2D Tessellation

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					1.0   What is tessellation?


      1.1    Definition


A tessellation is created when a shape is repeated over and over again covering a

plane without any gaps or overlaps. Another word for a tessellation is a tiling.


A dictionary* will tell you that the word "tessellate" means to form or arrange small

squares in a checkered or mosaic pattern. The word "tessellate" is derived from the

Ionic version of the Greek word "tesseres," which in English means "four." The first

tilings were made from square tiles.


A regular polygon has 3 or 4 or 5 or more sides and angles, all equal. A regular

tessellation means a tessellation made up of congruent regular polygons. [Remember:

Regular means that the sides of the polygon are all the same length. Congruent means

that the polygons that you put together are all the same size and shape.]


Only three regular polygons tessellate in the Euclidean plane: triangles, squares or

hexagons.We can't show the entire plane, but imagine that these are pieces taken from

planes that have been tiled. Here are examples of


                     a tessellation of triangles


                     a tessellation of squares


                     a tessellation of hexagons




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When you look at these three samples you can easily notice that the squares are lined

up with each other while the triangles and hexagons are not. Also, if you look at 6

triangles at a time, they form a hexagon, so the tiling of triangles and the tiling of

hexagons are similar and they cannot be formed by directly lining shapes up under each

other - a slide (or a glide!) is involved.


You can work out the interior measure of the angles for each of these polygons:


                    Shape                    Angle measure in degrees


                    triangle                 60

                    square                   90

                    pentagon                 108

                    hexagon                  120

                    more than six sides      more than 120 degrees


Since the regular polygons in a tessellation must fill the plane at each vertex, the interior

angle must be an exact divisor of 360 degrees. This works for the triangle, square, and

hexagon, and you can show working tessellations for these figures. For all the others,

the interior angles are not exact divisors of 360 degrees, and therefore those figures

cannot tile the plane.




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       1.2     Naming Conventions


A tessellation of squares is named "4.4.4.4". Here's how: choose a vertex, and then look

at one of the polygons that touches that vertex. How many sides does it have?


Since it's a square, it has four sides, and that's where the first "4" comes from. Now

keep going around the vertex in either direction, finding the number of sides of the

polygons until you get back to the polygon you started with. How many polygons did you

count?


There are four polygons, and each has four sides.




For a tessellation of regular congruent hexagons, if you choose a vertex and count the

sides of the polygons that touch it, you'll see that there are three polygons and each has

six sides, so this tessellation is called "6.6.6":




A tessellation of triangles has six polygons surrounding a vertex, and each of them

hasthree sides: "3.3.3.3.3.3".




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      1.3      Semi-regular Tessellations


You can also use a variety of regular polygons to make semi-regular tessellations. A

semiregular tessellation has two properties which are:


            1. It is formed by regular polygons.

            2. The arrangement of polygons at every vertex point is identical.


Here are the eight semi-regular tessellations:




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Interestingly there are other combinations that seem like they should tile the plane

because the arrangements of the regular polygons fill the space around a point. For

example:




If you try tiling the plane with these units of tessellation you will find that they cannot be

extended infinitely. Fun is to try this by yourself.


           1. Hold down on one of the images and copy it to the clipboard.

           2. Open a paint program.

           3. Paste the image.

           4. Now continue to paste and position and see if you can tessellate it.




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2.0       My Tessellation


                             Explanation’s About My Tessellation


Step 1 :
         Firstly, I draw a prefect square shape on the blank paper with pencil.




Step 2 :
         Then, I draw an additional shape on left side of the square.




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Step 3 :
      Next, I trace the shape I had drew in step 2, and copy it on the opposite side,
       inside the square.




Step 4 :
      After that, I draw another simple shape on the top of the square.




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Step 5 :
      I reflect back down the shape and we get a combination of the shape.
      The tessellation can look by the shape in square like an arrow face each other.




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Step 6 :
      Then, I trace the entire new shape, and I move it over horizontally if done
       correctly, the shapes will interlock perfectly.
      Create the tessellation pattern by repeating the shape as many times.




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Step 7 :
      Lastly, I colored the shape with a significant color.




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Why I choose red and blue color as my tessellation colour??

Firstly, red and blue are bright, cool and interesting colour. Then, this colour are the two
colour that can be found in our national flag. This colour remind me of important to live
with our nationality. Besides, many motivators stated red and blue are colour that can
approach somebody to become more spiritual and comfortable with many
circumstances. So, this is a big reason I choose this colour. Other than that, I like this 2
colour. It is very nice to see this colour combine to each others.




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REFLECTION

      Firstly, I would like to thank Mr Pusparajan as my Numerical Literacy lecturer to
give this assignment for me to handle it. Actually, there were a lot of problem to settle
this assignment. When I got the paper works, I was very anxious because I still had
many assignments to settle. If I had my own broadband to find some sources, maybe it
can decrease my anxious but I don’t had it. So, I must worked fast to settle other
assignment. I tried to understand the question first. I made a discussion with my friends
such as Ammar, Prabhu, Azman, Afiq and Zahir to settle this problem.

      I struggled to settle the problem. But, the other problem was arrived. It is because
of the aerobic exercise. I have to join the training every afternoon. Many days went
through for the aerobic training until 10/ 8 / 2009. On weekend I don’t had any training
and I had a little time to finish this assignment. Then, I struggled to finish this work.

      I felt very grateful I had finished my work but the other problem was I don’t have a
notebook to type the work as Mr Pusparajan wanted. My notebook was being formatted
at my friend who knew how to format the notebook. Luckily, my friend that had finished
his works would like to borrow me his notebook. I struggled to type the work because
the time had approached me.

      It was a grateful moment for me when I managed to settle the works on time. I
would like to thank my friends that always gave some courage for me to make this
works. I also would like to thank Mr Pusparajan again because let me to handle this
assignment. From this, I had learned many new things and my relationship with my
friends also become more tighten.

       Other than that, from the task given, I got a lot of more information regarding on
my topic. It is very useful for me because I learned the important of language for the
future. I hope by learning Numerical Literacy in English language can improve my
English. Thank you.




                                                          PREPARED BY AFIQ

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REFERENCES

  (a) Internet


     Accessed on 10th August 2009
        o    (images)

             http://web.ukonline.co.uk/polyhedra/tessellations/type14.htm

        o    (Totally Tessellated!)

             http://library.thinkquest.org/16661/

        o    (wasp article & picture)

             http://www.ppdl.purdue.edu/ppdl/weeklypics/8-23-04.html

        o    (art definitions)

             http://www.artlex.com/

        o    (dictionary)

             www.m-w.com



  (b) Book



  Sally Wehmeier, Colin Mclntosh, Joanna Turnbull and Michael Ashby, 2005.

     Oxford Advanced Learner’s Dictionary. United States: Oxford University Press.




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