there exists
no point located off at infinity at the ends of the cylinder, so we had to
put _cone points_ in quotes.) Since a typewriter can't make the standard
eight-on-its-side symbol for infinite, we'll abbreviate it as Inf.  So with
these conventions we get a new name for HOP imitating our plane symmetry
notation:

                                  Inf Inf




Now we turn to JUMP:

                    |         |
                    | Template|
                    |         |
                    |         |
                    |         |
                    |         |
                    |         |
  .....  o  .....  o| .....  o| .....  o  .....  o  .....  o  .....  o  .....  o  .....  o
 .     . o .     . o|.     . o|.     . o .     . o .     . o .     . o .     . o .     . o
  ...  . o  ...  . o| ...  . o| ...  . o  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o
     ..  o     ..  o|    ..  o|    ..  o     ..  o     ..  o     ..  o     ..  o     ..  o
                    -----------
     ..  o     ..  o     ..  o     ..  o     ..  o     ..  o     ..  o     ..  o     ..  o
  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o  ...  . o
 .     . o .     . o .     . o .     . o .     . o .     . o .     . o .     . o .     . o
  .....  o  .....  o  .....  o  .....  o  .....  o  .....  o  .....  o  .....  o  .....  o


This time we get a cylinder infinite in one direction and terminating at a
reflective edge in the other.  This suggests the notation


                                   Inf *





As for SIDLE, the template we get consists of an infinite strip with reflective
edges on each side. Now we should regard the ends as ``corner points'' or zero angle
and infinite order.

            |     |
            |     |
            |     |
  oooo oooo | oooo|oooo   oooo oooo   oooo oooo   oooo oooo   oooo oooo   oooo oooo
   ..   ..  |  .. | ..     ..   ..     ..   ..     ..   ..     ..   ..     ..   ..
  .  . .  . | .  .|.  .   .  . .  .   .  . .  .   .  . .  .   .  . .  .   .  . .  .
  .  . .  . | .  .|.  .   .  . .  .   .  . .  .   .  . .  .   .  . .  .   .  . .  .
  . .   . . | . . | . .   . .   . .   . .   . .   . .   . .   . .   . .   . .   . .
  . .   . . | . . | . .   . .   . .   . .   . .   . .   . .   . .   . .   . .   . .
  . .   . . | . . | . .   . .   . .   . .   . .   . .   . .   . .   . .   . .   . .
   .     .  |  .  |  .     .     .     .     .     .     .     .     .     .     .
            |     |
            |     |
            |     |

This suggests the notation

                                 * Inf Inf

Why just one * when the pattern has reflection lines of two types?  In the context
of plane symmetric patterns with finite templates, parallel lines never met, but
here we regard them as ``meeting at infinity'' with ``angle zero''.  So the two
line form parts of the same infinite boundary.   If this seems like cheating, or
seems arbitrary, have patience, the sense behind it all will emerge gradually.







Now a template for WALK reminds us of the template for HOP

                    |          |
                    |          |
                    |          |
  .....  o          |  .....  o|            .....  o             .....  o             .....  o
 .     . o          | .     . o|           .     . o            .     . o            .     . o
  ...  . o          |  ...  . o|            ...  . o             ...  . o             ...  . o
     ..  o          |     ..  o|               ..  o                ..  o                ..  o
               ..  o|          |    ..  o                ..  o                ..  o
            ...  . o|          | ...  . o             ...  . o             ...  . o
           .     . o|          |.     . o            .     . o            .     . o
            .....  o|          | .....  o             .....  o             .....  o
                    |          |
                    |          |
                    |          |

except now we must tape together with a twist.  The polyhedron we get
clearly contains a Mobius strip.  It also has an infinite order ``cone point'',
but we won't try to visualize it.  In any case we should give it the name

                                   Inf x




SPIN-JUMP presents no difficulties.  The template looks like

           |           |
           |           |
           |           |
  .....  o | o  .....  |  .....  o   o  .....     .....  o   o  .....     .....  o
 .     . o | o .     . | .     . o   o .     .   .     . o   o .     .   .     . o
  ...  . o | o .  ...  |  ...  . o   o .  ...     ...  . o   o .  ...     ...  . o
     ..  o | o  ..     |     ..  o   o  ..           ..  o   o  ..           ..  o
           ------------
     ..  o   o  ..           ..  o   o  ..           ..  o   o  ..           ..  o
  ...  . o   o .  ...     ...  . o   o .  ...     ...  . o   o .  ...     ...  . o
 .     . o   o .     .   .     . o   o .     .   .     . o   o .     .   .     . o
  .....  o   o  .....     .....  o   o  .....     .....  o   o  .....     .....  o


with all edges reflective.  We have, plainly visible, two order 2 corner points,
and a zero angle infinite order ``corner point'' at the end of the template.
Thus the name

                                 * 2 2 Inf




With SPIN-HOP a template might look

                                                            or like this.
              either like this

           |                       |                       |           |
           |                       |                       |           |
           |                       |                       |           |
           |                       |                       |           |
           |                       |                       |           |
  .....  o |              .....  o |              .....  o |           |  .....  o
 .     . o |             .     . o |             .     . o |           | .     . o
  ...  . o |              ...  . o |              ...  . o |           |  ...  . o
     ..  o |                 ..  o |                 ..  o |           |     ..  o
           -------------------------                       |           |
             o  ..                   o  ..                 | o  ..     |
             o .  ...                o .  ...              | o .  ...  |
             o .     .               o .     .             | o .     . |
             o  .....                o  .....              | o  .....  |
                                                           |           |
                                                           |           |


To form the polyhedron we would either crease the first template vertically and
tape the edges together, or crease the second template horizontally and tape.
Either way we get an infinite (semi-)rectangular pillowcase (this illustrates
why we emphasize polyhedra over templates) and the name

                                  2 2 Inf





Something similar happens with SPIN-SIDLE.  We could have a template

                                    |      |
                                    |      |
                                    |      |
                                    |      |
                                    |      |
  oooo   oooo    .       .     oooo | oooo |  .       .     oooo   oooo    .       .
   ..     ..    . .     . .     ..  |  ..  | . .     . .     ..     ..    . .     . .
  .  .   .  .   . .     . .    .  . | .  . | . .     . .    .  .   .  .   . .     . .
  .  .   .  .   . .     . .    .  . | .  . | . .     . .    .  .   .  .   . .     . .
  . .     . .   .  .   .  .    . .  |  . . | .  .   .  .    . .     . .   .  .   .  .
  . .     . .   .  .   .  .    . .  |  . . | .  .   .  .    . .     . .   .  .   .  .
  . .     . .    ..     ..     . .  |  . . |  ..     ..     . .     . .    ..     ..
   .       .    oooo   oooo     .   |   .  | oooo   oooo     .       .    oooo   oooo
                                    |      |
                                    |      |
                                    |      |
                                    |      |
                                    |      |


and again we should crease horizontally.  But now we have a reflective edge on the left
which we shouldn't tape, so we tape only on the right.  We'll get an order 2 cone point
on the right and an infinite order ``corner point'' at the end of the strip.  Hence
the name

                                  2 * Inf


                     THE PARALLELS WITH PLANE SYMMETRY
                     ---------------------------------_

A look at the names we've just derived shows that the satisfy a logic familiar to
us for plane symmetry.  We have two names which feature only gyration points:

                                  2 2 Inf
                                  Inf Inf

We can place a  *  in front of each name

                                 * 2 2 Inf
                                 * Inf Inf

and then move pairs of corner points left, across the  *  , to make a gyration point.

                                   Inf *
                                  2 * Inf

Finally we can replace a  *  with no corner points by a  x  :

                                   Inf x


The way this story parallels our previous one suggests that we're doing
something right.  In fact we can push the parallel further.  Returning
to the idea of building polyhedra by modifying a sphere and ``charging''
ourselves for various features, we could charge  $1 for a ``cone point'' Inf
and $.50 for a ``corner point'' Inf and we'd get exactly these 7 ways to
spend $2 and use one of the new coins.

Since a cone point Inf costs the same as a * , we can reuse our old
analysis.  We had three types with at least one * with no corner points:
**, *x, and 22*.  Changing one or more *'s to Inf's gives Inf Inf , Inf * ,
Inf x and 2 2 Inf.  Now we can enumerate the posibilities that use corner
point Inf from these, just as before.

So what's the status of all this??  We still haven't proved the completeness
of our classification.  We have no Euler theorem or Gauss-Bonnet theorem
for infinite polyhedra, so we can't simply lift the old argument.  Still,
everything appears to work the same way, and that deserves an explanation.