What's The Math Behind Tesselations?
There is a bit of geometry that goes on behind the construction of a tesselation. We will be using words like rotation, reflection, glide reflection and translation. We will be discussing each in detail so that noone will be lost.
Notice in the flash movie, we see first a circle that is being rotated by ninety degrees and we also see another circle that is appears on the opposite side of the line. We call this a reflection through that center line. It is just like looking into a mirror. The last point that the movie emphasizes is the translation or glide. Think about the glide as moving away from the original position in a straght line. The glide could be in any direction not just the one dipicted in the movie.
Rotation
In this image we can clearly see rotation by 90 degrees. By definition every rotation has a center and an angle of rotation. Here we see point P at the top of the circle before the rotation and then at the right side of the circle after the rotation through 90 degrees.
Translation
To translate an object means to move it without rotating or reflecting it. Every translation has a direction and a distance.
Reflection
Here we can see the reflection of the muliti-colored square
through the line L. To reflect an object means to produce its mirror image.
Every reflection has a mirror line. A reflection of an square is a backwards
square.
Glide
A glide reflection combines a reflection with a translation along the direction of the mirror line. Glide reflections are the only type of symmetry that involve more than one step.
Symmetries create patterns that help us organize our world conceptually.
Symmetric patterns occur in nature, and are invented by artists, craftspeople,
musicians, choreographers, and mathematicians.
In mathematics, the idea of symmetry gives us a precise way to think about
this subject. We will talk about plane symmetries, those that take place on
a flat plane, but the ideas generalize to spatial symmetries too.
Plane symmetry involves moving all points around the plane so that their positions relative to each other remain the same, although their absolute positions may change. Symmetries preserve distances, angles, sizes, and shapes.
For example, rotation by 90 degrees about a fixed point is an example of a
plane symmetry.
Another basic type of symmetry is a reflection. The reflection of a figure in the plane about a line moves its reflected image to where it would appear if you viewed it using a mirror placed on the line. Another way to make a reflection is to fold a piece of paper and trace the figure onto the other side of the fold.
A third type of symmetry is translation. Translating an object means moving it without rotating or reflecting it. You can describe a translation by stating how far it moves an object, and in what direction.
The fourth (and last) type of symmetry is a glide reflection.
A glide reflection combines a reflection with a translation along the direction
of the mirror line.
Resource Page: http://mathforum.org/sum95/suzanne/symsusan.html
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