The Wonderful World of Tessellations

I apologize for this being turned in so late. I was looking through my canvas the other day and I saw I didn't have a grade for this blog even though I remembered writing about it. I went to my blog and somehow I made the blog private and saved it in my drafts instead of publishing it. I apparently also forgot to email my link to you. I am so sorry for this being turned in so late! - Seth

For this blog I decided to explore the topic of tessellations.  The earliest time I can remember seeing tessellations was in 6^th grade.  I remember seeing a poster of a sketch by MC Escher. Here is a link to that piece of work: http://brettworks.files.wordpress.com/2012/04/escher_reptiles.jpeg

Anyway, I remember looking at the reptile tessellation in that sketch nearly everyday during my first year at the Academy.  Eventually I became fairly interested in tessellations and did some research of my own into the subject. I found out that there is only three tessellations that are composed of regular polygons that symmetrically tile.  These three shapes are triangles, squares, and hexagons. This is because each of these shapes has interior angles that are divisible by 360 (60, 90, and 120 respectively). Thus, only a shape with an interior angle that is a divisor of 360 can tessellate.  

            

One tessellation of this kind that is seen frequently in nature is the hexagonal tessellation. This is more commonly recognized as the honeycomb. Bees tessellate their hives for many reasons. One of the reasons is because it allows for maximum storage within a small space.

            

However, there are other types of tessellations. There are 8 tessellations that combine two or more regular polygons in the same order each vertex are known as semi-regular tessellations. I have also seen these referred to as Archimedean tessellations. Here is an awesome picture of semi-regular tessellated tile: http://euler.slu.edu/escher/upload/3/30/Semi-regular-Seville.JPG

            

Finally, there are 14 possible combinations for demi-regular tessellations. These are composed of the eight semi-regular and three regular tessellations. One place where this kind of tessellation is prevalent is in Islamic architecture and art. I found this neat picture: https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzbsBH5pOmBatDgnq6MJrjCafYEgAs3NFCrG7OgVU8oVu3UGsjZc9z57uWoHvKtfXAkxX-QP7BmD_I_4d7Jv4ZkI9LCUGz9UZXm7RkGaCC2HIgFdQObgcqpGFh1a87Y6vczqajqAPpMyg/s1600/DSC00101.JPG