Mathematics in Arts-Transformation
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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
10
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
5
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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
8
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
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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
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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
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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
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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
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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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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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