Monday, June 9, 2014

Weekly #2- Geometric Tessellations: Doing Math

In class we began working with different pattern blocks to make patterns that make a tessellation. This was a very interesting part to the semester because we were able to be creative and design our very own pattern.

I worked with semi-regular tessellations to create different patterns. Semi-regular tessellations include two or more regular polygons that have a similar pattern around each shape. In a semi-regular tessellation there will be no gaps within the shape. Below is the design I created during class.

To extend on the shape above, I have created a sketch of the overall pattern showing the tessellation created. As you can see the pattern works for the tessellation.

I found this website that shows how to use Paint program to create geometric tessellations on the computer. This website shows how to make a tessellation with an irregular shape. So I used Paint to create a tessellation using geometric figures.

I began looking at regular tessellations. A regular tessellation is a repeated pattern of a regular geometric shape where the vertices of each shape touch. Here is an example of a regular tessellation I created using a square.

As you can see this tessellation creates a pattern that looks similar to a checkers/chess board. 

I also found this website. This website says that there are exactly three regular tessellations. This seemed interesting to me, why are there only three regular shapes that allow a tessellation pattern.

I began thinking about these three regular shapes, the triangle, square, and the hexagon. It made me wonder if the reason why these three shapes work for a tessellation was because of symmetry, so I used an octagon to create a tessellation and below is my attempt to create a regular tessellation.

This concludes that there are only the three regular shapes that create these regular tessellations. So, then I began playing around with the regular shape the hexagon. Below is an example of a tessellation created from four hexagons stacked on top of one another.

I began the next tessellation using triangles and added a square in the center of the middle triangle for a visual effect.

I stacked the triangles up and created a larger triangle that could be stacked on top of one another. Shown below is the stacked triangles that I used to create the tessellation.

From here I realized that these stacked triangles could be flipped 180 degrees to fit exactly in the empty space of the figure. Shown below is the finished tessellation.
I also played around with the star shape found in Paint. Below is the tessellation I created.

But so what? What is so important about tessellations? I would say that tessellations are more common than you think. There are many examples where you can see tessellations intertwined throughout our everyday lives.

The bee hive shown below is an example of a tessellation found in nature.

This tiled floor is another example of a tessellation found in our everyday lives. It is crazy to think that even as a child I would notice the patterns on the floor and follow them as they repeated. The patterns I once saw are considered tessellations. I think that tessellations are very interesting and they are packed with lots of information.

I think it is important to have a student complete their very own tessellation, which allows them an opportunity to work and become familiar with the basic shapes found within their tessellation. Tessellations also provide examples of rotated images where students can learn about the different degrees and the mathematics involved in creating the tessellations. It is important to remember that tessellations are apart of math and it is also important to stop and enjoy their beauty.


  1. Content: how would you extend the pattern block design? What do all these examples have in common, or what did you get from making them?
    Consolidated: You could extend the what did you get idea to make a summary, or think about the 'so what' for these examples or consider 'now what' - what would you do next?

    1. I really like the idea of using pattern blocks to work with semi-regular tessellations. I agree with John Golden, in that you could extend the idea to have student think about the "so what". I am an 11th Grade math teacher and I have done a larger project with my students in which they have to design their own tessellation using Geometer's Sketchpad. This entails an understanding in transformations, interior angles of a polygon and I differentiated by creating different roles: some students had to design a mutated figure that would tessellate with an equilateral triangle, square, regular hexagon, irregular triangle, and irregular quadrilateral. I am stuck in how to make this project more authentic to the students though. Although it is true that tessellations can be found both in the natural world as well as in more synthetic (man-made) products/ art/architecture. I feel something is missing in my project that requires them to take it further than just designing their own. I am thinking about how I could create certain parameters in which the students will have to fill a finite plane of some shape and they will have to make some sort of prediction.. I'm still thinking about how to move forward on this though. As for the honey bees an interesting thing to look into is why do honey bees use regular hexagons rather than other regular polygon that tessellates-- it has to do with optimizing the amount of honey a regular hexagon stores. Here's a nice article that may give some ideas that students could look into to understand the purpose of tessellations in our natural world

    2. I would love to hear any input or ideas anyone has!!

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