Mathematics in Arts-Transformation

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4 CONTENTS NUM. TITLES/TOPICS PAGE NUMBER 1. Acknowledgement 2 2. Introduction of “Mathematics in Art” 3 3. Creating Of Design by Using Geometer Sketchpad 5 4. The Basic Block Design And The Eight Transforms 9 5. Artwork Produced And Detailed Explanations 10 6. Conclusion 22 7. Reflection 23 8. Bibliography 25 9. Appendices 26

Transcript of Mathematics in Arts-Transformation

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CONTENTS

NUM. TITLES/TOPICS PAGE NUMBER

1. Acknowledgement 2

2. Introduction of “Mathematics in Art”

3

3. Creating Of Design by Using Geometer Sketchpad

5

4. The Basic Block Design And The Eight Transforms

9

5. Artwork Produced And Detailed Explanations

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6. Conclusion 22

7. Reflection 23

8. Bibliography 25

9. Appendices 26

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ACKNOWLEDGEMENT

Firstly, I would like to thanks to all the people and parties who are helping me along

the process of completing this project especially to my lecturer, Mr. Lim Kang Chuan for giving

me the guide and advices to ensure that I do the best for this project.

A special thanks to my family, friends, and seniors who had helped me a lot in

gathering all the information and supports that is needed to complete this assignment. With this,

I thank you.

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INTRODUCTION

Spiders, cats, and birds are not artists. However, mans might inspired a bird's nest as being a

"work of art," and may find the patterns in the web made by spider and cat footprints on road.

Yet, the shape of a bird's nest may indeed be a form of communication for birds, just as "art" is

a form of communication for mans. What constitutes art in mathematics is a very complex.

When Jackson Pollock first experimented with expressing himself by flinging paint at a canvas,

many saw his activity as a form of self-indulgence rather than art. As another example, some

people collect maps and some of these maps are art, but not all maps are art. 

There is one artist whose work is having a mathematical quality. This artist was M. C.

Escher. The mathematical quality of his work is apparent even though Escher did not see

himself as having mathematical talent. Yet despite his lack of formal study of mathematics,

Escher approached many artistic problems in a mathematical way. Regular divisions of the

plane, called tessellations, are arrangements of closed shapes that completely cover the plane

without overlapping and without leaving gaps. Typically, the shapes making up a tessellation

are polygons or similar regular shapes, such as the square tiles often used on floors. Escher,

however, was fascinated by every kind of tessellation – regular and irregular – and took special

delight in what he called metamorphoses, in which the shapes changed and interacted with

each other, and sometimes even broke free of the plane itself.

His interest began in 1936, when he traveled to Spain and viewed the tile patterns used in

the Alhambra. He spent many days sketching these tilings, and later claimed that this “was the

richest source of inspiration that I have ever tapped.” In 1957 he wrote an essay on

tessellations, in which he remarked that in mathematical quarters, the regular division of the

plane has been considered theoretically. He though does this mean that it is an exclusively

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mathematical question? In his opinion, it does not. Mathematicians have opened the gate

leading to an extensive domain, but they have not entered this domain themselves. By their

very nature thay are more interested in the way in which the gate is opened than in the garden

lying behind it. They had shown that of all the regular polygons, only the triangle, square, and

hexagon can be used for a tessellation. (Many more irregular polygons tile the plane – in

particular there are many tessellations using irregular pentagons.) Escher exploited these basic

patterns in his tessellations, applying what geometers would call reflections, glide reflections,

translations, and rotations to obtain a greater variety of patterns. This are what we required to

apply the concept of transformations ( simple, combination, and repeatition) into our coursework

in creating a set of artwork designs based on the topic of “Enhancing Spatial Orientation” to

illustrate the concept of “Mathematics in Art” In creating the design for the master block, we just

used the basic shape of geometry instead of using Escher artwork which is he elaborated the

patterns by “distorting” the basic shapes to render them into animals, birds, and other figures.

These distortions had to obey the three, four, or six-fold symmetry of the underlying pattern in

order to preserve the tessellation.

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CREATING OF DESIGN BY USING GEOMETER SKETCHPAD (GSP)

To sketch a design on a master block by using the ICT skill, I used the GSP program by draw

each line symmetry and arcs one by one.

STEP 1:

Firstly, I began with drew a 4 cm square grid on the square grid graph as the basic shape for

the master block I designed. Then, I marked A, B, C and D at each corner or angle.

A B

D C

STEP 2:

Then, I sketched the 1st circle, about 3 cm radius and point A as centre of the circle. Then I

construct arc on the circle to get the curve EF. Next, I construct the 2nd circle which point A as

the centre with radius about 2.5 cm. Thus, I construct an arc on the circle also same as the 1 st

circle to get the curve GH. After that, I hide the both circles and left the both arcs inside the

master block.

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G EA B

D C

F

H

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STEP 3:

Then, I constructed symmetry to get line IJ. Hence, I construct another symmetry which at the

midpoint and perpendicular to IJ and bind to arc GH. Lastly, we would see the T shape had

been constructed on the master block.

G EA B

D C

F

H

J

I

K

STEP 4:

Next, I constructed 5 line of symmetry which is bind to the arc GH and we could see line KL,

KM, KN, KO, KP, KQ and KR had been formed which are line KR is perpendicular to line SL

and KL is perpendicular to line TR. Then, I constructed two line of symmetry which is line KS

and KT to form a small square on the master block.

G EA B

D C

F

H

J

I

K

O

L

R

NM

QP

T

S

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STEP 5:

Then, I constructed the 1st semi triangle at the bottom right side of the master block and mark at

each constructed point with UVC. Thus, I constructed an overturned small semicircle (2nd

semicircle) inside the 1st semicircle and mark at each constructed point with UWX.

G EA B

D C

F

H

V

J

I

K

O

L

R

NM

QP

T

S

U

X W

STEP 6:

After that, I constructed the 1st two symmetry lines which bind to the arc EF. Then, additional

two line that bind to the one side of 1st two symmetry lines, hence, we could see an upside down

Y shape is formed.

G EA B

D C

F

H

V

J

I

K

O

L

R

NM

QP

T

S

U

X W

Z

A1

B1

C1D1 E1

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STEP 7:

Lastly, I constructed three arcs in the master block. The 1st arc constructed which is I used point

B as the centre point by using a circle. Then, I construct an arc on the circle and hide the circle

to left the arc only in the master block. Next, I constructed the 2 nd arc by using the same way in

the opposite direction of the 1st arc. The 3rd arc constructed is at point C as the centre of circle

before I constructed the arc and hide the all the three circles to left the arcs only in the master

block. Hence, his was the last step. The three arcs constructed have radius about 0.5cm. Thus,

before I colorings the pattern, all the points are hidden.

I1G1

F1G EA B

D C

F

H

V

J

I

K

O

L

R

NM

QP

T

S

U

X W

Z

A1

B1

C1D1 E1

STEP 8:

After finished designing the pattern on the master block, I decided to paint the pattern by using

Paint computer program because it can be used easily to drawing picture. I had choose to used

the bright and passion colours to make the pattern looked beautiful and glowing. The choosen

of color is depend on the combination of transformation that I would have to do.

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THE BASIC BLOCK DESIGN AND THE EIGHT TRANSFORMS

From the design block, I had to apply the concept of translation, reflection, rotation, and

enlargement to 6 pieces of artwork. Firstly, in doing the artworks, the eight transform designs

blocks were given different code. To make sure easier for me to build all the artworks, I will

combined four blocks (different or same due to repetition) to be a new pattern.

BASIC BLOCK DESIGN

ROTATION

REFLECTION

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A

DCBA

DCB

270°180°90°0°

0° 90° 180° 270°

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ARTWORK PRODUCED AND DETAILED EXPLANATION

SINGLE TRANSFORMATION

The four blocks that I combined to be a new pattern is remarked with square number 1, 2, 3,

and 4 because it will be clear for me to make some explanations.

(1)

For the first artwork, I used pattern. This pattern is a single

transformation and done by using the reflection of basic block design. I used the same block for

the entire four square that I should combine to formed a new pattern which is square number 1

is using block A, square number 2 is also using block A, followed by square number 3 and 4 are

also placed by block A. Thus, I will make some repetitions and combine the pattern over and

over. Now, you could see a new shape is appeared. The shape is like many paper fans that

continued and also you could see many slanting A letter red in color when all the pattern

combined together.

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1 23 4

1 23 4

A AA A

A AA A

4

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(2)

For the second artwork, I used pattern. This pattern is done by

using a reflection of 270° rotation block. Yet, I used the same D block for the entire four blocks

that I should combine which is square number 1 is for D block, square number 2 is also D block,

followed by block number 3 and 4 are also using D block. Then, I will make some repetitions

and combined the pattern over and over. Now, you could see a new shape is appeared. The

shape is still same as the first artwork which is like paper fan that continued and also you could

see many overturned A letter red in color when all the pattern combined together.

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1 23 4

D DD D

D DD D

4

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COMBINATION OF TWO TRANSFORMATIONS

(3)

For the third artwork, which is combination of two

transformations, I used pattern. This pattern is done by using combination of two

blocks which are rotation of 90° (B block) and rotation of 270° (D block) from the basic block. I

used block B for square number 1, block D for square number 2, and again block B for square

number 3, and lastly, block D for square number 4. Then, I done some repetitions and

combined the pattern over and over. Now, you could see a new shape is appeared. We could

see the red Y shape is continued with the other block rotation and the circle pink in color bind

with the blue semi triangle and looks like a flying birds.

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B DB D

1 23 4

B DB D

4

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(4)

For the fourth artwork, I used pattern. This pattern is done by using

combination of two blocks which are reflection of rotation of 180° (C block) and reflection of

rotation of 270° (D block) from the basic block. I used block B for square number 1, block D for

square number 2, and again block B for square number 3, and lastly, block D for the square

number 4. Then, I done some repetitions and combined the pattern over and over. Now, you

could see a new shape is appeared. We could see by imagination images of many “little

chickens” are enjoying their spring season while lifting their leg to show their new high heel

shoe.

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C DC D

1 23 4

C D

C D

4

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COMBINATION OF THREE OR MORE TRANSFORMATIONS

(5)

For the fourth artwork, I used pattern. This pattern is done by

using combination of four blocks which are reflection of the basic block (A), rotation of 180° of

the basic block (C block), reflection of rotation of 180° (C block) and lastly, the basic block ( A

block). I replaced block A for square number 1, block C for square number 2, block C for

square number 3, and lastly, block A for square number 4. Then, I done some repetitions and

combined the pattern over and over same as the previous artworks. Now, you could see a new

shape is appeared. We could see the S shape that continued. By imagination, we also could

see it seems like the body snakes that creep on the blue sea water.

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A CC A

1 23 4

A CC A

4

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(6)

For the sixth artwork, I used two pattern which are

and .The first pattern is done by using combination of four blocks which are block C

(rotation of 180°) placed at square number 1, block A (reflection of the basic block) placed at

square number 2, block C (reflection of rotation 180°) placed at square number 3 and block A

(the basic block) that placed on the square number 4.

The second pattern is also done by using combination of four blocks which are

block C (rotation of 180°) placed at square number 1, block B (rotation of 90°) placed at square

number 2, block A (reflection of the basic block) placed at square number 3, and lastly, block A

(the basic block) that placed on the square number 4. For this second pattern, I had done

enlargement with the scale factor ½ and placed it at centre of the new pattern that will be

formed. The basic scale also had been used.

Then, I done some repetitions and combined the pattern over and over. Now, you

could see a new shape is appeared. We could see two circle images looks like wheel or the cut

of an orange fruit. We could also see many ribbon shapes formed, or them can be the images of

bugs or butterflies that fly in the air.

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1 23 4

C AC A

1 23 4

C BA A

C A

C A

C B

A A

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CONCLUSION20

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Throughout this coursework, I can conclude that Mathematics in Art is absolutely an interesting

topic and a lot of things have to be discovered. For example, we start by studying tiling. They

occur in many settings, and have a rich mathematical structure. The Platonic solids and

polyhedra have inspired people throughout the ages. The golden ratio has fascinated many

people, but we will take a critical look at whether it was really used in art and

architecture. Symmetry and patterns are important in ornamental art in all cultures. Among the

most famous are the Islamic patterns at Alhambra. Perspective originated in the Renaissance

and changed the way we look at the world. Many artworks are rich in mathematical structure.

We will look at the works of Escher and Holbein. Other beautiful applications of geometry

are kaleidoscopes, mazes and labyrinths, the fourth dimension and optical illusions. In the other

hand, this topic has been across the curriculum in primary school on the theme of the

application of mathematics in everyday life. In the class room procedure, tessellations can be

done by cut out of paper some congruent quadrilaterals with no to side parallel. Then, let the

students arrange them and see whether they can or not. It could be a surprise for them because

too many that any quadrilateral, convex, or concave can be arranged (do tessellation). Students

can also easily explore tessellation with graph or dot paper. Besides, the transformations is also

had been applied to the other effects of movements to primary school students such as turning,

sliding, or flipping in a variety of activities like jigsaw puzzles, printing or using mirrors. They find

recognizing transformations is much easier than constructing the transformations, due to need

them to visualize. Thus, it is really significance to apply the national numeracy strategy or the

better strategy to ease the transition to post-primary school, in due a lesson session would be

more effective and can increase the education development in our country.

REFLECTION

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We were given a Mathematics 1 Project and the title given is about to apply the concept of

simple transformations, combined transformations, and repeated transformation in creating a

set of artwork designs. This coursework need me to study and make some research about

design and at last create a design on a square master block by using ICT skill. In design the

master block I had applied the concept of translation, rotation and enlargement. Then, I

combined and repeated the transformation over and over to produce 6 set of artwork which is

based on single transformations, combinations of two transformations and combination of three

or more transformations. Firstly, in doing this coursework, I had tried to design my own idea

design. I tried to use different shapes of geometry such as triangle; a quarter of circle, square,

semicircle, and also line to make the master block are looked unusually. The first trial I failed

because when I tried to do some combination and transformations, the pattern formed looked

weird and not looked really good to be combined. Then, I did a new design for the master block

and I think it could be acceptable and attractive yet to me. Thus, I used the Geometer

sketchpad (GSP) program in order to complete my design rather than any other computer

program such as paint, or Microsoft office word because some of the design I had to

constructed is can only be constructed by using GSP. It is a long process because I had to

explore the program to construct my design. For example, I found the way to construct the arc

in my master block after exploring it by myself. Then, I had chosen the paint program to paint

my design; thus it will become beautiful. Besides, I also had overcome the way thus my block

will join well together. Furthermore, I tried to search the

relevant different materials in my understanding about this coursework.

I had used some opportunity to complete this coursework. Our lecturer had given some

useful briefing to us during the learning session in class. The briefing helps me a lot since we

don’t have to assume ourselves for what we must do. Then, we also had asked to our lecturer

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in class to ask about things that still seem blurring and make them clearer. Hence, I could make

sure; I went through a right guideline. Furthermore, I had learned something new which is

tessellation is a magic mathematics and impossible to be done at first, but Escher had prove it

is a possible. As a teacher, I would try to integrate this skill during the lesson in class. It is really

important to me, since it could make their understanding easier when we applied it to the lesson

session in class.

BIBLIOGRAPHY

1.http://www.mathacademy.com/pr/minitext/escher/index.asp

2.http://euler.slu.edu/escher/index.php/Tessellation_Art_Project

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3.http://euler.slu.edu/escher/index.php/Math_and_the_Art_of_M._C._Escher

4.http://euler.slu.edu/escher/index.php/Introduction_to_Tessellations

5.http://euler.slu.edu/escher/index.php/GSP_Introduction_Exploration

6. Claire Mooney, Mary Briggs, Mike Fletcher, Judith Mccullouch, Primary Mathematics: Teaching Theory and Practice, 2nd Edition, 2002, Learning Matters Ltd.

7. D.S. Macnab and J.A. Cummine, Teaching Mathematics 11-16 A Difficulty-Centred Approach, 1986, Basil Blackwell Ltd.

8. Max A. Sobel And Evan M. Maletsky, Teaching Mathematics, A Sourcebook Of Aids, Activities, And Strategies, 1991, Allyn And Bacon.

9. Pamela Cowan, Teaching Mathematics, 2006, Bell And Bain Ltd, Glasgow.

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APPENDICES

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