Mathematics in Arts-Transformation

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CONTENTS

NUM.1. 2. 3. 4. 5. 6. 7. 8. 9.

TITLES/TOPICSAcknowledgement Introduction of Mathematics in Art Creating Of Design by Using Geometer Sketchpad The Basic Block Design And The Eight Transforms Artwork Produced And Detailed Explanations Conclusion Reflection Bibliography Appendices

PAGE NUMBER2 3 5 9 10 22 23 25 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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INTRODUCTIONSpiders, 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 exclusively2

4 mathematical question? In his opinion, it does n ot. 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.A G

H

D

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B

C

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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.

A K

I

G

E

B

H

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.

A T

S I K

G R Q P O N

E

H

L M

5

B

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STEP 5:

Then, I constructed the 1st semi triangle at the bottom right sid e of the master block and mark at each constructed point with UVC. Thus, I constructed an overturned small semicircle (2nd semicircle) inside the 1 st semicircle and mark at each constructed point with UWX.

A T

F

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.

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THE BASI BLOCK DESIG AND THE EIGHT TRANSFORMS

From t

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lock, I

to

l

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concept of translation, reflection, rotation, and

enlargement to locks

pieces of art ork. Firstl , in doing t e art orks, t e eight transform designs o make sure easier for me to uild all the art orks, I ill

ere gi en different code.

combined four blocks different or same due to repetition) to be a new pattern. BASI B K ESI

A I 0 0 80 0

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B0 80 0

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ARTWORK PRODUCED AND DETAILED EXPLANATION

SINGLE TRANSFORMATION

1 2 3 4The 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)

1 2 3 4A A A A

A A A A

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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4

10

4

1 2 3 4 (2)

D D D D

For the second artwork, I used

D D D D

pattern. This pattern is done by

using a reflection of 27 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 th at continued and also you could see many overturned A letter red in color when all the pattern combined together.

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