esamen tesselation

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1.0 INTRODUCTION

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

covering a plane without any gaps or overlaps. Another word of tessellation is

tilling. Tilling means when you fit something to fill a flat space. For example of tiling

are at ceiling, floor and wall.

Steven (1994) says tessellation is a noun and it came from Latin word,

µtessera ¶ that means a square tablet ¶ or µa die used for gambilng ¶. Word tessera

may have been borrowed from Greek, tessares that gave meaning µfour ¶, since a

square tile has four sides. The diminutive of tessera was tessella , a small, square

piece of stone or a cubical tile used in mosaics. Since a mosaic extends over a

given area without leaving any region uncovered, the geometric meaning of the

word tessellate is to cover the plane with a pattern in such a way as to leave noregion uncovered. By extension, space or hyperspace may also be tessellated.

In tessellation, there are many types of dimension such as three

dimensions, two dimensions and polytopes dimensions ( n). Two dimensions (2 D)

tessellation also can be called by another name. A tilling of regular polygons.

According to Woo et al. (1999), tessellation is the breaking up of self -intersecting

polygons into simple polygons or more pr oper, polygon tessellation.



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2.0 REGULAR TESSELLATION

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.

R egular 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.

Below are the example of regular polygon.

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

has three sides: " 3.3.3.3.3.3 ".

Triangles

3.3.3.3.3.3

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

the sides of the polygons that touch it, there are three polygons and each has sixsides, so this tessellation is called " 6 .6 .6":

Hexagons

6.6.6



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There are four polygons, and each has four sides.

Squ ares

4.4.4.4

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 is involved.

All of the tesselation are self-intersecting by a vertex. A vertex is just a

corner point and from the pattern at each vertex can identify the shape of

tessletion.

Look at a vertex, there are shapes have meet. Three hexagons meet at this

vertex and a hexagon has six sides. So, this is called a ³ 6.6.6´ tessellation.



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To name a tessellation, go around a vertex and write down how many sides

each polygon has, in order such as like "3.12.12". And always start at the polygon

with the least number of sides, so "3.12.12", not "12.3.12". In these tilings, at each

vertex of each polygon, three or more polygons must meet.

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 it 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.

Here are the interior measure of the angles for each of these polygons:

shape

triangle

square

pentagon

hexagon

more than six sides

Angle measure in degrees

60

90

108

120

more than 120 degrees



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3.0 SEMI-REGULAR TESSELLATION

Other than regular polygon tessellation, there are semi-regular tessellation.

A semi-regular tessellation is made of two or more regular polygons which are

fitted together in such a way that the same polygons in the same cyclic order

surround every vertex. There are eight semi-regular tessellations which comprise

different combinations of equilateral triangles, squares, he xagons, octagons and

dodecagons. The pattern at each vertex must be the same.

According to information from website m athisfun.co m, there are eight type

of semi-regular tessellation. Here there are examples of eight type of semi-regular

tessellation.

3.3.3.3.6 3.3.3.4.4 3.3.4.3.4

3.4.6.4 3.6.3.6 3.12.12

4.6.12 4.8.8

Besides that, a semi regular tessellation or Archimedean tessellation has

two main properties. First, It is formed by two or more regular polygons, each with

the same side length. Second, each vertex has the same pattern of polygons

around it.



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4.0 DEMIREGULAR TESSELLATION

In regular and semiregular tessellations, the arrangement of the polygons at

each vertex point is the same. In demiregular tessellations the arrangement of the

polygons at each vertex point is not the same. They might include a combination

of two o r three ver tex p o in t ty pes . A vertex point in a demiregular tessellation

may be regular, semiregular, or nonregular. There are at least fourteen

demiregular tessellations. Below is the example of demiregular tessellatio n.

A demiregular tessellation, also called a polymorph tessellation. Some

authors define them as orderly compositions of the three regular and e ight

semiregular tessellations which is not precise enough to draw any conclusions

from, while others defined them as a tessellation having more than one transitivity

class of vertices which leads to an infinite number of possible tilings.

(Sumber : http://mathworld.wolfram.com)



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5.0 NONREGULAR TESSELLATION

Nonregular tessellation shapes are shapes where the interior angles add up

to 360 degrees. M.C. Escher made these shapes famous in his works. These are

tessellations with nonregular simple convex or concave polygons. All triangles and

quadrilaterals will tessellate. Some pentagons and hexagons will. Non-regular

tessellations are those in which there is no restriction on the order of the polygons

around vertices. There is an infinite number of such tessellations.

Examples of nonregular tessellations.

Tessellations can be created by performing one or more of three basic

operations, transla t io n , r ot a t io n and reflec t io n , on a polyiamond. See Figure:

(Sumber: http://www.mathpuzzle.com)



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MY TESSELLATION DESIGN





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EXPLAINATION OF MY TESSELLATION



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To describe the steps of creating my tessellation design, I totally use

translation technique that was being popular by M.C Esher. A translation is a

movement is a specific direction, without turning or reflecting. Use a tessellation

tracer to draw a square, make a change to one of its sides and use tracing paper

to copy the modified square. Slide the original paper so that the change istranslated to the opposite side and copy it, as shown below.

Then, make an octagon shape on the square shape, as shown below.

The sides of square abcd can be completely redrawn since sides ab, bc, cdand ad are all distinct types of sides. Notice that a new type of same shape is

formed after the sides are translated. This same process can be applied to a

square when two adjacent sides are changed.

Start with a square and change two adjacent sides, trace the changes and

translate these changes to the opposite sides as shown below. Form the

tessellation by tracing this shape at least 6 to 12 times and shade or color

alternating shapes.

a b

cd



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Use a new sheet of tracing paper to copy this shape until at least 6 shapes

are traced. Shade or color the tessellation as shown below.

In my 2 D tessellation design, the basic 2 D shapes of my tessellations are

irregular octagon and isosceles triangle. There are another 2 D shape that formed

base on both octagon and isosceles triangle due to no gaps and overlap between

all shapes. There are big square, and rhombus. Below is the table of shape that I

used in tessellation design.

Name 2 D Shapes

Octagon

Rhombus





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B s i s tha t, t here are a r i h t ang le tr iang le tha t f ormed due to ross line in

the rhombus shape . S o tha t, the o lour of isosce les tr iang le is a lso in red co lour

too .

Isosce les tr iang le

ex t is the square shapes f orm base on irregu lar oc tagon and a cross line

f rom the rhombus shape . hus , to exp la in the co lour of square shape f rom my tesse lla tion is in b lue and red co lour in the s ide corner is isosce les tr iang le shape .

S quare



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REFLECTION

Assalamualaikum wbt. This Math task has been given to our class from our

lecturer . He gave us on a first brief about the task by that time. From this task, we

must doing in individual. I must create and draw a 2 -D tessellation design

manually without using any grap hics item from technology like computers and

internets, copy the designing and paste then print it. From that I had realize to put

my brain on designing by myself is really difficult and we will not easily to give our

own designing to other stranger people who want it easily. Now, I know what the

feeling of master minder in any work when people do the duplicates.

At the beginning I ¶ve got problem to finish this task. I didn ¶t know meaning of

the tessellation and how to draw a 2-D tessellation design. So I met my senior

Math and told her my problem. She thought me how to make a 2-D tessellationand she also gave an example of 2-D tessellation. After she gave information and

teach me how to make and draw a tessellation, so I started make a draft. At the

first, it so difficult but with comment from my friends, I can finish this tessellation

design. We fill very happy because our tessellation really unique and beautiful.

At the same time, I look for other information of 2 D tessellation from the

internets and a few books. When I read the informat ion again and again to

understand about it more, I knew that my design of tessellation was wrongbecause some of the polygon shapes were have not a vertex that intercept with

another polygon shapes. So that, I make a new one with other way that more

easily by using A4 colour papers because I knew that my colouring technique was

not good at all.

Furthermore, I feel very excited to do this assignment. My first draft of

tessellation, I design it using paper and pencil (drawing skills). I love drawing. But

then when times to colour it, I have a problems because I hate colouring. So, I

think and think how to solve this designing problem using another way. So that I

get the ideas by using A4 papers that is more neatly for my assignment differ from

other friends. That how I did it. Thanks to my friend Ain Nadia because show me

the great ideas on designing this 2 D tessellation



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Other than that, I also learn something that important in doing this task or

another assignment too for this semester. That are confident of ourselves.

Honestly, when doing this assignment task, I always want to compare on how to

describe the tessellation. Actually, there is no need because we have a good

explanations from lecturer. All we have to do is read again and agai n theinformation to understand it by our own and do the task in your way and not

according same like your friend. Be confident.

Last but not list, I am very happy cause can finished this task with flying

colours. The most obviously that I realized I had change in myself after finishing

this tasks, it had totally changed my perception tessellation because there are a

thousand and one way how to do it. Now, I really enjoyed tessellation topics and I

becoming to love art too.

As the conclusion, for me, this task is very good for student in order to

improve their mathematics skill on designing more and developed their interest

on this subject most and also the art too.



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BIBLIOGRAFI

Sumber Buku:

Britton, J. Sy mm etr y and T essellations: I nvestigating P atterns. Englewood Cliffs,

NJ: Prentice-Hall, 1999.

Critchlow, K. Order in Sp ace: A D esign S ource Book. New York: Viking Press,

1970.

Cundy, H. and Rollett, A. Mathe m atical Models , 3 rd ed. Stradbroke, England:

Tarquin Pub., pp. 60-63, 1989.

Pappas, T. "Tessellations." T he J oy of Mathe m atics. San Carlos, CA: Wide World

Publ./Tetra, pp. 120-122, 1989.

Kraitchik, M. "Mosaics." §8.2 in Mathe m atical R ecreations. New York:

W. W. Norton, pp. 199-207, 1942.

Sumber In terne t :

T essellation . http://www.mathsisfun.com. accessed on 29 th August 2010

T essellation P atterns . http://mathworld.wolfram.com. Accessed on 29 th August

2010.

Defination of T essellation . http://mathforum.org. Accessed on 29 th August

2010.

T essellation Design. http://gwydir.demon.co.uk. Accessed on 29 th August

2010.

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