Part 14: Strip Patterns 

Strip or frieze patterns

 We now consider subgroups G of Iso(2) which contain a translation T such that every other translation in G is a power of T. They are the symmetry groups of repeated patterns in a line, also known as strip or frieze patterns.

  First consider the frieze patterns having only direct isometries.

  1. The simplest possibility is G = <T> = {Tn : n in Z}. So G iso Z. One strip having this isometry group is  

    2.  

     
     
     

         ... R R R R ...
     

  2. Let R be a half-turn, and let G = { TnRi : n in Z, i = 0 or 1}.  

    3.  

     
     
     

    A strip pattern having G as isometry group is

         ...p p p p ...
        ....d d d d....

    Now  consider patterns having indirect symmetries.
     
     

  3. Let M be reflection in the line of T, and let G = { TnMi : n in Z, i = 0 or 1}. Note that TM = MT, so G is abelian. In fact G is iso to the direct product of Z and Z(2). 

    4.  

     
     
     

    A strip pattern having G as isometry group is

         ...q q q q ...
        ....d d d d....

     Note that TM is also a glide.
     

  4. Let L be a reflection in a line perpendicular to the direction of T, and let G = { Tn Li: n in Z, i = 0 or 1}. 

    5.  

     
     
     
     
     

    A strip pattern having G as isometry group is

         ... d b d b d b ...

     Note that the distance between consecutive mirrors is half the length of T.
     

  5. Suppose G contains T, R and M as above so G = { TnRi Mk: n in Z, i = 0 or 1, k = 0 or 1}. 

    6.  

     

    A strip pattern having G as isometry group is

     ...qp qp qp qp ...
    ....db db db db....
     

  6. Let G contain T, R and L as above, so G = { TnRi Lk: n in Z, i = 0 or 1, k = 0 or 1}.

    7.  

     

     A strip pattern having G as isometry group is

     ...qp db qp db qp ...

      Note that the distance between consecutive rotation points is half the length of T.

     Note that this pattern also has vertical reflection symmetries, with the mirrors half way between the rotation points.
     

  7. Finally let H = TM be a glide and let G = <H> 
    8. = {Hn : n in Z}. So again G iso Z, but a strip having G for isometry group is 
     

     
     
     

        .... p p p p ...
        .......b b b b ...
     
     

Note that these give seven distinct patterns. If you try to adjoin any product of rotations, horizontal or vertical reflections to these, you get one of these seven types again. Hence these seven exhaust all possibilities.

 For example, the most general isometry has the form TnMiLk : n in Z, i = 0 or 1, k = 0 or 1. This is because ML = R = LM.
 
 

Recognition Table for Strip Patterns

Type p-rotation Horizontal Reflection Vertical Reflection Proper glide
1 No No No No
2 Yes No No No
3 No Yes No No
4 No No Yes No
5 Yes Yes Yes No
6 Yes No Yes No
7 No No No Yes

 
 

 

   .

Examples

 For some better pictures of the strip patterns, click here. .
. Table of Contents.

To Continue .. 

To return to Page 1 .

 Last update May 24, 2000

Author: Phill Schultz, schultz@maths.uwa.edu.au

 .