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.


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.



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.


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

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
pm bd
cm bp|pb
cmm bdpq|pqbd
pg bp
pgg bp|dq
pmg bd|qp
pmm bd|pq
p4 bb+|q+q
p4g bdp+b+|pqq+d+|p+b+bd|q+d+pq