Tessellations and Symmetries

Date: 06/07/97 at 21:51:15
From: Joey Fang
Subject: Is it possible to make a tessellation with glide reflection, 
rotation, and translation all in one object?

Dr. Math,

My homework is to make a tessellation with a rotation, reflection, and 
translation all in one shape. Is this possible? I've used the 
TesselMania program from MECC, but it doesn't design 3-way 
tessellations, only 2-way ones.  My teacher said it was possible, but 
I'm stuck trying to make one.  Please help me out.  

Thanks a lot,
Joey

Date: 06/10/97 at 00:05:37
From: Doctor Sarah
Subject: Re: Is it possible to make a tessellation with glide 
reflection, rotation, and translation all in one object?

Hi Joey -

With the help of Suzanne Alejandre, whose Web tutorials on 
tessellations are hosted by the Math Forum, we passed your question 
along to an expert, Prof. Susan Addington of the Math Department at 
California State University, San Bernardino - see her home page:

 http://www.math.csusb.edu/faculty/susan/home.html   

Prof. Addington is the Director of the California Math Show, a 
portable, interactive math exhibit based on the idea of symmetry. She 
and technology teacher Alejandre have collaborated on web pages about 
symmetry in tessellations:

  http://mathforum.org/sum95/suzanne/wheremath.html   

Here's how Prof. Addington answered your question:

Yes, you can have tessellations that include three types of 
symmetries, or even all four types (reflection, rotation, translation, 
glide reflection).

There are two main ideas in the reason why:

1. In a symmetric pattern, if you have two symmetries, then you have 
   the combination (also called composition or product) of the 
   symmetries. For example, if your pattern has symmetries of rotation 
   by 45 degrees and rotation by 90 degrees, then it also has rotation 
   by 45 + 90 = 135 degrees. This pattern would also have rotation by 
   45, 90, 135, 180, 225, 270, 315, and 0 degrees. (Why?)

2. All 4 types of symmetry can be made by combining reflections.

   - A rotation is the composition of two reflections in intersecting 
     lines.
   - A translation is the composition of two reflections in parallel 
     lines.
   - A glide reflection is the composition of a translation and a 
     reflection, so it is the composition of 3 reflections.

For some exercises in understanding these facts, see:

  http://mathforum.org/sum95/suzanne/rex.html   

So if you build a tessellation that has, say reflections in two 
parallel lines and a perpendicular line, then the combinations will 
give you translations, glide reflections, and a 180-degree rotation.

TesselMania doesn't include any tessellations with reflection 
symmetry. You might want to try the free program Kali from the 
Geometry Center. You can run it over the web at

  http://www.geom.uiuc.edu/java/Kali   

or download a Macintosh version from:

  http://www.geom.uiuc.edu/software/download/kali.html   

A page with full information about all possible combinations of 
symmetries (there are 17 combination types) is:

  http://aleph0.clarku.edu/~djoyce/wallpaper/   

(Some parts are fairly technical (college level), but you can get an 
overview of the subject.)

-Doctor Sarah,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/