Weekly 2: Doing Math

Working with Tessellations

Tessellations can be defined as a repeated pattern on a flat that uses geometric shapes with no spaces or gaps. When I think of ones that I see in everyday life, they are present on floors, wall tiling, and tapestry.  There are patterns present everywhere.  The more intricate the pattern, the more angles and shapes become a factor.  The way the shapes interact is absolutely fascinating.  Below I used GeoGebra to test these interactions.  For the most part, I used circles (midpoints) and lines to create grids/boxes. 

This was the starting shape to
the first tessellation.

When overlapped, this pattern is created.  

For the second pattern, I used the same size circles, but with different frequency and placement.
It's amazing how similar, yet different the two look. 

This is what one segment of the pattern looks like blown up.
 It appears to be composed of individual curved shapes
and not the product of circles and lines.  

Next, I put lines through every vertex in every direction.  The lines created a horizontal and vertical grid with their diagonals. The new shapes that were created make this pattern more dynamic and visually appealing.

Tessellations introduce and explore many parts of mathematics.  Students can work with these at many different levels.  An important lesson that students can learn is how shapes can interact.  In the examples above, we could see that overlapping circles can create ovals that make a floral pattern.  Then, moving to the next example, we can see that a slight shift in position can create a pattern completely unlike the rest.  Students can be creative and make discoveries on their own. They can answer questions like: Is this a linear pattern?  What happens if one more shape is added? How could lines be utilized?  What if only circles or squares were used?  Can you make different shapes from the beginning?  What types of ideas have you explored?

These patterns provide an inside look at the role of angles, lines, and circles.  When simple shapes are put together and overlapped in a repeating patterns, it appears to create a complex image.  This was the greatest mathematical lesson that I gained.  Not only can a simple object be made to appear complex, but knowing that complexity can always be made simpler.