Basic Tessellation: [Bio-hazard Typhoon]

Master Tile

In order to create our master tile, we used the line method.

Step 1: Create a line with a design of your choice. Make sure it is a continuous line so that when you draw it you never have to pick up your pencil.

Step 2: Choose one of the endpoints of your line to be the center for rotation. We choose the endpoint where the star is located on our line created in step 1.

Step 3: Rotate your line around this point 60 degrees. Step 4: Now �nd the midpoint of the line segment created when you connect the two ends of our orginal line and the rotated line that are adjoined. (The two stars indicate the ends that need to be connected in order to �nd the midpoint.)

Step 4: Create a line segment that is from one endpoint of our original line to the midpoint that we just found. Design this line any way you desire.

Step 5: Use the midpoint just found as the new center of rotation. Rotate the newly designed line 180 degrees around this point. *You have now created your own master tile.

In order to make our tesselation we �rst started out with our single master tile.

1) We then choose a single point of rotation, marked by the star.

2) From here we copied our master tile and then selected the tile to rotate 60 degrees about this point. We continued this form of rotation six times until we formed a hexagon from our master tile.

3) The next step we took in created our tessela-tion was choosing one of the tiles in our hexagon and rotating the �gure 180 degrees at the midpoint we had found earlier when originally designing our master tile. (The star indicates the center of rotation.)

4) We followed these patterns of rotations over and over again until we had created a full page of our master tiles.

60°60°

60°60°60°

60°

180°

Advanced Tessellation: [Implode/Explode]

Master Tile

In order to create our master tile, we had done the following procedure.

Step 1: We created a line of any length and designed it in any way that we wanted to. (The length that we choose ultimately decides the length of all the other sides because we based our design o� of a square.)

Step 2: We translated this line to the right. To determine the appropriate length to move it, we used a vector that is equal to the length of our newly created line.

Step 3: Then create a segment that connects the points where the stars are located on our diagram to the right. Be creative when designing this line.

Step 4: Once you have it created you are going to re�ect this line across the vertical line that runs through the midpoint. (you will have to �nd the midpoint of the line segment you created)

Step 5: Now translate the re�ected line down. To determine the appropriate length to move it, you must use a vector that is equal to the length of the sides that are already formed.

Step 6: You have now created your master tile. Every side of your tile should be of equal length. It will resemble a square.

In order to make our tesselation we �rst started out with our single master tile.

1) We then found the midpoint of the top line segment in our master tile which is marked by the star.

2) Create a vertical re�ection line through our master tile.

3) Create a copy of your master tile, right on top of the orginial. Re�ect the copy across the vertical line.

4)Translate the copy downwards. Use the same vector length that you have already used when orginally creating the master tile.

5)Continue this process of re�ecting and translating till you are satis�ed with the number of tiles.

6)Finally, translate your row of tiles to the right with the same vector distance you previously used. Do this process until you have �lled the page as you desire.

7) The result is a glide re�ection tesselation.

KatieRubenCalebAugspurger

November20,2013

1) Although tessellations have been around for centuries, they have not been studied in

mathematics for that long. A tessellation, or tiling, is when you fit shapes together to completely

cover the plane. The most basic tessellations use regular polygons. For example, regular

tessellations are made by repeating a single regular polygon over and over again, while semi-

regular tessellations are configured using two or more regular polygons. But these tiling’s can be

made with other shapes as well, even shapes that are not polygons. One example is a method

called the slice method. In this method, you start with a figure that you know tessellates, such as

a square or kite. Then, cut out an interesting pattern from one side, and translate it directly to the

opposite side. This will tessellate the entire plane. Here is an example of this slice method:

Figure 1 www.tessellations.org/methods-diy-papercut.shtml

There is quite an opportunity to encourage mathematical engagement in students. While

it may seem as if tessellations are simply forms of art, there is mathematical reasoning behind it.

For example, when students try to completely fill a page, they will be forced to use different

isometries (rotations, translations, reflections, and glide reflections). Younger students can get

exposure to these new concepts, while as students get older, they can be challenged with new

problems. For example, students could be asked to determine which single polygons could tile a

page and why. Or why any triangle and any quadrilateral can tile that page. Tessellations can

also be an effective way to introduce direct and indirect isometry, especially with younger

students. When students try to fill a page with tiles, the figure sometimes needs to be flipped

KatieRubenCalebAugspurger

November20,2013over (reflected). This can then help students understand direct versus indirect isometry. In their

explorations, they will deal with concepts such as the interior angles and what the word ‘angle’

means. Many times, students will learn a book definition of terms and never truly understand

what they mean. Activities such as these lend themselves to student exploration, and if properly

encouraged by teachers, can lead to a much deeper understanding of geometry.

2)Tessellations can be found in many different places, including being found in nature.

For example, if you look at a honeycomb, you will see

that bees create hexagons that completely cover the

honeycomb. Because regular hexagons can tile the entire

plane by themselves, there is very little wasted space.

Hence, hexagons are very efficient for honey bees to use.

What makes this even more intriguing is the fact that

hexagons are one of three regular polygons that tile the entire place by themselves and are the

most efficient in terms of area to perimeter (or volume to surface area). Simply speaking, it is

very remarkable what bees have accomplished. Another example of tessellations in society is

their use in the creation of stained glass. Some religions, including the Islamic religion, forbid

the use of representational objects on religious monuments.

The idea was that people would come to think of the image

as what they were worshipping. Because of these rules,

artists would use abstract designs, many times using

tessellations. They produced many very beautiful stained

glass windows. To the right is an example of a stained glass

tessellation.

Figure2www.Martygumblesworth.wordpress.com

Figure3www.mathpuzzle.com

KatieRubenCalebAugspurger

November20,2013 3) SEE ATTACHED PAGES FOR STEP-BY-STEP CONSTRUCTION OF

TESSELATIONS.

4) The tesselation we created called bio-hazard typhoon uses only one type of isometry

in two different manors. The isometry that we used was a rotation where we initially choose a

point on our master tile to be the center of rotation. In step 1 of our explanation of the basic

tessellation construction, we choose the first center of rotation. After we created our master tile,

we rotated the tile 60 degress about that point. After 6 rotations about that point, we needed to

find a new point of rotation to continue tiling the plane. We found this point by finding the

midpoint of one of the exterior sides of out master tile (refer back to step 4 in our explanation of

the basic tessellation construction). Instead of our 60 degree rotation, we now rotated it 180

degrees about that point. Combining these two types of rotations and continuing this process,

will tile the entire Euclidean plane. The type of symmetry inherent in our final tessellation was

rotational and translational. If you pick our first point of rotation, we have rotational symmetry

of any multiple of 60 degrees. However, if you pick any other vertex of the master tile, we have

rotational symmetry of any multiple of 120 degrees. The way in which we can translate our

entire tesselation is by choosing the center of one hexagon which was the point of rotation for 60

degrees, to the center of a neighboring hexagon. The line segment created gives us our vector of

translation. Our final tessellation has periodic tiling. We know this is true because there is a

repeating pattern. If you look at our group of 6 master tiles, you will see that it repeats

throughout the entire plane in a mosaic pattern. A periodic tiling is noted by having the

capability of being translated across the plane. If translation symetry can not occur then it will

not be a periodic tiling. Since our basic tesselation is based off of a regular hexagon, then it is a

periodic tiling.

KatieRubenCalebAugspurger

November20,2013 For our advanced tessellation, we used a translation and a glide reflection to create our

master tile. We based our master tile on a square because we knew that a square tiles the entire

plane. We started off by creating an arbitrary line segment that we used as a side of our square.

Then, we created an artistic design that started and ended at the endpoints of the line segment.

The length of that line is now the vector for translation. We had made a copy of this line segment

and translated to the right the length of this vector. To finish our master tile we needed to create

the final two edges of our tile. To do this we connected corresponding vertices from the original

line segment to the translated segment (using an artistic design). Then we made a copy of this

new segment, reflected it across its midpoint and then translated downwards the same vector

length we had used before (glide reflection). To tile the page we found the midpoint of our top

segment which was used to find our vetical line of reflection. We followed the process of a glide

reflection concerning our entire master tile in which we reflected across this vertical line and

translated down the same length of the vector we had used to orginally create the tile. We

repeated this step over several times until we were satisfied. Then, we translated the row of tiles

to the right using the exact same vector defined above. We repeated this process until we had

filled the entire Euclidean plane. The type of symmetry involved in this tesselation is

translational symmetry. When looking at the entire tesselation we can translate it down two

vector lengths and it will be the same picture or we can translate the tesselation left or right just

one vector length and it will be the same as well. In addtion you can translate any multiple of

those vectors. Our final tesselation is periodic because there is a repeating mosaic pattern. We

alos see that our tesselation involves translation symetry and can tile a page using translations.

5) We found this project to be fun and interesting. We enjoyed being creative and

designing our own tessellations. In addition, it was nice to be able to work independently to

KatieRubenCalebAugspurger

November20,2013develop our understanding of tessellations. Probably the most difficult part for us was getting a

basic understanding of what the rules are for creating a tiling. Neither of us had particularly

worked with tessellations, so it was slow starting off. Once we got a basic understanding though,

we were able to use our creativity to design our tessellation.

This project is also quite beneficial for us as we continue to develop as teachers. From

experience, we both know that teachers will sometimes just assign projects for ‘busy’ work. As

a student, it is difficult to learn anything from those experiences. As future teachers, we are in a

very unique situation. We are learning to become teachers, but yet we are still students. This

allows us to think about what is most beneficial to students and what gets students engaged.

Projects like this give us an opportunity to see what works and what does not work with projects

that can be used in high schools. What we learn can then be beneficial in our own classes in just

a couple years. In particular for a project like this, we want to be able to show the mathematical

reasoning behind a tessellation and not just the fact that it can be a beautiful piece of artwork.

6.)

"Martholomew Gumblesworth."Martholomew Gumblesworth. N.p., n.d. Web. 19 Nov. 2013.

<http://www.martygumblesworth.wordpress.com>

"Tessellations - M. C. Escher and how to make your own Escher style art." N.p., n.d. Web. 19

Nov. 2013. <http://www.tessellations.org/index.htm>

Tidd, Hillary. "Transformational Geometry: Tessellations." N.p., n.d. Web. 19 Nov. 2013.

<http://jwilson.coe.uga.edu/EMAT6680Fa07/Tidd/Instructional%20Unit.html >

"Tessellation." Tessellation. N.p., n.d. Web. 18 Nov. 2013.

<http://www.mathsisfun.com/geometry/tessellation.html>

KatieRubenCalebAugspurger

November20,2013"Tessellations." Tessellations. N.p., n.d. Web. 18 Nov. 2013.

<http://www.csun.edu/~lmp99402/Math_Art/Tesselations/tesselations.html>

"North Texas Institute for Educators on the Visual Arts." N.p., n.d. Web. 19 Nov. 2013.

<http://art.unt.edu/ntieva/pages/about/newsletters/vol_14/no_1/>

"MathPuzzle.com." MathPuzzle.com. N.p., n.d. Web. 19 Nov. 2013. <www.mathpuzzle.com>