# Symmetry Group Links and Notation

There are many great resources on the 17 plane symmetry groups.

I made great use of a couple of on-line apps while writing my Symmetry Tile plug-in.

## Morenaments

This great Java applet can either be used here on-line or downloaded as a `jar`

file and run locally.

It allows you to draw on a canvas that automatically completes the chosen symmetry group.Be sure to investigate the menu on the top-right. Selecting tile and/or cell under the grid menu will show these on the pattern canvas and help with seeing the structure underlying a particular symmetry. Look in the manual under help for more information. Be sure to try dragging the coloured dots in the tile view around. These can show how the same symmetry pattern can have different shaped cell or tile.

## Kali

This on-line Java applet can be used here. Its much easier to do straight lines in this app. It uses the orbifold notation for the symmetry groups.

## Books

The two books I consulted the most while working on this plug-in were:

- "Handbook of Regular Patterns: An Introduction to Symmetry in Two Dimensions" by Peter S. Stevens
- "Designing Tessellations: The Secret of Interlocking Patterns" by Jinny Beyer

All of these references use different notations and descriptions for the symmetry groups. I've summarised them in the following table for easy reference. The Symmetry Tile plug-in uses the notation in the left most column.

## Notation for Symmetry Groups

Crystallography | full | Terrazo | Jinny Beyer's description | Orbifold | Peter S. Stevens's description |
---|---|---|---|---|---|

p1 | p1 | Gold Brick | Translation | o | Two Nonparallel Translations |

p2 | p211 | Hither & Yon | Midpoint or Half-Turn Rotation | 2222 | Four Half-Turns |

pm | p1m1 | Wings | Mirror | ** | Two Parallel Mirrors |

pg | p1g1 | Card Tricks | Glide | xx | Two Parallel Glide Reflections |

pgg | p2gg | Honey Bees | Double Glide | 22x | Two Perpendicular Glide Reflections |

pmm | p2mm | Prickly Pear | Double Mirror | *2222 | Reflections in Four Sides of a Rectangle |

pmg | p2mg | Lightning | Glided Staggered Mirror | 22* | A Mirror and a Perpendicular Reflection |

cm | c1m1 | Crab Claws | Staggered Mirror | *x | A Reflection and a Parallel Glide Reflection |

cmm | c2mm | Spider Web | Staggered Double Mirror | 2*22 | Perpendicular Mirrors and Perpendicular Glide Reflections |

p4 | p4gm | Pinwheel | Pinwheel or Quarter-Turn Rotation | 442 | Quarter-Turns |

p3m1 | p3m1 | Winding Ways | Mirror and Three Rotations | *333 | Reflections in an Equilateral Triangle |

p3 | p3 | Storm at Sea | Three Rotation | 333 | Three Rotations through 120° |

p4g | p4gm | Primrose Path | Mirrored Pinwheel | 4*2 | Reflections of Quarter-Turns |

p4m | p4mm | Sunflower | Traditional Block | *442 | Reflections on the Sides of a 45°-45°-90° Triangle |

p6 | p6 | Whirlpool | Six Rotation | 632 | Sixfold Rotation |

p31m | p31m | Monkey Wrench | Three Rotations and a Mirror | 3*3 | Refections of 120° Turns |

p6m | p6mm | Turnstile | Kaleidoscope | *632 | Refections in the Sides of a 30°-60°-90° Triangle |

The symmetry groups that can be made with rectangular or square cells can be defined using the "bdpq" notation. If the "bdpq" string contains a plus sign then the cell must be square. The strings for all the symmetry groups start with a "b". Each of the symmetry groups could be created with an alternative string starting with one of the other letters.

The 32 strings here are this produced by the Symmetry Tile plugin when "all square cells" is selected and "Multiple" is set to "Yes". When "Multiple" is set to "No" only one string from each symmetry group is used.

Symmetry Group | bdpq string |
---|---|

p1 | b |

p2 | bq |

b|q | |

bq|qb | |

pm | bd |

b|p | |

cm | bp|pb |

bd|db | |

cmm | bdpq|pqbd |

bd|qp|db|pq | |

bqpd|pdbq | |

bd|pq|db|qp | |

pg | bp |

b|d | |

bd+|d+b | |

bp+|p+b | |

pgg | bp|dq |

bq|dp | |

bp|qd | |

pmg | bd|qp |

b|p|d|q | |

b|q|d|p | |

bdpq | |

bqpd | |

bq|pd | |

pmm | bd|pq |

p4 | bb+|q+q |

bq+|b+q | |

p4g | bdp+b+|pqq+d+|p+b+bd|q+d+pq |

bdd+q+|b+p+pq|d+q+bd|pqb+p+ | |

bb+p+d|q+qpd+|p+dbb+|pd+q+q | |

bq+d+d|pp+b+q|d+dbq+|b+qpp+ |

## Comments !