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Tessellations and Symmetries

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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

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

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

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

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/
```
Associated Topics:
High School Geometry
High School Symmetry/Tessellations

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