10182

# Border Patterns

A border pattern, or frieze pattern, is a design obtained from a motif by repeating that motif infinitely often in two opposing directions along a strip. Border patterns are traditionally oriented horizontally, thus the patterns are understood to continue both to the left and right without end. A border pattern falls into one of seven symmetry types. This Demonstration provides one example of each type to let you explore the effects of reflection, rotation, and translation on those patterns.

### DETAILS

When you select the "show highlighted motif" option, one motif appears in blue and the one immediately to its left appears in red. This lets you see the action of a reflection, rotation, or translation more easily. The axes of reflection used go through the center point of the blue motif, and that point is the center of the rotations as well. Note that in the case of a border pattern of type mg the vertical axis over which the border pattern has symmetry is not through the center of the motif (see Snapshot 5).
All border patterns have translational symmetry. Border patterns are classified according to which other symmetries they exhibit. The seven symmetry types of a border pattern are:
11: no other symmetry
m1: reflection over a vertical axis, but not a horizontal axis
1m: reflection over a horizontal axis, but not a vertical axis
12: rotation by 180 degrees, but no reflection symmetry
mm: reflection over both a horizontal axis and a vertical axis (which implies rotational symmetry)
1g: glide reflection, but no reflection
mg: reflection over a vertical axis and glide reflection (which implies rotational symmetry)

### PERMANENT CITATION

Contributed by: Marc Brodie
(Wheeling Jesuit University)
 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Curriculum Standards

US Common Core State Standards, Mathematics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.