classification of wallpaper group in islamic arts

Transcription

SIM UNIVERSITY
SCHOOL OF SCIENCE AND TECHNOLOGY
CLASSIFICATION OF WALLPAPER
GROUP IN ISLAMIC ARTS
STUDENT
: KAM JIEWEN
(W0605338)
SUPERVISOR
: QUEK WEI CHING
PROJECT CODE : JAN2011/MTH/015
A project report submitted to SIM University
in partial fulfilment of the requirements for the degree of
Bachelor of Mathematics (Honours)
November 2011
JAN2011/MTH/015
i
ABSTRACT
W
allpaper group is the centre of this project research while
Islamic Art is the scope. Islamic Art created by Muslim artists,
visible in architecture, books, accessories, household items and
on countless other things, were collected for analysis. These collected art
decorations which might be repeated patterns may fall under one of the
seventeen wallpaper groups in Euclidean plane geometry.
During the
research, it has been realised that some of the art decorations fall into rosette
or one of the seven frieze groups in Euclidean plane geometry. Therefore,
research of rosette and frieze groups is included into the project at a later
phase. It has also been discovered that majority of the collected patterns fall
into certain frieze or wallpaper groups whilst some frieze or wallpaper groups
are displayed rarely.
For those uncommon frieze and wallpaper patterns, Geometer‘s Sketchpad®
is used to construct them. From constructing these patterns, it establishes a
better understanding and identification of wallpaper and frieze groups. The
last part of the project will focus on applying the knowledge to education.
Using some of the constructed patterns, notes and worksheets are created to
teach upper primary students transformations (translation, reflection and
rotation). High progress students will also learn some frieze and wallpaper
groups by means of simpler explanations. Subsequently, they will create their
own frieze and wallpaper patterns.
MTH499 CAPSTONE PROJECT FINAL REPORT
KAM Jiewen, W0605338
JAN2011/MTH/015
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ACKNOWLEDGEMENT
M
y project objectives would not have been achievable without
the guidance and help of a number of individuals who have
contributed their valuable assistance and support in the
preparation and completion of this project research.
First and foremost, my deepest and utmost gratitude to Mr. Quek Wei Ching,
my supervisor whose valuable knowledge, experience and constant support I
will always remember. He has been very patient, supportive and encouraging
in supervising and providing research materials and recommendations to the
completion of my project.
Last but not least, I am heartily thankful to my family for their
encouragement and support especially my husband, Marcus.
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Table of Contents
ABSTRACT ...................................................................................................... I
ACKNOWLEDGEMENT .............................................................................. II
LIST OF FIGURES ...................................................................................... VI
LIST OF TABLES ..................................................................................... VIII
INTRODUCTION............................................................................................ 1
CHAPTER ONE: PROJECT DEFINITON
1.1 PROJECT OBJECTIVE ............................................................................... 2
1.2 OVERALL OBJECTIVE .............................................................................. 2
1.3 PROPOSED APPROACH AND METHODOLOGY ........................................... 3
1.4 SKILLS REVIEW ....................................................................................... 5
CHAPTER TWO: INVESTIGATION OF PROJECT BACKGROUND
2.1 ART BACKGROUND .................................................................................. 6
2.2 KNOWLEDGE BACKGROUND .................................................................... 8
CHAPTER THREE: PROJECT PLAN
3.1 PLAN DESCRIPTION ............................................................................... 14
3.2 RISK ASSESSMENT AND MANAGEMENT ................................................. 17
CHAPTER FOUR: LITERATURE REVIEW
4.1 CLASSIFICATION OF FRIEZE GROUPS ................................................... 19
4.2 CLASSIFICATION OF WALLPAPER GROUPS ............................................ 26
4.3 RELATED WORK .................................................................................... 31
CHAPTER FIVE: DATA COLLECTION AND RESULTS
5.1 CLASSIFICATION OF DATA UNDER FRIEZE GROUPS .............................. 34
5.2 CLASSIFICATION OF DATA UNDER WALLPAPER GROUPS ...................... 37
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CHAPTER SIX: APPLICATION OF GEOMETER’S SKETCHPAD®
SOFTWARE
6.1 CONSTRUCTION OF FRIEZE PATTERNS.................................................. 48
6.2 CONSTRUCTION OF WALLPAPER PATTERNS .......................................... 49
CHAPTER
SEVEN:
APPLICATION
OF
KNOWLEDGE
TO
PROFESSION
7.1 OBJECTIVES OF LESSON ........................................................................ 53
7.2 CREATION AND CONSOLIDATION OF ACTIVITIES................................... 56
CHAPTER EIGHT: PROBLEMS AND DISCUSSIONS .......................... 57
CHAPTER NINE: CONCLUSIONS AND RECOMMENDATONS ....... 59
CHAPTER TEN: CRITICAL REVIEW AND REFELCTIONS ............. 61
BIBLIOGRAPHY .......................................................................................... 65
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APPENDICES
APPENDIX A GANTT CHART ........................................................................ 73
APPENDIX B LESSON PLAN ON TRANSFORMATION ..................................... 74
APPENDIX C LESSON PLAN ON FRIEZE AND WALLPAPER GROUP............... 76
APPENDIX D1 TRANSFORMATION STUDENTS’ NOTES................................. 78
APPENDIX D2 TRANSFORMATION STUDENTS’ WORKSHEETS ..................... 79
APPENDIX E FRIEZE AND WALLPAPER GROUP STUDENTS’ WORKSHEETS . 85
APPENDIX F FREQUENCY TALLY OF FRIEZE AND WALLPAPER GROUPS .... 89
APPENDIX G1 CAPSTONE PROJECT MEETING LOG (1) ........................... 91
APPENDIX G5 CAPSTONE PROJECT MEETING LOG (5) ......................... 100
APPENDIX G10 CAPSTONE PROJECT MEETING LOG (10) ..................... 110
GLOSSARY.................................................................................................. 112
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LIST OF FIGURES
Figure 1 Five types of plane lattices ............................................................. 9
Figure 2 Euclidean mirror-reflection ......................................................... 10
Figure 3 Rotational symmetry .................................................................... 10
Figure 4 Rotational and reflection symmetry combined .......................... 10
Figure 5 Translational symmetry ............................................................... 11
Figure 6 A complex combination of symmetries ....................................... 11
Figure 7 Cyclic Group Order 4 ................................................................... 20
Figure 8 Dihedral Group 8m ...................................................................... 21
Figure 9 Dihedral Group 8m ...................................................................... 21
Figure 10 Dihedral Group 8m .................................................................... 22
Figure 11 Frieze Group r1m ....................................................................... 34
Figure 12 Frieze Group r2mm .................................................................... 35
Figure 13 Frieze Group r2mm .................................................................... 35
Figure 14 Frieze Group r2........................................................................... 35
Figure 15 Frieze Group r11g....................................................................... 36
Figure 16 Frieze Group r2mg ..................................................................... 36
Figure 17 Wallpaper Group p .................................................................... 37
Figure 18 Wallpaper Group pm ................................................................. 38
Figure 19 Wallpaper Group pg .................................................................. 38
Figure 20 Wallpaper Group cm.................................................................. 39
Figure 21 Wallpaper Group p2 .................................................................. 39
Figure 22 Wallpaper Group p2mm ............................................................ 40
Figure 24 Wallpaper Group p2mg ............................................................. 41
Figure 25 Wallpaper Group p2gg .............................................................. 41
Figure 26 Wallpaper Group c2mm ............................................................ 42
Figure 27 Wallpaper Group c2mm ............................................................ 42
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Figure 32 Wallpaper Group p4gm ............................................................. 45
Figure 36 Frieze Group r............................................................................. 48
Figure 37 Frieze Group r11m ..................................................................... 48
Figure 39 Wallpaper Group p31m ............................................................. 50
Figure 40 Wallpaper Group p3m1 ............................................................. 51
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LIST OF TABLES
Table 1 Risk Assessment ............................................................................. 17
Table 2 Risk Management........................................................................... 18
Table 3 Two Infinite Families of Rosette Groups Summary ................... 20
Table 4 Frieze Groups Symmetry............................................................... 24
Table 5 Wallpaper Groups Summary ........................................................ 29
Table 6 Mathematics Syllabus Primary 4 Geometry/Symmetry and
Tessellation ................................................................................................... 54
Table 7 Primary 4 Mathematics Lesson Objectives ................................. 55
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INTRODUCTION
A
t some point of my teaching profession, I remember having
difficulties tessellating a shape given in one of my students‘
exercises. Then I pause to ponder the difficulties in tessellating
certain shapes. It prompts me to consider the ways (how) to differentiate
shapes that can tessellate and aspects that (what) differentiates it.
Undoubtedly, with a deeper understanding of wallpaper groups, as a teacher, I
can make use of this knowledge to better teach my students in the topics
―Symmetry‖ and ―Tessellation‖ as many students face difficulties in
completing a symmetric figure with respect to a given horizontal or vertical
line of symmetry, designing and making patterns and tessellating given a
basic unit shape.
Along with this project, besides gaining deeper knowledge into the
mathematics work done by famous mathematicians on wallpaper groups, its
application can be beneficial in teaching students techniques or strategies to
assist their understanding of the topics required in the Singapore syllabus. It
would also be challenging and interesting to construct designs which were
thought impossible or difficult to construct before commencing this project.
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CHAPTER ONE: PROJECT DEFINITON
1.1 Project Objective
The primary objective of this project is to investigate a wallpaper pattern
mathematically using the concept of wallpaper groups and to classify
different authentic two-dimensional Islamic Art patterns into different
wallpaper groups. The secondary objective is to construct geometric designs
of the different wallpaper groups using Geometer‘s Sketchpad®.
Then,
activity worksheets are to be produced for upper primary students by utilising
the geometric designs.
1.2 Overall Objective
Wallpaper group can be an interesting and fascinating subject area to research
in. It is exhibited in wallpaper designs, moulding, jewellery, ornaments of all
kinds, patterns in fabric and in tiling and many other man-made items. It
exhibits symmetry of one type or another. Many applications of symmetry
can be seen in scientific areas like the arrangement of atoms and molecules in
crystals and also in theoretical physics. A list of decorative possibilities can
be created by classifying and specifying symmetry groups (Henle,
1997:pp.247-248).
Mathematicians realised the association between
analytical geometry and symmetry in the nineteenth century.
They
formulated group theory to study the symmetry of mathematical objects
subsequently (Mortenson, 1999:pp.73).
Crystallographers
conducted
study
of
symmetry
groups
(Morandi,
2003:pp.1). Twenty years later from the nineteenth century, Fedorov, the
Russian crystallographer and M. Schoenflies classified the 17 planar and 320
spatial symmetry groups.
4-dimensional patterns classification was
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completed in the seventies but the problem is open in higher dimensions
(Arzhantseva, Bartholdi, Burillo and Ventura, 2007:pp.51).
Morandi said that the symmetry group of a wallpaper pattern is a wallpaper
group. To research into symmetry group, geometric restrictions have to be
determined so to create tessellations (Morandi, 2003:pp.2-4).
In the process of achieving the primary and secondary objectives of this
project, the formation of a wallpaper pattern and the basis of pattern
construction will be determined. In addition, study of the rosette and frieze
groups is encompassed into the research as some of the collected art
decorations might fall into one of these groups. It brings great opportunity to
learn and appreciate the history of distinctively fine visual arts and be
exposed to many rare and unique arts created by different well-known
respected artists.
1.3 Proposed Approach and Methodology
The fundamentals are of the five different lattices (parallelogram, rectangular,
rhombic, square and hexagonal) and symmetries of the plane. Symmetry is a
planar transformation which is self-coincident after moving the pattern
(Joyce, 1994). Each isometry of the plane, plane symmetry movement, is a
distance-preserving transformation of a plane. It is one of the following:
identity, translation, rotation, reflection, glide-reflection (The Geometry
Center, 1995). Lattices and isometries will be explained later in Section 2.2
Knowledge Background.
Asche and Holroyd (1994:pp.10) states that a group of plane isometric
movement, which is the symmetry group of wallpaper patterns is a wallpaper
group. A group of symmetries for a given pattern is called a symmetry group.
Given two elements S, a translation, and T, a reflection, of a group, when they
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are combined in succession, the resultant is the product of the two. Therefore
ST is a reflection followed by a translation (Lockwood and Macmillan,
1978:pp.98). In this case, it is combining transformations of the plane leaving
a pattern invariant. Thus their product is a symmetry of the pattern since each
transformation movement is a symmetry. The product is not enough to give a
group. In order to have a group, the operation must satisfy certain axioms
(Joyce, 1994; Kaplan, 2009:pp.12).
Holme (2010:pp.451) states that a discrete group of plane isometries is a
symmetry group. A rosette is a symmetry group containing rotations and/or
reflections but not translations.
A frieze group has rotations and/or
reflections but one-directional translations and their inverses. A wallpaper
group has rotations and/or reflections and no parallel translations.
An
explanation of the symmetry groups and its classification is presented later in
Section 2.2 Knowledge Background.
A detailed study of wallpaper groups will be presented in Chapter Four
Literature Review of the report. The symbols used for these groups which
were
developed
by
the
International
Union
of
Crystallographers
(Arzhantseva, Bartholdi, Burillo and Ventura, 2007:pp.55) are necessary for
familiarisation.
With a conceptual understanding of wallpaper groups, collected Islamic Art
patterns, originating from the South East Asia, will be studied. There will be
an attempt to create designs inspired by contemporary or classical wallpaper
patterns. In the application phase, wallpaper patterns will be constructed after
mastering the use of Geometer‘s Sketchpad®.
Ultilising the constructed
patterns, activity worksheets will be designed for teaching purposes. The
worksheets will allow students to identify transformations and also complete
symmetric figures and/or create frieze and wallpaper patterns.
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1.4 Skills Review
Progress of the project can be measured against the scheduled plans in the
Gantt chart. These are the deliverables: proposal, classification of wallpaper
patterns, interim report, construction of wallpaper patterns, applying
constructed patterns to profession and final report. They are executed in the
different phases. In all phases, literature review is necessary for analysis and
investigation.
The key skills necessary to complete the project are the understanding of the
mathematical concept of frieze and wallpaper group, application of concept to
classify different collected art patterns, constructing frieze and wallpaper
patterns using Geometer‘s Sketchpad® and practical application into creating
educational worksheets for upper primary students.
The enthusiasm in the independent study of mathematics, adequate
visualisation skills and efficient time management are vital factors to
achieving success in this project. Good report writing and presentation skills
also contribute much to the preparation of proposal, interim and final report.
Reading up on report writing should improve on appropriate report structure,
approach and sentence structure; prepare and rehearse for the presentation.
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CHAPTER TWO: INVESTIGATION OF
PROJECT BACKGROUND
2.1 Art Background
The term ‗Islamic art‘ describes art created by Muslim artist for Muslim
patrons otherwise it is about Fatimid, Iranian, Maghrebi or Mughal art. It
ranges from many things like architecture to book-production and for
decoration such as glass, jewellery, metalwork, pottery and textiles (Khalili,
2006:pp.9).
Muslim artist reserved the key elements of the traditional
convention and then gave details upon them to create new designs which
emphasised the importance of unity, logic and order.
One of the basic
characteristics of these designs is that the patterns are made up of repeated
geometric elements (Hens, 2004:pp.10).
Although the created designs stressed the importance of unity, logic and
order, the explanation of a theory or an approach to the significance and
artistic importance of the arts still has proved estranged to Islamic culture.
For the most part of the earliest centuries, Muslim attitudes towards the arts
remain a hypothesis due to the scarcity of documentary proof. Even though
art is created in culturally Islamic countries, it is difficult to extract elements
indicative of a truly ‗Islamic‘ art (Mozzati, 2010:pp.22).
However, Papadopoulo (1979:pp.41) mentioned that Arabic Muslim art
employs mathematical figures as it is greatly influenced by their fine
mathematicians.
Beneath all the art also lies the concept of Platonism.
Platonism makes use of Numbers and mathematical Forms to represent the
most fundamental reality and its beauty. Geometrical decoration in the mind
of the Muslims held a spiritual and philosophical value and not for decoration
purpose. But, these decorations of nature became more abstract until they
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resembled a mathematical spiral. Muslim abstract painting had geometric
form as its basis – in particular, the spiral and the arabesque.
Museum With No Frontiers (2007:pp.19) reveals perhaps that there is a close
relation between Islamic and Western art as ‗Islamic‘ ornamentation descend
from late Roman decoration.
Nevertheless, Islamic art was not just a
continuance of late Roman art but it is an artistic fusion of selected forms
made use in new ways or new social, philosophical purposes.
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2.2 Knowledge Background
The seventeen wallpaper groups are the only possible symmetry groups for a
network figure.
Hermann-Mauguin-like symbols or orbifold notation are
frequently used to represent them (Zwillinger, 1995:pp.260).
Wallpaper patterns possess translational symmetries of some lattice.
By
repeating a pattern on one of the five different plane lattices, figures of
network symmetry are created. The only five different kinds of plane lattices
that will give us five kinds of network figures are parallelogram lattice
symmetry group, rectangular lattice symmetry group, rhombic lattice
symmetry group, square lattice symmetry group and hexagonal lattice
symmetry group (Asche and Holroyd, 1994:pp.5-10; Clair, 2011:pp.1).
Figure 1 below shows the five different plane lattices.
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parallelogram lattice
rectangular lattice
rhombic lattice
square lattice
hexagonal lattice
Figure 1 Five types of plane lattices
(Extracted from Asche and Holroyd, 1994:pp.10)
Besides translational symmetry, network figures have other additional
symmetries such as rotations, reflections or glide reflections. Transforming a
pattern to look exactly the same after a transformation is symmetry of a
pattern (Joyce, 1994).
An example of reflection symmetry, which also means mirror symmetric, is
shown in Figure 2. Rectangles are invariant under two different reflections.
They are another example of mirror symmetric (Kaplan, 2009:pp.11-12).
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Figure 2 Euclidean mirror-reflection
(Inspired by Lockwood and
Macmillan, 1978:pp.107)
Created using Geometer‘s Sketchpad®
Figure 3 Rotational symmetry
(Inspired by Lockwood and
Macmillan, 1978:pp.35)
Created using Geometer‘s
Sketchpad®
The pinwheel in Figure 3 is a fourfold rotational symmetry as the minimal
rotation of 90o leaves it invariant. It has to be repeated four times to get its
identity (Horne, 2000:pp.8).
Figure 4 Rotational and reflection symmetry combined
(Inspired by Horne, 2000:pp.11-13)
Created using Geometer‘s SketchPad®
Figure 4 is an example of having more than one kind of symmetries. It is a
twelvefold rotational symmetry as the minimal rotation of 30o leaves it
invariant. It has to be repeated twelve times to get its identity. It also has
reflection symmetry when it is under reflection in twelve different mirror
lines. This still leaves the figure invariant (Mortenson, 1999:pp.73-74).
Figure 5 exhibits only part of the whole design there. It extends infinitely far
in both horizontal directions.
The arrow below the figure indicates a
translational symmetry. It is also a twofold rotational symmetry. In addition,
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an infinite number of twofold rotocenters are exhibited in the figure. In fact,
they show two different kinds (Henle, 1997:pp.245).
Figure 5 Translational symmetry
(Extracted from Henle, 1997:pp.245)
Figure 6 A complex combination of symmetries
(Extracted from Henle, 1997:pp.245)
Figure 6 shows only part of the whole design there. It extends infinitely far in
both horizontal and vertical directions. It is much more complicated but
possible. The two arrows, pointing in two different directions indicates twodimensional translational symmetry. A number of other symmetries have
been exhibited here as well. They are twofold and fourfold rotocenters of
rotational symmetry and both horizontal and vertical reflection lines which
are indicated by dashed lines. An additional feature exhibited is the glide
reflection lines. It is indicated by a long-dashed line. Glide reflection is a
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reflection in the long-dashed line followed by a translation along the direction
of the long-dashed line (Solyom, 2007:pp.158-163).
From the examples above, Henle (1997, pp.232-248) defines as follows:
―DEFINITION Let F be a figure in a geometry.
The set of all
transformations T such that T(F) = F is called the symmetry group of
F and is noted σ(F), The elements of σ(F) are called symmetry
elements of F.‖
It will be Euclidean plane geometry where my project research is based on. It
is possible to find and classify all symmetry groups by adding a technical
simplifying assumption (Henle, 1997:pp.246-247).
―DEFINITION Let G be a group of transformations of the complex
plane, and let z be a complex number. The set {Tz : T ∈ G} is the
orbit of z under the action of G.
A group G is a discrete group if every orbit of G has only a finite
number of points inside any circle.‖
By restricting research to discrete groups, all Euclidean plane symmetry
groups can be listed. A list of the 17 wallpaper groups possible in such a case
is provided under Section 4.2 Classification of Wallpaper Groups (Amer,
2010:pp.68; Henle, 1997:pp.247).
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According to their fixed elements, discrete symmetry groups are classified
and Henle (1997, pp.232-248) defines as follows:
―DEFINITION The full Euclidean group E+ is the group of motions
of the Euclidean plane consisting of the transformations of the special
Euclidean Group E (rotations and translations), plus reflections, and
all transformations that are combinations of reflections, rotations, and
translations.‖
―DEFINITION Let G be a subgroup of the full Euclidean group E+.
Then, G has a fixed point z, if Tz = z for all T in G. G has a fixed line
λ if T(λ) = λ for all T in G.
A figure F in Euclidean geometry (C, E+) has a singular point (or
singular line) according as its symmetry group σ(F) has a fixed point
(or fixed line).‖
Figures with singular elements are easily recognised such as Figures 3 and 4.
They have singular points whilst Figure 5 has a fixed centre horizontal
singular line and no point on the line can be fixed. But Figure 6 has no
singular elements at all (Horne, 2000:pp.14).
Henle (1997, pp.232-248) states:
―CLASSIFICATION OF SYMMETRY GROUPS Let F be a figure in
the geometry (C, E+). There are three cases:
(a) F has a singular point. Then F is called a rosette.
(b) F has a singular line, but no singular point. Then F is called a
frieze.
(c) F has no singular element. Then F is called a network.‖
Therefore Figures 3 and 4 are rosettes, Figure 5 is a frieze and Figure 6 is a
wallpaper (Horne, 2000:pp.14).
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CHAPTER THREE: PROJECT PLAN
3.1 Plan Description
After selecting the project title "Classification of Wallpaper Group in Islamic
Arts", phase one of the project ―Project Planning and Preparation‖ can be
commenced.
From the confirmation date of the project to the project
proposal deadline, an extensive in-depth reading and literature review on the
study of frieze and wallpaper groups has to be done in order to write the
project proposal.
Several books and websites are recommended by the
project supervisor. A project schedule is plotted using a Gantt chart. The
project proposal is targeted to finish a week before submission deadline so
that necessary refinements can be made upon review by supervisor.
―Requirements Analysis‖ is the second phase of the project. While waiting
for supervisor‘s evaluation on the submitted proposal, more extensive
literature research will be done on the topic for the following three weeks.
UniSIM library, its online resources such as e-journals, and National Library
are useful sources. It is needed to work on some examples and exercises in
the materials if possible. After the supervisor has evaluated the proposal,
refinements to the methods and approaches to achieving project objectives
will be made.
The third phase of the project ―Implementation of Methods‖ would take
approximately three weeks as it requires visiting of mosques, museums or art
centres in order to gather authentic patterns for analysis and classifying them
into one of the seven frieze or seventeen wallpaper groups. The focus of the
project is on classifying patterns originating from South East Asia region in
order to limit the extent of the research. Supervisor review on the wallpaper
patterns collected, analysis and classification is essential to the validity of the
work.
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In the fourth phase of the project ―Investigations‖, the main focus here is to
validate the analysis and classification done in the previous phase. Numerous
mathematicians have used different methods to prove the classification such
as the study of point groups (Asche and Holroyd, 1994:pp.13-24), symmetry
groups (Kaplan, 2009:pp.12-16), group cohomology (Morandi, 2003:pp.3346).
However, this research will not be studying of the proofs to the
classifications. This phase also includes preparing of interim report. This
report will be reviewed by the supervisor prior to submission to UniSIM. The
whole process will take approximately five weeks.
―Refinement of Model‖ is the fifth phase of the project and this phase centres
modelling wallpaper patterns using Geometer‘s Sketchpad® and refinements
to the patterns.
It would require approximately six weeks constructing
wallpaper patterns employing the knowledge and understanding acquired in
previous phases. Familiarising with the use of the software provided by the
supervisor is necessary. It can be accomplished more effectively by working
on examples and exercises in activity sheets, provided by the supervisor.
There will be a presentation of the several constructed wallpaper patterns to
the supervisor for his review and validation.
This sixth phase ―Application of Model‖ focuses on applying the created
frieze and wallpaper patterns to profession. Some of the frieze and wallpaper
patterns will be selected as the students‘ learning outcomes. Teaching ideas,
students‘ activities, worksheets and other resources will be created or adapted
from reliable sources in order to meet the learning outcomes. Then, advices
from the supervisor will be sought based on the resources created. More
literature review and research is also required to corroborate the effectiveness
of these lesson ideas and resources. This phase will take approximately seven
weeks.
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In ―Project Report and Poster Design‖ phase, research, findings,
investigations and analysis have to be reported in the final year report. There
should be sufficient time set aside for preparation of poster and presentation
as well. A period of twelve weeks is required. There will also be regular
meetings with the supervisor to corroborate work done.
The Gantt chart is reflected in Appendix A.
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3.2 Risk Assessment and Management
Following are some of the risks identified that might deter project completion
and ways it can be managed or reduced.
Probability of Event Scale
Severity of Event Scale
1 – Rare
1 – Negligible
2 – Unlikely
2 – Marginal
3 – Possible
3 – Moderate
4 – Likely
4 – Critical
5 – Definite
5 – Catastrophic
Table 1 Risk Assessment
(Kirkham, 2004:pp.58-59)
No. Categories: Risks Identified:
Probability Severity
1
Health
scale
scale
2
5
3
4
or 3
4
Unforeseen
medical
reasons
such as accidents
2
Software
Loss of data
3
Self
Inadequate
improper
planning
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Table 2 Risk Management
(Kirkham, 2004:pp.58-59)
No. Categories: Risks
1
Health
Risk Control:
Implementation:
Identified:
Strategies to reduce risks
Action by
Unforeseen
1.1 Avoid high risk sports Self
medical
prior
reasons such
completion
as accidents
to
1.2 Avoid
high
project
travelling
risk
to
countries
(SARS, H1N1)
2
Software
Loss of data
2.1 Always back up files Self
by
saving
updated
copies into multiple
medias such as thumb
drive,
portable
hard
disk, self email
2.2 Ensure
anti-virus
programme is up-todate
3
Self
Inadequate or 3.1 Supervisor review on Self
improper
scope and depth of supervisor
planning
project
and
3.2 Adhere to deliverable
deadlines
3.3 Supervisor review on
deliverables
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CHAPTER
REVIEW
Page 19 of 117
FOUR:
LITERATURE
4.1 Classification of Rosette and Frieze Groups
The different lattices, transformations and classifications of symmetry groups
had been mentioned in Section 2.2. In this section, a clearer explanation into
rosette and frieze groups would be presented.
Given a figure, it can have no symmetry. Therefore, to leave the figure
invariant, it would only be the identity transformation. For this symmetry
group, the International Symbol is 1. By applying geometric transformations
to figures with no symmetry, figures with symmetry can be achieved
(Schattschneider, 1978:pp.443).
Cyclic symmetry is generated by applying one rotation repeatedly to a figure
without symmetry until it returns back to its original position. All these
rotations have to be about the fixed point. This is one type of rosette (Ronan,
2006:pp.34).
A rosette can have reflections added to rotational symmetry but cannot
contain translations or glide reflections. The combination of rotations and
reflections will create more reflection lines (Ronan, 2006:pp.34). This kind
of symmetry is called dihedral symmetry and for this symmetry group, the
International Symbol is nm. n is the order of the rotorcenter of the group and
the number of reflections (Schattschneider, 1978:pp.439).
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Table 3 Two Infinite Families of Rosette Groups Summary
(Grünbaum and Shephard, 1992:pp.333)
1. Cyclic Groups
Only rotational symmetry-symbols:1, 2, 3, …
2. Dihedral
n-fold rotocenter + reflection-symbols: m, 2m, 3m, …
Groups
Below is an example of cyclic symmetry of 4.
Figure 7 Cyclic Group Order 4
Turkey 20th century AD: Calligraphic piece. [paper] (Asian Civilisation
Museum Collection).
Legend
Centers of Rotation:
4-fold
8-fold
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The three pictures, Figure 8, 9 and 10, are examples of dihedral symmetry of
8m.
Figure 8 Dihedral Group 8m
Singapore 1824 AD: Flooring. [tiles] (Masjid Sultan Compound).
Singapore 1824 AD: Flooring. [tiles] (Masjid Sultan Compound).
Legend
4-fold
8-fold
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Singapore 1824 AD: Grill. [metal] (Masjid Sultan Compound).
Legend
4-fold
8-fold
In Mozzati‘s (2010:pp.310) Islamic Art book, he states that:
―A frieze is a continuous band which is made of different materials, is
usually affixed to a wall in architecture.‖
This definition is comparable to the mathematics definition of frieze which its
infinite translations are in a single direction similar to a continuous band.
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Translational symmetry appears in friezes. Following pictures are examples
of frieze groups.
r is used as an International Symbol to represent
translational symmetry in a single direction. Friezes of different symmetry
types are generated when other symmetry elements are added to r, the basic
translational symmetry (Henle, 2010:pp.250).
Table 4 will give you an idea about the only seven different frieze groups
(Belcastro and Hull, 2002:pp.94-98).
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Table 4 Frieze Groups Symmetry
(Collins and Liu, 1998:pp.7-8; Livio, 2005:pp.206)
1. r
Only translational symmetry in one dimension
Representation: … b b b b b b b b b b …
2. r11m
Translational symmetry and a parallel reflection
Representation: … b b b b b b b b b b …
…pppppppppp…
3. r1m
Translational symmetry and a perpendicular reflection
Representation: … b d b d b d b d b d …
4. r2mm
Translational
symmetry
with
perpendicular
and
parallel
reflections
Representation: … b d b d b d b d b d …
…pqpqpqpqpq…
5. r2
Translational symmetry and perpendicular twofold axes
Representation: … b q b q b q b q b q …
6. r11g
Only glide reflection symmetry in one direction
Representation: … b
…
7. r2mg
b
p
b
p
b
p
b
p
…
p…
Glide reflection symmetry and perpendicular reflections
Representation: … b d
…
bd
qp
bd…
qp
…
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According to Table 4, the letter ‗b‘ is used for identifying the orientation and
position of the fundamental generator as it has no symmetry by itself. The
letters ‗d‘, ‗p‘, ‗q‘ and ‗b‘ are the four images after isometries (Livio,
2005:pp.206).
Symbol r, which appears in all the seven frieze groups, is the symbol for
translational symmetry in one dimension. Its direction is parallel to the yaxis. The characters after the symbol r will indicate any extra symmetry
element, example ―r11g‖, ―r2mg‖.
―1‖ appearing as the first character after r indicates that there is no rotational
axis however if ―2‖ appears, it indicates that there is a twofold rotational axis.
―1‖ appearing as the second character after r indicates that there is no
perpendicular reflection plane but if ―m‖ appears, it indicates that there is a
perpendicular reflection plane.
―m‖ appearing as the third character after r indicates that there is a parallel
reflection plane. If ―g‖ appears, it indicates that there is glide reflection plane
(Drager, 2011:pp.22; Henle, 2010:pp.252).
Studying of the rosette (no translations) and frieze (translation in only one
direction) groups can benefit the research in the classification of wallpaper as
they are closely related to wallpaper groups (McCallum, 2001). Wallpaper
are translations in more than one direction. Their symmetry group is evident
in the discrete and translational symmetries of rosette and frieze respectively
(Klette and Rosenfeld, 2004:pp.461; Schattschneider, 1978:pp.439).
In the next section, classification of wallpaper groups will be discussed.
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4.2 Classification of Wallpaper Groups
The options for wallpaper patterns are constraint by the underlying
Crystallographic Restriction (only rotations of order 2, 3, 4 and 6 may leave
the wallpaper pattern invariant) and the ways the four kinds of isometries
interact with one another (Baloglou, 2002).
The first two wallpaper groups to be introduced are p and p2 which are drawn
on a parallelogram lattice.
A parallelogram consists of its identity and
rotations of order 2 but no reflections (Baez, 2008). Hence wallpaper group
p, may also be known as p1, is generated solely from translational symmetry.
There are no rotations, reflections or glide reflections. And wallpaper group
p2 has rotations of order 2 only. There are no reflections or glide reflections
(Bart and Clair, 2011).
The next five wallpaper groups, p1m, p1g, p2mm, p2mg and p2gg, are based
on a rectangular lattice. A rectangle consists of its identity, rotations of order
2 and reflections in both the horizontal and vertical directions (Collins and
Liu, 1998:pp.11). Therefore, wallpaper group p1m, also known as pm, has
reflections whose axis is horizontal but no rotations. Wallpaper group p1g,
also known as pg, has glide reflections in one direction whose axis is
horizontal but no rotations or reflections (Bart and Clair, 2011). Wallpaper
group p2mm has reflections in both horizontal and vertical directions. It also
has rotations of order 2 (Kung, 2009:pp.64). Wallpaper group p2mg has
reflections whose axis is horizontal and glide reflections whose axis is
vertical. It also has rotations of order 2. And wallpaper group p2gg has glide
reflections in two directions whose axes are both horizontal and vertical. It
also has rotations of order 2 (Shakiban, 2006:pp.2-4).
The next two wallpaper groups, c1m and c2mm, are based on a rhombic, may
also be known as centered rectangular lattice. A rhombus consists of its
identity, rotations of order 2 and reflections in both the horizontal and vertical
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directions. Hence, wallpaper group c1m, also known as cm, has reflections
and glide reflections in one direction and no rotations (Kung, 2009:pp.67).
And wallpaper group c2mm has reflections with axes in both the horizontal
and vertical directions. It also has rotations of order 2 (Clair, 2011:pp.2).
The next three wallpaper groups, p4, p4mm and p4gm, are based on a square
lattice. A square consists of its identity, rotations of order 4 and reflections in
the horizontal, vertical and diagonal directions (Shakiban, 2006:pp.5-6).
Wallpaper group p4 has rotations of order 4 only and no reflections.
Wallpaper group p4mm has reflections in the horizontal, vertical and
diagonal directions. It also has rotations of order 4. And wallpaper group
p4gm has rotations of order 4 and glide reflections with axis in the vertical
direction. Its rotation axis is not on the reflection axis (Bart and Clair, 2011;
Sanderson, 2001).
The last five wallpaper groups of the seventeen, p3, p3m1, p31m, p6 and
p6mm, are based on a hexagonal lattice. A hexagon consists of its identity,
rotations of order 2, 3 and 6 and reflections axes making angles of ±30o from
the horizontal axis (Clair, 2011:pp.3). Wallpaper group p3 has rotations of
order 3 only but none of order 6. There are no reflections or glide reflections.
Wallpaper group p3m1 has rotations of order 3 only but none of order 6. It
has 3 reflection axes perpendicular to the sides. Wallpaper group p31m has
rotations of order 3 only but none of order 6. It has 3 reflection axes parallel
to the sides (Schattschneider, 1978:pp.448; The Mathematical Association of
America, 1998). Wallpaper group p6 has rotations of order 6 only and no
other symmetries. Wallpaper group p6mm has rotations of order 6. It has 6
reflection axes where 3 reflection axes are perpendicular to the sides and the
other 3 reflection axes parallel to the sides (Clair, 2011:pp.3).
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The symbols used in Table 5 for the listing of the seventeen wallpaper groups
were developed by the International Union of Crystallographers (IUC) in
1952 (Arzhantseva, Bartholdi, Burillo and Ventura, 2007:pp.55; Clair,
2011:pp.1).
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Table 5 Wallpaper Groups Summary
(Holme, 2010:pp.460-468; The Math Forum @ Drexel and The Textile
Museum, 2011)
1. p
purely parallelogram translational symmetry
2.
p2
parallelogram translational symmetry + twofold axes
3.
p1m
rectangular translation lattice + reflections in one direction
4.
p1g
rectangular translation lattice + glide-reflection in one direction
5.
p2mm rectangular translation lattice + reflections in two directions
6.
p2mg
rectangular translation lattice + reflections in one direction and
perpendicular glide-reflections
7.
p2gg
rectangular translation lattice + glide-reflections in two
perpendicular directions
8.
c1m
centered rectangular / rhombic translation lattice + alternating
reflections and glide-reflections in one direction
9.
c2mm centered rectangular / rhombic translation lattice + alternating
reflections
and
glide-reflections
in
two
perpendicular
directions
10. p4
square translation lattice + fourfold axes and twofold axes
11. p4mm square translation lattice + fourfold axes + reflection lines
through the fourfold axes
12. p4gm
square translation lattice + fourfold axes + glide-reflection
lines through the fourfold axes
13. p3
hexagonal translation lattice + threefold rotation axes
14. p3m1
hexagonal translation lattice + threefold axes + reflection lines
through the threefold axes
15. p31m
hexagonal translation lattice + threefold axes + reflection lines
that miss some of the threefold axes
16. p6
hexagonal translation symmetry + sixfold axes
17. p6mm hexagonal translation symmetry + sixfold axes + reflection
lines
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Symbol p is used for wallpaper groups with parallelogram, rectangular,
square or hexagonal lattices whilst symbol c is used for wallpaper groups
with rhombic lattice.
The digits ‗1‘, ‗2‘, ‗3‘, ‗4‘ or ‗6‘ appearing as the first character after p or c
indicates maximum rotations of order n.
The character ―m‖ appearing as the second character after p or c indicates that
there is a reflection on the horizontal axis. The character ―g‖ indicates that
there is a glide reflection on the horizontal axis. ‗1‘ indicates no symmetry
axis.
The character ―m‖ or ―g‖ appearing as the third character after p or c
indicates that there is a symmetry axis at an angle to the horizontal axis, with
the angle dependent on the maximum rotations of order n. ‗1‘ indicates no
symmetry axis (Edwards, 2002; Schattschneider, 1978:pp.443).
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4.3 Related Work
This project investigates patterns mathematically using the concept of rosette,
frieze and wallpaper groups and classifies different authentic twodimensional Islamic Art pattern into the different respective groups.
In
Grünbaum and Shephard (1992:pp.331) journal, they analysed an even more
complicated kind of patterns named interlace patterns which are made up of
identical or two shapes of strand and are frequently occurred in Islamic and
Moorish art. Some of these patterns are very complicatedly interweave that
its repeat is not visible. It was explained in their journal the several ways in
measuring these complications and relating them to symmetry properties.
They provided an explanation of the frequent appearance of these patterns in
Islamic art as their final point.
However, Collins, Liu and Tsin (2004:pp.354,367) developed computer
algorithms from crystallographic groups which can analyse real patterns.
Their computational model for periodic patterns falls under three main
components. It would extract the translational lattice of a periodic pattern,
then classifying the pattern into one of the symmetry groups (frieze or
wallpaper) before capturing meaningful motifs perceptually. On the contrary,
in this research, if the repeated pattern extends infinitely in both horizontal
and vertical directions, it belongs to one of the seventeen wallpaper groups.
If not, it might be in the frieze group for extending infinitely in the horizontal
direction only.
When the repeated pattern is recognised as one of the
wallpaper groups, the number of maximum rotation order (1, 2, 3, 4, or 6) is
determined. From there, the classification of the pattern into one of the
wallpaper groups is decided from the pattern‘s constitution of any reflection,
glide-reflection or rotation.
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There was another computational model proposed by Djibril and Thami
(2008:pp.2,13). Their model extracts symmetry properties to classify and
index Islamic geometrical patterns (IGP) image. They used symmetry group
theory to model IGP patterns and these IGP patterns are classified into three
main categories, F (frieze), W (wallpaper) and R (rosette). The algorithm
would automatically extract fundamental region of the pattern and
characterise it into one of the three main categories. Then the feature vector
was created by combining the symmetry information of patterns with their
fundamental region histogram information so to process a new query on an
image from IGP images database.
While classifying patterns into wallpaper groups, different types of
symmetries and groups such as rosette and frieze are acquired and mentioned
in this project. Lockwood (1973:pp.14-15) briefly introduced cyclic, dihedral
symmetry with its repetition in two or more directions and glide reflection
movement in one or two directions.
He had also mentioned how these
symmetries and movements relate to the five different lattices and did a
listing of the seventeen types of patterns.
Although Belcastro and Hull
(2002:pp.94) have presented a combinatorial proof for frieze patterns without
using group theory, this project is not required to prove any of the patterns.
In McLeay‘s article (2006), she had presented her collected data which was
classified into the seven different frieze groups. Although the names of the
frieze groups were different from the ones presented in this report, the
symmetries in each group are the same as the ones presented in Table 4. Her
collected data were photographs of decorations of railings and balustrades in
the countries she had visited.
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The next focus to this project is to create geometric designs of the different
wallpaper groups using Geometer‘s Sketchpad®. Other authors have written
in their paper about using other software to create geometric designs in their
local context.
In Majewski and Wang (2009), the authors presented an
overview of the Chinese lattices history and classification done by Daniel
Sheets Dye. They examined Chinese lattices found in wooden windows and
doors from a transformation geometry angle and by concentrating purely on
planar symmetry groups. Then they showed the classification of 2D patterns
into the seventeen symmetry groups with different names.
Finally, they
modelled patterns chosen from Chinese lattices with Dynamic Geometry
software but the examples shown in that paper were created using Geometer‘s
Sketchpad® too. In another earlier published paper written by Majewski and
Wang (2008), they modeled Chinese lattices using another software named
MuPAD, a modern Computer Algebra System from Germany.
Last of all, activity worksheets for upper primary students are to be produced
by utilising the geometric designs created from this project. The authors,
Colgan and Sinclair (2000:pp.2,7), also extended investigation of frieze
pattern into the classroom. The paper describes how they have included
paper folding and mathematics using paper dolls to explore symmetry,
transformation groups and coding system. Their unit was designed for 11-14
year old students.
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CHAPTER FIVE: DATA COLLECTION
AND RESULTS
5.1 Classification of Data into Frieze Groups
After presenting frieze and wallpaper groups explanations in Sections 4.1 and
4.2, the collected data are classified into the different groups. Most of the
data collected were pictures taken from mosques while some were taken from
museums in Singapore, non Muslim state, Kuala Lumpur and Muslim state,
Terengganu in Malaysia.
This section will classify collected data which are patterns that only extends
horizontally into different frieze groups.
Frieze group r1m, as illustrated in Figure 11, is of translational symmetry in
one dimension and a perpendicular reflection.
Figure 11 Frieze Group r1m
Legend
Axes of Reflection:
axis of reflection
axis of glide-reflection
2-fold
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The following Figures 12 and 13 are of translational symmetry with
perpendicular and parallel reflections and it is categorised as frieze group
r2mm.
Figure 12 Frieze Group r2mm
Singapore 1824 AD: Exterior Wall. [concrete] (Masjid Sultan Compound).
Figure 13 Frieze Group r2mm
China 1392 AD: Parapet of Xian Mosque. [wooden replica] (Islamic
Civilisation Park Collection).
Next, frieze group r2 has translational symmetry and perpendicular twofold
axes. Figure 14 depicts such a frieze group.
Figure 14 Frieze Group r2
Singapore 1824 AD: Carpet. [wool] (Masjid Sultan Compound).
Legend
Axes of Reflection:
axis of reflection
2-fold
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Then, there is frieze group r11g which has only glide reflection symmetry in
one direction and its example is shown in Figure 15.
Figure 15 Frieze Group r11g
For the frieze group r2mg, Figure 16 illustrates glide reflection symmetry and
perpendicular reflections.
Figure 16 Frieze Group r2mg
Singapore 1824 AD: Carpet. [wool] (Masjid Sultan Compound).
Legend
Axes of Reflection:
axis of reflection
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5.2 Classification of Data into Wallpaper Groups
In Section 5.1, collected data which are borders are categorised into frieze
groups. In this section, collected data (network) classified into wallpaper
groups will be shown.
Patterns that extend infinitely in both horizontal and vertical directions might
fall into one of the seventeen wallpaper groups. The identification of these
patterns as one of the seventeen different wallpaper groups is first determined
by the maximum number of rotation order.
If the maximum rotation order is 1, then it could be classified as p, pg, pm or
cm as in Figure 17, 18, 19 or 20.
Figure 17 is an example of wallpaper group p as it does not possess any
reflections or glide reflections.
Figure 17 Wallpaper Group p
Syria 706-715 AD: Exterior Wall of The Great Umayyad Mosque. [scaled
replica] (Islamic Art Museum Malaysia Collection).
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For a pattern with maximum rotation of order 1 and reflections, it is type pm
and is shown in Figure 18.
Figure 18 Wallpaper Group pm
Israel 692 AD: Kubbah As-Sakhrah. [scaled replica] (Islamic Civilisation
Park Collection).
Type pg has only glide reflections in one direction and maximum rotation
order of 1 but no reflections. Figure 19 shows one example.
Figure 19 Wallpaper Group pg
Iran 17th century AD: Textile Fragment. [silk brocade] (Islamic Art Museum
Malaysia Collection).
Legend
Axes of Reflection:
axis of reflection
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Figure 20 illustrates wallpaper group cm. It has maximum rotation order of 1
and also reflections and glide reflections in one direction.
Figure 20 Wallpaper Group cm
Russia 16th Century AD: Kul Sharif Mosque. [scaled replica] (Islamic
Next, the criterion is maximum rotation order of 2. If this is met, it could be
p2, p2gg, p2mm, p2mg or c2mm as shown in Figure 21 to 27. With that
criterion and no other reflections or glide reflections, it is classified as p2 as
shown in Figure 21.
Figure 21 Wallpaper Group p2
Singapore 2011 AD: Wrapping paper. [paper] (A shop selling Malay and
Muslim artefacts outside Masjid Sultan).
Legend
Axes of Reflection:
axis of reflection
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With the same mentioned criterion, p2mm also has reflection axes on both
horizontal and vertical directions. Examples are shown in Figure 22 and 23.
Figure 22 Wallpaper Group p2mm
Egypt 1848 AD: Mohammed Ali Mosque. [scaled replica] (Islamic
Malay Archipelago 20th century AD: Songket Shawl. [silk woven with
supplementary-weft gilt thread] (Islamic Art Museum Malaysia Collection).
Legend
Axes of Reflection:
axis of reflection
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Figure 24 shows an example of a wallpaper group with the same criterion as
before. It is p2mg and has reflection axis and perpendicular glide reflections.
Figure 24 Wallpaper Group p2mg
Uzbekistan 1127 AD: Kalyan Minaret. [scaled replica] (Islamic Civilisation
Park Collection).
Applying the same criterion to type p2gg, as shown in Figure 25, it has two
glide reflection axes, one of them is perpendicular to the other.
Figure 25 Wallpaper Group p2gg
Malaysia 2008 AD: Masjid Krystal. [footpath] (Masjid Krystal).
Legend
Axes of Reflection:
axis of reflection
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c2mm is the last type that satisfies the criterion of maximum rotation order of
2. It has reflection axes in two directions, each direction perpendicular to the
other, as displayed in Figures 26 and 27.
Figure 26 Wallpaper Group c2mm
Indonesia 19th century AD: Ceremonial Cloth. [silk weft ikat with
supplementary-weft gold threads] (Islamic Art Museum Malaysia Collection).
Figure 27 Wallpaper Group c2mm
Tunisia 670 AD: The Great Mosque Qairawan. [scaled replica] (Islamic
Legend
Axes of Reflection:
axis of reflection
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If the maximum rotation order is 4, then it could be p4, p4gm or p4mm as in
Figure 28 to 32. Figure 28 illustrates p4 with only maximum rotation order 4.
Kazakhstan 1399 AD: Exterior Wall of Khodja Ahmad Yassaviy Mosque &
Mausoleum. [scaled replica] (Islamic Art Museum Malaysia Collection).
Figures 29, 30 and 31 depicts the same criterion as p4 but they have reflection
axes horizontally, vertically and diagonally.
India 18th century AD: Jali. [stone] (Islamic Art Museum Malaysia
Collection).
Legend
Axes of Reflection:
axis of reflection

4-fold
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Turkey 1557 AD: Suleyman Mosque. [scaled replica] (Islamic Civilisation
Park Collection).
Afghanistan 17th Century AD: Khwaja Abu Nasr Parsa Mosque. [scaled
replica] (Islamic Civilisation Park Collection).
Legend
Axes of Reflection:
axis of reflection

4-fold
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Last pattern that has the same criterion as mentioned before is p4gm. It is
depicted in Figure 6 and 32. It has reflection axis and its rotation axis is not
on the reflection axis.
Figure 32 Wallpaper Group p4gm
Malaysia 2008 AD: Masjid Krystal. [footpath] (Masjid Krystal).
Legend
Axes of Reflection:
axis of reflection

4-fold
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Patterns grouped under p6mm have maximum rotation order of 6. p6 has the
same criterion as p6mm. The difference is that p6mm has 6 reflection axes
but none for p6. Figures 33 to 35 have illustrated type p6mm.
Pakistan 1673 AD: Badshahi Mosque. [scaled replica] (Islamic Civilisation
Park Collection).
Eurasia 14th century AD: Pair of Tile Panels. [underglaze painted with
calligraphy] (Islamic Art Museum Malaysia Collection).
Legend
Axes of Reflection:
axis of reflection

6-fold
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Saudi Arabia 12th Century AD: The Prophet’s Mosque Nabawi. [scaled
replica] (Islamic Civilisation Park Collection).
So, type p6 with only rotations is shown in Figure 41 in the next chapter.
The remaining wallpaper groups p3, p3m1 and p31m have maximum rotation
order of 3. They are mentioned and illustrated in the following chapter as
well.
Legend
Axes of Reflection:
axis of reflection

6-fold
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CHAPTER
SIX:
GEOMETER’S
APPLICATION
OF
SKETCHPAD®
SOFTWARE
6.1 Construction of Frieze Patterns
From the classification of the collected data in Section 5.1, only five out of
seven frieze groups (r1m, r2mm, r2, r11g and r2mg) were identified. The
absence of frieze patterns, r and r11m, are illustrated in Figure 36 and 37.
They are constructed using Geometer‘s Sketchpad®.
Figure 36 Frieze Group r
(Inspired by Morandi, 2003:pp.2)
Figure 37 Frieze Group r11m
(Inspired by Colgan and Sinclair, 2000:pp.6)
Legend
axis of reflection
2-fold
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6.2 Construction of Wallpaper Patterns
Two constructed frieze patterns were presented in Section 6.1. This section
will illustrate rare wallpaper patterns absence from Section 5.2. Only 13 out
of the 17 wallpaper groups mentioned in Section 5.2 namely p, p2, p1m, p1g,
p2mm, p2mg, p2gg, c1m, c2mm, p4, p4mm, p4gm and p6mm were
identified from the collected data. The absent wallpaper patterns, p3, p3m1,
p31m and p6, are shown in Figure 38 to 41. They are constructed using
Geometer‘s Sketchpad®.
Wallpaper groups p3, p3m1 and p31m have maximum rotation order of 3.
p3 has only rotations as shown in Figure 38.
(Inspired by Edwards, 2002)
Legend
axis of reflection

3-fold
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Figure 39 is of type p31m which also has maximum rotation of order 3. In
addition, it has 3 reflection axes parallel to the sides.
Figure 39 Wallpaper Group p31m
(Inspired by Blanco and Harris, 2011:pp.35)
Legend
axis of reflection

3-fold
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Similar to p3 and p31m is type p3m1 which also has maximum rotation of
order 3. However, its 3 reflection axes are perpendicular to the sides as
illustrated in Figure 40.
Figure 40 Wallpaper Group p3m1
(Inspired by Blanco and Harris, 2011:pp.37)
Legend
axis of reflection

3-fold
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As mentioned in Section 5.2, p6 has maximum rotation of order 6 only. It is
depicted in Figure 41.
(Inspired by Collins and Liu, 1998:pp.49)
Legend
axis of reflection

6-fold
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CHAPTER SEVEN: APPLICATION OF
KNOWLEDGE TO PROFESSION
7.1 Objectives of Lesson
Using the knowledge of frieze and wallpaper groups as well as their patterns
constructed in Sections 6.1 and 6.2, it is aimed to achieve one of the project
objectives stated in Section 1.1. It is to produce activity worksheets for upper
primary students of High Progress (HP).
The target level for the design of the worksheets is HP primary 4 students.
Only at primary 4, the students will learn sub-topics Symmetry and
Tessellation which are under the main topic Geometry. These sub-topics are
most relevant to the project research as compared to other sub-topics like
Angles, Nets.
The worksheets will require primary 4 students to understand new
Mathematics skills that are not in the syllabus. Hence, HP primary 4 students
are selected.
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The following table is a segment of the primary 4 Mathematics Syllabus.
Table 6 Mathematics Syllabus Primary 4 Geometry/Symmetry and
Tessellation
Ministry of Education (2006:pp.24)
Geometry
Symmetry
Include:

identifying symmetric figures

determining whether a straight line is a line of symmetry of
a symmetric figure

completing a symmetric figure with respect to a given
horizontal/vertical line of symmetry

designing and making patterns
Exclude:

finding the number of lines of symmetry of a symmetric
figure

rotational symmetry
Tessellation Include:

recognising shapes that can tessellate

identifying the unit shape in a tessellation

making different tessellations with a given shape

drawing a tessellation on dot paper

designing and making patterns
Although primary 4 students were only expected to learn Mathematics skills
as listed in the syllabus above, the worksheets designed in this project will
allow HP primary 4 students to learn the different transformations, some
classifications of frieze and wallpaper groups.
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For the selected students to be able to create their own frieze and wallpaper
patterns eventually, they have to learn the fundamentals behind these created
patterns.
The fundamentals are the different transformations needed to
generate the patterns. Some patterns might be complicated as it consists more
than one symmetry.
Due to the age and ability of the target group, students will only create
patterns of these frieze groups, r1m and r2 and wallpaper groups, p2mm and
p4. Before the students are able to create the selected patterns, they will need
to go through two lessons on the topics, Transformation and Frieze and
Wallpaper group. Table 7 presents planned lesson objectives of the two
lessons.
Table 7 Primary 4 Mathematics Lesson Objectives
Topic: Geometry
Transformation SIO: Pupils will be able to

identify whether a symmetric figure is a reflection,
rotation or translation.

determine whether a straight line is the line of
symmetry or whether a point is the point of rotation of a
symmetric figure.

complete a symmetric figure with respect to a given a
horizontal/vertical line of symmetry or a point of
rotation.

complete a symmetric figure by translating.
Frieze and
SIO: Pupils will be able to
wallpaper

design and make patterns of frieze group r1m and r2.
group

design and make patterns of wallpaper group p2mm and
p4.
The next section will highlight the activities which the students can do to
meet
the
stated
objectives
ultimately.
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7.2 Creation and Consolidation of Activities
Lesson objectives have been presented in Section 7.1. This section is about
the planned student activities. The student activities can range from online
lessons, interactive games/manipulatives to hands-on games/manipulatives,
notes or worksheets.
Appendix B and C present lesson plans with lesson objectives and
accompanied student activities.
Activities reflected in Appendix B are
adapted from Moore (2011).
The next chapter will state challenges and problems encountered in the course
of meeting the project objectives and present discussed resolutions.
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CHAPTER EIGHT: PROBLEMS AND
DISCUSSIONS
S
tudying the seventeen wallpaper patterns is a new concept to grasp.
Initially, it had presented an adequate level of difficulty in
comprehending its classification.
Acquiring this new knowledge
through literature research and extensive reading is useful in the identification
of the collected data.
From the start of the project to preparing the interim report, the main
difficulty was collecting authentic wallpaper patterns that were within
Southeast Asia relating to Islam. Majority of the patterns collected were
categorised under frieze groups.
After thorough discussion with project
supervisor, acquiring Islamic wallpaper patterns beyond Southeast Asia were
acceptable. These wallpaper patterns also need not be directly related to
Islamic art. Hence, while travelling in June to mosque and museums in Kuala
Lumpur and Terengganu, many wallpaper patterns beyond Southeast Asia
had been collected. This had brought much more data samples and variety to
the research.
The wallpaper patterns collected were very complicated and elaborated.
There were many added decoration to the generating region and it was made
vibrant with many colours. This would require more skills to distinguish the
type of group it would be classified into. For an easier approach, disregard
the colours and analyse the patterns in black and white.
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Another challenge faced was when there was a necessity to become skillful at
using Geometer‘s Sketchpad® to construct certain frieze and wallpaper
patterns absent from the collected data. Many patterns collected fall into
common groups. A number of frieze and wallpaper patterns were much rarer
and less commonly available therefore acquiring them was an anticipated
problem. Sufficient time for familiarisation had to be spent to get acquainted
with the new software in order to construct these absent patterns.
Local religious practice disallows non Muslims to enter all the mosques in
Terengganu except Masjid Krystal. Therefore the collection of patterns had
become even more limited. Furthermore, in the mosque, women are not
allowed into the main hall for prayers. For this reason, collection of data
could only take place outside the main hall or building.
Creation of the students‘ worksheet was planned to be completed by
beginning of September.
When the worksheets were ready, it was
inappropriate to carry out the lesson and activities with experimental students
as the teachers were busy preparing the students for examinations in October.
Colleagues and Head-of-Department were occupied with other year-end
duties. For that reason, it was inappropriate to seek their assistance.
After putting forward these challenges and problems, the following chapter
will offer some recommendations as well as encapsulate the main points of
this research.
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CHAPTER NINE: CONCLUSIONS AND
RECOMMENDATONS
T
he report began by presenting classifications of frieze and wallpaper
groups. Frieze and wallpaper patterns were subsequently collected
from mosques and museums. They were analysed and classified
into the different frieze and wallpaper groups. Some collected data had no
patterns while the rests were identified.
Although many patterns were
collected, it was found that some of the frieze and wallpaper groups were
lacking from the identification. Therefore, Geometer‘s Sketchpad® was used
to construct these lacking patterns. These constructed patterns were later
utilised to create activity worksheets for students.
The main conclusion that can be drawn from this research is patterns or
borders found on flat surfaces may not fall into any one of the seven frieze or
seventeen wallpaper groups unless the pattern repeats itself. If the identified
generating region repeats horizontally, it will then fall into one of the seven
frieze groups.
Likewise, if the identified generating region repeats both
horizontally and vertically, it will fall into one of the seventeen wallpaper
groups. After that, the maximum rotation order is identified first followed by
the detection of any reflection or glide-reflection axes to further categorise the
pattern into one of the seventeen wallpaper groups.
It is interesting to discover that majority of the collected patterns fall into the
frieze groups, r2mm and r1m, and wallpaper groups, p4mm and c2mm whilst
frieze group, r and r11m and wallpaper groups, p3, p3m1, p31m and p6, are
much more rare and difficult to collect (Refer to Appendix F). McLeay
(2006) also mentioned the rarity of r11m in her article.
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Collecting patterns of seventeen different wallpaper groups might deem
challenging if collection is limited to the scope of Islamic Arts and within
Southeast Asia. Besides viewing exhibits related to Islamic Arts and from
Southeast Asia, the three museums visited have also displayed traditional and
cultural exhibits from East Asia, Central Asia and Middle East. In the light of
this, it is recommended that the collection of data could be extended outside
Southeast Asia. In addition, the research could expand beyond Islamic Arts.
In this report, lessons and activities have been designed and created. On the
other hand, it was not conducted in class due to clashes with the examination
preparation period. Consequently, it is proposed that seeking a colleague or
Head-of-Department could aid in vetting the lessons and activities planned.
During the collection of data in Terengganu, it was found that non Muslims
are only allowed into Masjid Krystal. In addition, in Masjid Krystal, only
males are allowed into the main hall for prayers. It is suggested that seeking
the help of a Muslim friend could help take photographs of the interiors of
other mosques.
This report introduced the classification of the seven frieze and seventeen
wallpaper groups. For an extension to this research, it is proposed that the
classification could be substantiated with evidence. An investigation on the
proof of the seven frieze and seventeen wallpaper groups could be carried out.
In this report, a brief introduction of Islamic art background had been
highlighted. The key feature was that the geometrical design behind Islamic
art held a spiritual and sacred meaning.
Based on this finding, it is
recommended that the research could be developed further by investigating
the origin to such meaning behind Islamic art and probably also exploring
into the method of designing and forming these patterns by the various artists.
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CHAPTER TEN: CRITICAL REVIEW
AND REFELCTIONS
M
y project‘s objectives: 1) to investigate a wallpaper pattern
mathematically using the concept of wallpaper groups; 2) to
classify different authentic two-dimensional Islamic Art
patterns into different wallpaper groups; 3) to construct geometric designs of
the different wallpaper groups using Geometer‘s Sketchpad®; 4) to produce
activity worksheets for upper primary students by utilising the geometric
designs have been met.
Related works have been discussed in the report and comparisons have been
made.
Although there were numerous studies on Frieze and Wallpaper
groups, none had presented classifications of real data collected from the five
museums/mosques, Masjid Sultan, Asian Civilisation Museum, Islamic Art
Museum Malaysia, Masjid Krystal, Islamic Civilisation Park or a frequency
tally to conclude the most and least common groups. In addition, Geometer‘s
Sketchpad® was used to construct some patterns in this paper. It is not
observed in other related works that they had used this software in their
papers except in the work of Majewski and Wang (2009). Furthermore,
activities were put together in lesson plans, notes and worksheets were
created for use unlike in other works.
This paper will interest students
researching in this area who need a variety of real patterns classified into the
different frieze and wallpaper groups and the manner for their identification.
Teachers will also find this paper useful if they would like to extend their
students‘ interest in geometry as there are ready lessons to be used directly.
The notes and worksheets are attached for adaptation or direct printing for
their students‘ use.
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In this report, new proof or proof of the classification is not presented.
Despite this, it had listed accurate and valid references to support the
classification of the seven frieze and seventeen wallpaper groups. The report
provided introductory knowledge to allow the reader to better appreciate the
classifications.
An Islamic background to the art is also put forward to
enhance the report.
The classifications to the collected patterns were
explained. Reflection and glide-reflection lines were drawn to show the way
the patterns tessellate. The issue of certain lacking frieze or wallpaper groups
was resolved by constructing those patterns using Geometer‘s Sketchpad®.
The constructed patterns were explained, with reflection and glide-reflection
lines clearly shown as well. The conclusion on the least and most common
frieze and wallpaper groups was determined with a sample size of over 200
patterns which was considerably sufficient with respect to the five places
where the data was collected.
Before the submission of my proposal, ample time was spent discussing the
objectives of my project and the contents of my proposal. The activities
needed to do to meet the objectives were identified. Based on the objectives
and activities, the Gantt chart was drawn up to schedule the work. In order to
meet the criteria and targets for the different phases of the project, my
supervisor had agreed to meet at least once a month to monitor work done.
Spending a minimum of 13 hours weekly on this project was also one of the
expected criteria. Problems or risks that might hinder my project delivery
were also discussed to minimise the risks of not meeting the targets set at
each phase. From these discussions, it had taught me to be systematic in my
planning, stay focused with my objectives and activities, be disciplined in
meeting deadlines and always be prudent in dealing with my soft copies.
Every meeting session is recorded in meeting logs in order to review the
decisions made after the discussion as well as to document the next intended
activity to undertake (Refer to Appendix G1-G10).
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Prior to the submission of the interim report, a thorough understanding of the
classification of frieze and wallpaper groups and sufficient number readings
to support ones understanding have to be achieved. The next short term goal
was to collect data from mosques and museums. Then the collected data was
identified and classified. It is crucial to read up on other similar projects or
researches completed by others in order to make comparisons and
improvements could be applied. My project supervisor recommended some
journals and conference papers and shared methods of researching.
Subsequently, retrieving more of such resources was done independently.
From the above discussions, I had learnt to manage the project by setting
numerous short term goals, as done in Gantt chart. After reading journals, I
realised that many others had researched on the same topic but had applied it
in many ways. I was then able to do a comparison on the work others had
done and objectives I had set to accomplish. Citing references had also made
me appreciate the importance of affirming and ascertaining my reading and
newly acquired knowledge.
It had also made me realised that the
classification of wallpaper patterns is related to physics. Furthermore, there
were many different terms used by different mathematicians and physicists in
naming these patterns while referring to the same group. Researching into
Islamic art background had also made the research an interesting and
meaningful one. The emphasis of the created designs was based on unity,
logic and order yet there was no sufficient documentary proof on the theory to
the artistic meaning of the designs. It was fascinating to discover that the
Arabic designs were influenced by mathematicians or philosophers and a
probable relation between Islamic and Western art.
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After the submission of the interim report, my next goal was to have my
classification verified by exploring other people‘s work along with obtaining
supervisor‘s valuable feedback and knowledge.
My supervisor had also
approved my constructed patterns and the way I had made use of the
constructed patterns to create students‘ notes and worksheets. Goals at each
phase was achieved therefore there was a clear indication that my project was
on task.
This project had given me an opportunity to learn new Mathematical concepts
and apply concepts to real data. I gained knowledge of using a new software
to construct patterns and it had helped me to better understand the
construction of patterns under the different groups. Along with this skill, I
could also apply my knowledge to my work and impart this knowledge to my
students through carefully selected activities. In my view, I had also picked
up writing skills, the correct method to cite references and researching skills
to check the reliability and credibility of websites and documents found. I
had a chance to appreciate and be exposed to one of the dominant religions in
the world, Islam – with a rich culture that spills over thousands of years
spanning across continents.
This is in line with the government‘s
propaganda, religious harmony. And, this project had also given me the
opportunity to travel to places such as Terengganu, mosques and Islamic
museums that would never have crossed my mind. This project had indeed
benefitted me in many ways.
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APPENDICES
Appendix A Gantt Chart
Jan-Feb
WBS
1
1.1
Plan
Start
Classification of Wallpaper Group in Islamic Art
Project planning & prep
30-Jan
Initial meeting with supervisor
30-Jan
Literature research part 1
6-Feb
Development of plan
6-Feb
Development of proposal
13-Feb
Supervisor review
27-Feb
Refinement of proposal
27-Feb
Tasks
Plan
Days FTEs Hrs. 1/23 1/30
Finish
607
6-Mar
5-Feb
5 0.02 1
5-Mar 20 0.30 39
12-Feb
5 0.05 2
5-Mar 15 0.30 29
6-Mar
6 0.02 1
6-Mar
6 0.30 12
1.2 Project Management
1.2.1 Requirements analysis
Supervisor review
Refinement of approaches
7-Mar
7-Mar
20-Mar
20-Mar
26-Mar
19-Mar
26-Mar
26-Mar
9
5
5
0.30
0.05
0.30
18
2
10
1.2.2 Implementaton of methods
Collection of wallpaper pattern
Classification of wallpaper pattern
Supervisor review
27-Mar
27-Mar
27-Mar
3-Apr
10-Apr
16-Apr
9-Apr
2-Apr
9-Apr
16-Apr
10
5
5
5
0.30
0.30
0.30
0.05
20
10
10
2
1.2.3 Investigations
Analysis & application
Preparation of interim report
Supervisor review
17-Apr
17-Apr
17-Apr
24-Apr
1-May
7-May
7-May
30-Apr
7-May
7-May
14
10
10
5
0.30
0.30
0.30
0.05
27
20
20
2
1.2.4 Refinement of model
Mastery of the use of software
Designing of wallpaper
Supervisor review
Refinement of design
8-May
8-May
15-May
29-May
12-Jun
19-Jun
25-Jun
11-Jun
28-May
11-Jun
18-Jun
25-Jun
20
10
10
5
5
0.30
0.30
0.30
0.05
0.30
39
20
20
2
10
1.2.5 Application of model
Create design
Supervisor review
Validate design
Supervisor review
26-Jun
26-Jun
26-Jun
17-Jul
24-Jul
7-Aug
13-Aug
6-Aug
16-Jul
23-Jul
6-Aug
13-Aug
30
15
5
10
5
0.30
0.30
0.02
0.30
0.02
59
29
1
20
1
1.3
14-Aug
14-Aug
14-Aug
28-Aug
4-Sep
2-Oct
9-Oct
23-Oct
13-Nov
1-Oct
3-Sep
3-Sep
1-Oct
8-Oct
22-Oct
13-Nov
35
15
5
20
5
10
16
0.30
0.30
0.02
0.30
0.02
0.30
0.30
68
29
1
39
1
20
31
Project report & poster design
Development of project report
Supervisor review
Preparation of poster
Supervisor review
Refinement of report & poster
Finalise report & poster
2/6
Feb-Mar
2/13 2/20 2/27
3/6
3/13 3/20 3/27
Mar-Apr
4/3
4/10 4/17 4/24
5/1
Apr-May
May-Jun
Jun-Jul
5/8
6/5
7/3
5/15 5/22 5/29
6/12 6/19 6/26
8/7
8/14 8/21 8/28
Aug-Sep
9/4
Sep-Oct
Oct-Nov
9/11 9/18 9/25 10/2 10/9 10/16 10/23 10/30 11/6 11/13
#
#
1
1
1
1
Note: Calendar week dates begin on Mondays.
Jul-Aug
7/10 7/17 7/24 7/31
#
JAN2011/MTH/015
Page 74 of 117
Appendix B Lesson Plan on Transformation
Subject: Mathematics
Level/Ability: Primary 4 / HP
Topic: Transformation

identify whether a symmetric figure is a reflection, rotation or translation.

determine whether a straight line is the line of symmetry or whether a point is the point
of rotation of a symmetric figure.

complete a symmetric figure with respect to a given a horizontal/vertical line of
symmetry or a point of rotation.

complete a symmetric figure by translating.
Description: Pupils will learn translation, rotation and reflection through using of interactive
websites, hands-on activities and class discussions.
Background knowledge: Pupils have an understanding of identifying symmetric figures,
determining whether a straight line is a line of symmetry of a symmetric figure and
completing a symmetric figure with respect to a given horizontal/vertical line of symmetry.
Duration: 60 minutes
Activities
1.
Duration

Introduction:

Get pupils to try to form as many figures out of
Resources
15 min
tangram
the 14 figures shown using the tangram shapes.

Interactive
manipulatives:
Lead pupils to slide, rotate or flip to form the
http://nlvm.usu.edu
figures.
/en/nav/frames_asi
d_268_g_1_t_3.ht
ml

Interactive
whiteboard
2.

Development:
Pupils are given tangram shapes to share. Pupils will
30 min
Tangram cutouts:
have hands-on as the teacher explains the following:
http://www.creativ
1) Translate (slide)
eimaginations.net/

PAGE7.HTML
Demonstrate how to slide the tangram shape in
various directions.

Lead pupils to notice that no matter which
direction the tangram shape moves, the shape
remains the same.
JAN2011/MTH/015

Page 75 of 117
Write ―A slide is to translate.‖ on the
whiteboard. Tell pupils that a slide is to
translate.
It is where the shape moves to
another place but the shape remains the same.
2) Rotate (turn)

Write ―A turn is to rotate.‖ on the whiteboard.

Demonstrate rotations by rotating any one of the
tangram shapes 360 degrees then 90 degrees.

Talk about how the tangram shapes may look
different when rotated but the shape remains the
same.
3) Reflect (flip)

Demonstrate horizontal and vertical flips.

Encourage pupils to talk about what happens
when the tangrams are reflected.

Lead them to notice that when a tangram shape
is flipped, it becomes a mirror-image.
3.

Conclusion:


Review the concepts of reflecting, translating,
15 min
Teacherprepared notes
and rotating.
and
Give out pupils‘ notes and worksheets.
worksheets
(Appendix D1 and
D2)
4.
Extension:
Pupils may do the following activities to get familiar with the new terms learnt.
Online Lesson: http://www.misterteacher.com/translation.html#definition
Interactive Games:

http://www.onlinemathlearning.com/transformation-game.html

http://www.harcourtschool.com/activity/icy_slides_flips_turns/

http://www.mathsonline.co.uk/freesite_tour/gamesroom/transform/golftrans.html

http://www.mathsonline.co.uk/nonmembers/gamesroom/transform/postshape.html

http://www.innovationslearning.co.uk/subjects/maths/activities/year3/symmetry/sh
ape_game.asp
Interactive Quiz: http://www.misterteacher.com/transformationsquiz.html
Teacher may prepare the following class game found in the link below.
http://www.ntcschool.com/sec/math/t_resources/gamezone/pdfs/mac3_04/class_ch06.pdf
JAN2011/MTH/015
Page 76 of 117
Appendix C Lesson Plan on Frieze and Wallpaper Group
Subject: Mathematics
Level/Ability: Primary 4 / HP
Topic: Frieze and Wallpaper Group

design and make patterns of frieze group r1m and r2.

design and make patterns of wallpaper group p2mm and p4.
Description: Pupils will learn the frieze group r1m and r2 and the wallpaper group p2mm
and p4 through class discussions and the use of an interactive website.
Background knowledge: Pupils have an understanding of what translation, rotation and
reflection is.
Duration: 60 minutes
Activities
1.
Duration
Resources
10 min

Introduction:

Show pupils some pictures of patterns e.g.
Figure 14 and 28.


2.
Interactive
Whiteboard
Ask pupils to identify translation, rotation or
Figure 14 frieze
reflection if any found in the patterns.
group of r2mm
Tell pupils that Figure 14 is an example of a
Figure
frieze pattern and Figure 28 is an example of a
wallpaper group of
wallpaper pattern.
c2mm
28
–
Development:


Write the word ―frieze pattern‖ and ―wallpaper
30 min
Figure 12 frieze
pattern‖ on the whiteboard.
group of r1m –
Tell pupils that a frieze pattern is a pattern that
reflection
extends in a one direction either horizontally or
translation
and
vertically and a wallpaper pattern is a pattern
that extends in both directions.
Figure 16 frieze
Relate the meanings of frieze and wallpaper to
group
Figure 14 and 28.
rotation

Show pupils Figure 12 and Figure 16.
translation

Ask pupils if Figure 12 and 16 are frieze or

Figure
wallpaper and explain if possible.

Lead
pupils
to
identify
the
transformation in Figure 12 and 16.

Show pupils Figure 24 and 30.
kind
of
of
r2
–
and
24
wallpaper group of
p2mm – rotation
order 2, reflection
in 2 directions and
JAN2011/MTH/015

Page 77 of 117
translation
Ask pupils if Figure 24 and 30 are frieze or
wallpaper and explain if possible.


Lead pupils to identify the different kinds of
Figure
transformation in Figure 24 and 30.
wallpaper group of
Use the interactive website to reinforce the
p4 – rotation order
frieze groups r1m and r2 and wallpaper groups
4 only
p2mm and p4.

Frieze
30
and
wallpaper
interactive
illustrations:
http://www.science
u.com/geometry/ha
ndson/kali/index.c
gi?group=wt

Frieze
interactive
illustrations:
http://www.licm.co
m/interactive_gam
es.php?id=2
3.

Conclusion:


Review the concepts of frieze groups r1m and
15 min
Teacherprepared notes
r2 and wallpaper groups p2mm and p4.
and
Give out pupils‘ notes and worksheets.
worksheets
(Appendix E)
JAN2011/MTH/015
Page 78 of 117
Appendix D1 Transformation Students’ Notes
TOPIC: TRANSFORMATION
Page 1 of 1
Pupils‘ Notes
There are three kinds of transformations:
1) Reflection
2) Rotation
3) Translation
After each transformation, the figure stays the same size and shape.
Below shows an example for each transformation:
1) Reflection:
It is a mirror image similar to your reflection. It can also be thought of as
having to flip an image over to the other side. Let‘s take a look at the
examples below.
2) Rotation:
It is like a turn on its centre or about a point. Mathematicians will say ―rotate
the figure about point x‖ Let‘s take a look at the example below.
3) Translation:
It is like a slide. The figure still remains the same shape and orientation but is
moved; there is no need to turn or flip it.
JAN2011/MTH/015
Page 79 of 117
Appendix D2 Transformation Students’ Worksheets
TOPIC: TRANSFORMATION
Worksheet
Name: _________________________ Class: ______
Page 1 of 6
Date:____________
Section A: For Questions 1 to 8, identify each transformation by writing
reflection, rotation or translation in the box provided.
1.
2.
3.
4.
5.
6.
7.
8.
JAN2011/MTH/015
Page 80 of 117
Page 2 of 6
Section B: For Questions 9 to 13, choose the correct answer and write its
number in the brackets (
) provided.
9. Which of the following lines is the line of symmetry of the figure shown
below?
1
2
3
4
(
)
below?
1
2
3
4
(
)
JAN2011/MTH/015
Page 81 of 117
Page 3 of 6
11. Which of the following points is the point of rotation of the figure shown
below?
1
2
3
4
(
)
12. Which of the following points is the point of rotation of the figure shown
below?
1
2
3
4
(
)
below?
1
2
3
4
(
)
JAN2011/MTH/015
Page 82 of 117
Page 4 of 6
Section C: For Questions 14 to 19, do as instructed in the question.
14. Complete the symmetric figure below by drawing its reflection given line
x as the line of symmetry.
x
15. Complete the symmetric figure below by drawing its reflection given line
y as the line of symmetry.
y
JAN2011/MTH/015
Page 83 of 117
Page 5 of 6
16. Complete the symmetric figure below by drawing its rotation given point
x as the point of rotation.
x
17. Complete the symmetric figure below by drawing its rotation given point
y as the point of rotation.
y
JAN2011/MTH/015
Page 84 of 117
Page 6 of 6
18. Complete the symmetric figure below by drawing its translation.
19. Complete the symmetric figure below by drawing its translation.
JAN2011/MTH/015
Page 85 of 117
Appendix E Frieze and Wallpaper Group Students’ Worksheets
TOPIC: Frieze and Wallpaper Patterns
Worksheet
Name: _________________________ Class: ______
Page 1 of 4
Date:____________
For Sections A and B, only use either one of the given flag figures below to
create your patterns.
or
Section A: For Questions 1 and 2, create each of the two frieze patterns, r1m
and r2.
Frieze Pattern r1m:
Example A1
or
1) Use either one of the given flag figures above to create your frieze pattern
r1m.
JAN2011/MTH/015
Page 86 of 117
Page 2 of 4
Frieze Pattern r2
Example A2
or
2) Use either one of the given flag figures above to create your frieze pattern
r2.
JAN2011/MTH/015
Page 87 of 117
Page 3 of 4
Section B: For Questions 3 and 4, create each of the two wallpaper patterns,
p2mm and p4.
Wallpaper Pattern p2mm
Example B3
3) Use either one of the given flag figures above to create your wallpaper
pattern p2mm.
JAN2011/MTH/015
Wallpaper Pattern p4
Page 88 of 117
Page 4 of 4
Example B4
4) Use either one of the given flag figures above to create your wallpaper
pattern p4.
JAN2011/MTH/015
Page 89 of 117
Appendix F Frequency Tally of Frieze and Wallpaper Groups
Frieze
Groups
Masjid
Sultan
Asian
Civilisation
Museum
6
5
4
6
9
1
Islamic
Art
Museum
Malaysia
Masjid
Krystal
Islamic
Civilisation
Park
Total no.
of Frieze
Types
2
1
14
1
8
20
r
r11m
r1m
r2mm
r2
r11g
r2mg
Wallpaper
Groups
10
3
1
1
1
Masjid
Sultan
Asian
Civilisation
Museum
p
p2
1
2
Islamic
Art
Museum
Malaysia
1
Masjid
Krystal
1
1
2
1
3
p2mm
3
1
3
8
15
3
3
1
p2gg
c1m
2
c2mm
2
8
1
1
1
4
2
12
24
3
p4
4
12
3
6
16
1
p4gm
3
4
p2mg
11
Total no. of
Wallpaper
Types
9
p1g
p4mm
Islamic
Civilisation
Park
9
p1m
4
49
1
p3
p3m1
p31m
p6
p6mm
No
3
6
5
13
18
3
2
14
JAN2011/MTH/015
Page 90 of 117
pattern
rosette
4
1
1
Total
54
29
38
17
2
8
69
207
JAN2011/MTH/015
Page 91 of 117
Appendix G1 CAPSTONE Project Meeting Log (1)
1
Date
12 February 2011
2
Time
1450hr – 1600hr
3
Duration
1 hr 10 mins
4
Venue
UniSIM
5
Student
Kam Jiewen (W06605338)
Name(PI)
6
7
Project Name
Classification of Wallpaper Group in Islamic Arts
(Project Code)
(Jan2011/MTH/015)
Review of
Summary of previous meeting
Previous
This is the first meeting.
Meeting and
8
progress
Progress
Minutes of
Meeting was attended by:
current meeting
1) Mr Quek Wei Ching (supervisor)
2) Ms Kam Jiewen
1) Discussed on the objective of the project
2) Asked about the recommended books or software to reference or
use.
3) Discussed on the approach to do project, possible areas to include
in research
4) Discussed on points to include in Part 2 of Project Proposal
5) Discussed on the number of hours to spend per week on the
project
6) Discussed on the frequency to meet supervisor
9
Action items/
1) To learn about wallpaper group and classify Islamic Art into the
Targets to
various groups. If possible, sketch some geometric designs based
achieve
on the wallpaper group.
2) Recommended books:
―Modern Geometries – The Analytic Approach‖ by Micheal Henle
(borrowed)
―Experiencing Geometry‖ by David Henderson
Software: Geometer‘s Sketchpad® (may decide to buy and claim it
from UniSIM) Recommended to read ―Modern Geometries‖
JAN2011/MTH/015
Page 92 of 117
Chapter 21 – Discrete Symmetry, pg 244 – 265 and to complete
reading in 2 weeks, 29/02/2011
3) I may want to uncover some history behind the art or applying it
to my work (as a teacher), teaching of tessellation.
4) I may include the history of why some art are designed or how I
can better teach primary school students to tessellate to make
project research slightly more interesting
5) To spend a minimum of 13 hours weekly
6) To meet supervisor a minimum of once monthly
10
Other
1) -
comment/Areas
to improve
11
Reference
1.
Journals -
materials
2.
Reference books – ―Modern Geometries – The Analytic
Approach‖ by Micheal Henle, ―Experiencing Geometry‖ by
David Henderson
3.
Reports of past years project @
http://sst.unisim.edu.sg:8080/dspace
12
How did you
4.
Website –
5.
Resources – The Geometer‘s Sketchpad®
6.
Others -
7.5
progress so far?
(10 Excellent,
1 Poor)
JAN2011/MTH/015
Page 93 of 117
1
Date
21 February 2011
2
Time
1800hr – 1850hr
3
Duration
50 mins
4
Venue
Singapore Poly T1813 Level 1
5
Student
Name(PI)
6
7
Project Name
(Project Code)
(Jan2011/MTH/015)
Review of
Previous
-
Meeting and
Progress
progress
-
Discussed on the Project Proposal and supervisor‘s expectations
Have been doing Literature Review on wallpaper groups and
working on the Project Proposal
8
Minutes of
current meeting
2) Ms Kam Jiewen
1) Discussed whether there is a need to prove why there are only 17
classifications.
2) Discussed whether there is a need to do a Literature review on
Group/Number Theory or read up on the textbooks, Linear
Algebra/Abstract Algebra.
3) A professor from New Mexico State University has shown proof
on the only 17 classifications using Group Cohomology and
Spectral Sequences. Others have used other ways to arrive to the
17 classification. Discussed whether I should briefly mention
other people‘s work.
4) Since wallpaper group is a collection of symmetry groups,
discussed whether I should mention symmetry groups in my
proposal.
5) Discussed if Tessellation can be included into my project and if I
can also mention the restrictions to tessellating.
6) Discussed if I can use Excel to generate my Gantt chart instead of
using OpenProj.
7) Discussed about the points for risk assessment of PART 3: Project
JAN2011/MTH/015
Page 94 of 117
Plan.
9
Action items/
1) Using of Group Cohomology and Spectral Sequence to prove
Targets to
(which was done by a professor) might be difficult, I can try to
achieve
understand what he had done to extend my research or just focus
on my main objectives. Others have used algebra to show
classification. Alternatively, I can use Geometry to show the 17
classifications.
2) I need not read up on the textbooks Linear Algebra / Abstract
Algebra. Reading up on Modern Geometries and GROUPS &
GEOMETRY BLOCK THREE UNIT GE4 Wallpaper patterns
might be deemed sufficient for now.
3) I may briefly mention on other people‘s work in my proposal to
show awareness of other people‘s research.
4) I may touch on symmetry groups since it is the basic theory
behind wallpaper group.
5) The concept of tessellation comes before wallpaper group. It can
be included into the project.
6) I may use Excel to generate my Gantt chart if I already have a
template.
7) I will mention what factors might hinder me in reaching my
project objectives and what I can do about to reach my objective
ultimately.
10
Other
comment/Areas
1) -
to improve
11
Reference
1.
Journals –
materials
2.
Reference books – GROUPS & GEOMETRY BLOCK THREE
UNIT GE4 Wallpaper patterns – MZS336 Unit GE4 Mathematics
and Computing – A third-level course, TheOpenUniversity
3.
4.
Website – http://www2.spsu.edu/math/tile/symm/ident17.htm
5.
Resources – THE GEOMETER‘S SKETCHPAD® VERSION 4
and Supervisor-provided Activity worksheets
6.
Others – A sample of Examination Timetabling – Research Plan
by Ryan Chan Jun Neng, NUS High School of Math & Science
12
How did you
8
JAN2011/MTH/015
Page 95 of 117
progress so far?
(10 Excellent,
1 Poor)
JAN2011/MTH/015
Page 96 of 117
1
Date
30 March 2011
2
Time
1640hr – 1720hr
3
Duration
40 mins
4
Venue
5
Student
Name(PI)
6
7
Project Name
(Project Code)
(Jan2011/MTH/015)
Review of
Previous
-
Meeting and
Progress
progress
-
Discussed in details on the Project Proposal
Have been doing literature review on classification of wallpaper
groups and collecting sample data (wallpaper design) for
classification
8
Minutes of
current meeting
2) Ms Kam Jiewen
1) Discussed whether there is a need to cite references for the interim
report.
2) Discussed about the comments on my project proposal.
3) Discussed on the wallpaper designs which I have collected.
4) Clarified on the objectives which I should focus on.
5) Explained some of the symmetry groups‘ algorithm to me.
9
Action items/
Targets to
achieve
1) Citing references is necessary in the writing of the interim report
of 4 pages.
2) I have to include the creating of the student‘s activity worksheet
into my Gantt chart. I also need to cite references on work of
others who have researched on similar projects. Some of the
wallpaper designs which I have collected do not fall into any of
the 17 groups. I might not have enough sample designs for
classifications. Thus, need to create some using GSP.
3) Some wallpaper designs collected are complicated due to the
colours so there is a need to simplify the design in order to
classify them. As wallpaper designs are limited, I may use designs
JAN2011/MTH/015
Page 97 of 117
outside South East Asia or non Islamic instead of just Islamic Art
in South East Asia.
4) I should focus on understanding the 17 groups so as to correctly
classify wallpaper designs collected. Employing the knowledge
and skills learnt to create student exercise worksheets.
5) I may investigate the proof of the 17 groups.
10
Other
comment/Areas
1) N.A.
to improve
11
Reference
1.
materials
Journals – A Computational Model for Periodic Pattern
Perception Based on Frieze and Wallpaper Groups by Yanxi Liu,
Member, IEEE, Robert T. Collins, Member, IEEE and Yanghai
Tsin
2.
Reference books –
3.
12
How did you
4.
Website – N.A.
5.
Resources – N.A.
6.
Others – N.A.
8
progress so far?
(10 Excellent,
1 Poor)
JAN2011/MTH/015
Page 98 of 117
1
Date
30 April 2011
2
Time
1500hr – 1540hr
3
Duration
40 mins
4
Venue
5
Student
Name(PI)
6
7
Project Name
(Project Code)
(Jan2011/MTH/015)
Review of
Previous
-
Meeting and
Progress
progress
-
Discussed in details on the Interim Report
Have been researching, classifying collected data and reading up
on other related work
8
Minutes of
current meeting
2) Ms Kam Jiewen
1) Discussed on the length of the Interim Report and the inclusive of
pictures.
2) Discussed about the difficulties in getting wallpaper patterns in
Singapore.
3) Discussed about the points included in the Interim Report.
9
Action items/
1) Length of the report should be kept within 4 pages only. Pictures
Targets to
can be attached in the appendix instead of in the 4-page report.
achieve
Source of the pictures have to be stated as well. It was also
mentioned that some of the frieze patterns can be repeated
vertically in layers to make wallpaper.
2) Suggested visiting the mosque in Kuala Lumpur where the prime
minister goes to pray. Also suggested visiting the museum in
Kuala Lumpur to get more wallpaper.
3) Affirmed my work. Approved of the inclusion of comparing and
contrasting other related work to the project. Length of the history
and background of Islamic art was appropriate. To include
reflection and glide reflection symmetry lines if possible. Also to
include rotational points.
JAN2011/MTH/015
10
Page 99 of 117
Other
comment/Areas
1) N.A.
to improve
11
Reference
1.
Journals – N.A.
materials
2.
Reference books – N.A.
3.
12
How did you
4.
Website – N.A.
5.
Resources – N.A.
6.
Others – N.A.
8
progress so far?
(10 Excellent,
1 Poor)
JAN2011/MTH/015
Page 100 of 117
1
Date
28 May 2011
2
Time
1400hr – 1440hr
3
Duration
40 mins
4
Venue
Singapore Polytechnic T1832 Level 3
5
Student
Name(PI)
6
7
Project Name
(Project Code)
(Jan2011/MTH/015)
Review of
Previous
-
Meeting and
Progress
progress
-
Discussed the details of the Interim Report
Have been researching, classifying collected data and planning
upcoming trips to Muslim states to collect more data required for
classification
8
Minutes of
current meeting
2) Ms Kam Jiewen
1) Discussed on the classification of collected data from Masjid
Sultan (Singapore) mosque and Asian Civilisation Museum.
2) Discussed on the progress of the project.
3) Discussed the plans for June holidays, collecting more data for
classification.
9
Action items/
1) Supervisor has affirmed the classification of data into the different
Targets to
frieze group in the interim report as accurate. He has also affirmed
achieve
most of the classification of data into the different wallpaper
group as accurate except for one. He has shared valuable feedback
and knowledge on the inaccurate classification and the features of
some wallpaper group.
2) Commented that the progress is according to schedule as planned
in the Gantt chart. As for the future write-ups of the report, he
advised not to mention about non-Euclidean wallpaper.
3) Informed him of my plans to go to :
a) National Mosque and the museum, across the mosque, Kuala
Lumpur and;
JAN2011/MTH/015
Page 101 of 117
b) Masjid Abidin, Masjid Tengku Tengah, Masjid Krystal and
Islamic Civilisation Park across the mosque, Terengganu.
10
Other
comment/Areas
1) N.A.
to improve
11
Reference
1.
Journals – N.A.
materials
2.
3.
12
How did you
4.
Website – N.A.
5.
Resources – N.A.
6.
Others – N.A.
8.5
progress so far?
(10 Excellent,
1 Poor)
JAN2011/MTH/015
Page 102 of 117
1
Date
30 June 2011
2
Time
1835hr – 1925hr
3
Duration
50 mins
4
Venue
SIM HQ Level 5
5
Student
Name(PI)
6
7
Project Name
(Project Code)
(Jan2011/MTH/015)
Review of
Previous
-
Meeting and
progress
Discussed the classification of the collected data from Masjid
Sultan (Singapore)
Progress
-
Have been researching, classifying collected data and collecting
data from Muslim states (Kuala Lumpur and Terengganu) in June
8
Minutes of
current meeting
2) Ms Kam Jiewen
1) Discussed the classification of collected data from National
Mosque and the museum which was across the mosque in Kuala
Lumpur
2) Discussed coming plans for the project
9
Action items/
1) Supervisor has affirmed most of the classification of data into the
Targets to
different wallpaper group as accurate. He has shared valuable
achieve
feedback and knowledge on the inaccurate classification and the
features of some wallpaper group.
2) Informed him of my plans:
to start creating wallpaper (uncommonly-found) using Geometer‘s
Sketchpad®
a)
to start creating student worksheets
b) to state the common and uncommon wallpaper displayed in
mosques and museums into the report
c)
to mention in the report that only one mosques in Terengganu
allow non-Muslim to enter, that is Masjid Krystal.
10
Other
1) N.A.
JAN2011/MTH/015
Page 103 of 117
comment/Areas
to improve
11
Reference
1.
Journals – N.A.
materials
2.
3.
4.
Website – N.A.
5.
Resources – N.A.
6.
Others – Islamic Arts Museum Malaysia (KL) and Masjid Negara
(KL)
12
How did you
7.5
progress so far?
(10 Excellent,
1 Poor)
JAN2011/MTH/015
Page 104 of 117
1
Date
30 July 2011
2
Time
1710hr – 1830hr
3
Duration
80 mins
4
Venue
SIM HQ Level 1
5
Student
Name(PI)
6
7
Project Name
(Project Code)
(Jan2011/MTH/015)
Review of
Previous
1) Discussed the classification of collected data from the
Meeting and
National Mosque in Kuala Lumpur and the museum across it
progress
2) Discussed coming plans for the project
Progress
Sketchpad®
8
Minutes of
current meeting
2) Ms Kam Jiewen
9
Action items/
1) Supervisor has affirmed the creation of the different wallpaper
Targets to
created using Geometer‘s Sketchpad®.
achieve
He has shared valuable feedback and knowledge on some of the
created wallpaper.
2) He has also helped clarified matters on project claims.
3) Informed him of my plans:
a)
to start creating student‘s worksheets
b) to start writing the final project report
10
Other
comment/Areas
1) N.A.
to improve
11
Reference
1.
Journals – N.A.
materials
2.
3.
4.
Website – N.A.
5.
Resources – Geometer‘s Sketchpad®.
JAN2011/MTH/015
12
How did you
Page 105 of 117
7.5
progress so far?
(10 Excellent,
1 Poor)
JAN2011/MTH/015
Page 106 of 117
1
Date
31 August 2011
2
Time
1800hr – 1835hr
3
Duration
35 min
4
Venue
Singapore Polytechnic Block 18 T1813
5
Student
Name(PI)
6
7
Project Name
(Project Code)
(Jan2011/MTH/015)
Review of
Previous
1) Supervisor has affirmed the creation of the different wallpaper
Meeting and
using Geometer‘s Sketchpad® and shared valuable feedback and
progress
knowledge on some of the created wallpaper.
2) Discussed coming plans for the creation of student‘s worksheets
and the writing of the final project report.
Progress
-
Have been creating student‘s worksheets using created designs
from Geometer‘s Sketchpad®
-
Have been planning the contents of final project report and
writing of the project‘s abstract.
8
Minutes of
current meeting
2) Ms Kam Jiewen
9
Action items/
1) Supervisor has affirmed the project contents and its sequential
Targets to
flow. It has been clarified that the final project report is a
achieve
continual of a combination of the project proposal and interim
report. Final project report is to be continued based on the
discussed contents with Supervisor.
2) Supervisor has approved the drafted ideas of the project abstract.
3) It has been clarified that the acknowledgements need not include
authors / creators of books / websites that assist in writing of the
report.
4) Supervisor has approved the creation of the student‘s worksheets
teaching them transformation. Worksheets created are progressive
in difficulty. It has been proposed and approved by the supervisor
JAN2011/MTH/015
Page 107 of 117
that the following worksheets to be created will expose students to
only some wallpaper designs. Ultimately, the students will be able
to create some wallpaper designs on their own.
10
Other
comment/Areas
1) N.A.
to improve
11
Reference
1.
Journals – N.A.
materials
2.
3.
12
How did you
4.
Website – N.A.
5.
Resources –
®
, Microsoft Document
7.5
progress so far?
(10 Excellent,
1 Poor)
JAN2011/MTH/015
Page 108 of 117
1
Date
30 September 2011
2
Time
1745hr – 1835hr
3
Duration
50 min
4
Venue
Singapore Polytechnic Library
5
Student
Name(PI)
6
7
Project Name
(Project Code)
(Jan2011/MTH/015)
Review of
Previous
1) Supervisor has affirmed the project contents and its sequential
Meeting and
progress
flow.
2) Supervisor has approved the drafted ideas of the project abstract.
3) Supervisor has approved the creation of the student‘s worksheets
teaching them transformation.
Progress
-
8
Have been drafting final
Minutes of
current meeting
2) Ms Kam Jiewen
9
Action items/
1) Clarified the required number of words for:
Targets to

Problem and Discussion – 500 words
achieve

Conclusion and Recommendations

o
Conclusion – 1 paragraph / 250-300 words
o
Recommendations – 500-600 words
Critical Review and Reflection – 500 words / 1-2 pages
o
Reflect on ones progress in meeting log but
need not make reference to it.
Supervisor has advised to follow the guidelines of the handbook
closely.
2) Discussed on the points on the following chapters:

Problem and Discussion

Conclusion and Recommendations

Critical Review and Reflection

Glossary
JAN2011/MTH/015
Page 109 of 117

o
wallpaper group table
o
Need not include definitions
Appendices
o
Figures, Gantt chart, worksheets and meeting
logs
3) Objectives shall be modified to include research and classification
of frieze patterns
10
Other
comment/Areas
1) N.A.
to improve
11
Reference
1.
Journals – N.A.
materials
2.
3.
12
How did you
4.
Website – N.A.
5.
Resources – Geometer‘s Sketchpad® , Microsoft Document
8
progress so far?
(10 Excellent,
1 Poor)
JAN2011/MTH/015
Page 110 of 117
1
Date
24 October 2011
2
Time
1840hr – 1930hr
3
Duration
50 min
4
Venue
Singapore Polytechnic Library
5
Student
Name(PI)
6
7
Project Name
(Project Code)
(Jan2011/MTH/015)
Review of
Previous
-
Meeting and
progress
of some chapters of the final report.
Progress
-
8
Minutes of
current meeting
Supervisor has clarified issues on the criteria and subject matters
Have been drafting final project report
2) Ms Kam Jiewen
9
Action items/
1) Clarified on the need to reference ideas and resources used in the
Targets to
lesson plans. Supervisor has advised to include these references
achieve
into the report.
2) Supervisor has approved the project claims. The claim form has
since been submitted.
3) Had compiled the collected data into a frequency tally.
Supervisor has affirmed the tally and it is to be included into the
report. A summary statement or conclusion can be drawn from
the table. It is not required to include all 207 pictures into the
report as the file would be too large. At least 1 picture of each
group to be included.
4) Supervisor has affirmed the planned activities and created
resources for the students.
5) Clarified some issues about the inclusion of implementation
phase into the Gantt Chart. Supervisor mentioned that it is not
necessary to include.
6) Shared some findings and research regarding frieze and
wallpaper groups.
JAN2011/MTH/015
10
Page 111 of 117
Other
comment/Areas
1) N.A.
to improve
11
Reference
1.
Journals – N.A.
materials
2.
3.
12
How did you
4.
Website – N.A.
5.
Resources – Geometer‘s Sketchpad®, Microsoft Document
8
progress so far?
(10 Excellent,
1 Poor)
JAN2011/MTH/015
Page 112 of 117
GLOSSARY
Arabesque
an ornamental motif, correlated with Islamic art, of leaves and branches
combined with geometric decoration, human figures, patterns of flowers and
plant and animal forms (Savory, 1994:pp.89).
Axiom
mathematical assumption that is true without the need for formal
demonstration (Easwaran, 2007:pp.2-3; Toida, 2009).
Balustrade
a railing supported by balusters, which forms an ornamental parapet to a
balcony, bridge or terrace (Oxford Dictionaries, 2011).
Dihedral group
a
group
of isometries
(reflections
and
rotations)
(Schattschneider,
1978:pp.439)
Euclidean geometry
the study of plane and solid figures on the basis of axioms and theorems
employed by the Greek mathematician Euclid (Encyclopædia Britannica, Inc.,
2011).
Discrete group
a group that has only a finite number of points inside any circle (Henle,
1997:pp.246-247).
JAN2011/MTH/015
Page 113 of 117
Fatimid Dynasty
political and religious dynasty that dominated an empire in North Africa and
subsequently in the Middle East from ad 909 to 1171. It took its name from
Fāṭimah, the daughter of the Prophet Muḥammad, from whom the Fāṭimids
claimed descent (Encyclopædia Britannica, Inc., 2011).
Four-dimensional space
any three-dimensional body moving in time and leaving a trace of its
movement, the temporal four-dimensional body (Norton, 2008).
Frieze group
pattern that has rotations and/or reflections but one-directional translations
and their inverses (Holme, 2010:pp.451).
Group Cohomology
a theory that intertwines algebraic invariants and topological aspects (Adem,
2007:pp.317)
Hermann-Mauguin-like symbols
international symbols that describe crystal classes from the symmetry subject
matter (Nelson, 2010).
High Progress (HP)
high ability
Invariant
entities, properties, quantities, relationships, etc., that are unaltered by
particular transformations (Brummelen and Kinyon, 2005:pp.248).
JAN2011/MTH/015
Page 114 of 117
Islam
literally ―surrender,‖ ―submission‖ to the will of God; the religion
promulgated by Muhammad and followed today by about one-quarter of the
world‘s population (Hens, 2004:pp.46).
Isometry
symmetry movement, a distance-preserving transformation (The Geometry
Center, 1995).
Lattice
a grid structure resulted from connecting all translation of matching points
(Bart and Clair, 2011).
Maghrebi script
maghrebi also spelled maghribi, in calligraphy, Islamic cursive style of
handwritten alphabet that developed directly from the early Kūfic angular
scripts used by the Muslim peoples of the Maghrib, who were Westerninfluenced and relatively isolated from Islam as it was absorbed into the
eastern part of North Africa. The script they developed is rounded, with
exaggerated extension of horizontal elements and final open curves below the
register (Encyclopædia Britannica, Inc., 2011).
Moorish
to describe a member of a NW African Muslim people of mixed Berber and
Arab descent who conquered the Iberian peninsula in the 8th century, but
were finally driven out of their last stronghold in Granada at the end of the
15th century (Oxford Dictionaries, 2011).
Mosque (Masjid)
literally ―place of prostration,‖ where Muslims gather for prayer; a new
mosque is built where the calls to prayer from the nearest mosque can no
longer be heard (Hens, 2004:pp.46).
JAN2011/MTH/015
Page 115 of 117
Mughal art
art of the Mughals was similar to that of the Ottomans in that it was a late
imperial art of Muslim princes. Both styles were rooted in several centuries
(at least from the 13th century onward) of adaptation of Islāmic functions to
indigenous forms (Encyclopædia Britannica, Inc., 2011).
Muhammad
(b. Mecca, Arabia, ca. 570 A.D., d. Medina, 632 A.D.) recognised as ‖the
messenger of God‖ by the Muslims, he was an Arab merchant who preached
the Islamic faith, began receiving divine revelations about 610 A.D., and was
forced to leave with his followers from Mecca to Medina in 622 A.D. (Hens,
2004:pp.46).
Muslim
a follower of Islam, literally ―one who surrenders,‖ hence, one who has direct
access to his/her God (Islam having no priesthood) (Hens, 2004:pp.46).
Orbifold
a quotient space of the two-dimensional Euclidean space with respect to a
finite group (Kung, 2009:pp.57).
Platonism
any philosophy that derives its ultimate inspiration from Plato. Platonism can
be said to have in common is an intense concern for the quality of human
life—always ethical, often religious, and sometimes political, based on a
belief in unchanging and eternal realities, which Plato called forms,
independent of the changing things of the world perceived by the senses.
Platonism sees these realities both as the causes of the existence of everything
in the universe and as giving value and meaning to its contents in general and
the life of its inhabitants in particular (Encyclopædia Britannica, Inc., 2011).
JAN2011/MTH/015
Page 116 of 117
Point group
in crystallography, listing of the ways in which the orientation of a crystal can
be changed without seeming to change the positions of its atoms. These
changes of orientation must involve just the point operations of rotation about
an axis, reflection in a plane, inversion about a centre, or sequential rotation
and inversion. Only 32 distinct combinations of these point operations are
possible, as demonstrated by a German mineralogist, Johann F.C. Hessel, in
1830. Each possible combination is called a point group (Collins and Liu,
1998:pp.3).
Reflection
mirror symmetric (Kaplan, 2009:pp.11-12)
Rosette
a symmetry group containing rotations and/or reflections but not translations
(Holme, 2010:pp.451).
Rotation
a figure, without symmetry, is repeated about a fixed point until it returns
back to its original position (Ronan, 2006:pp.34).
Specific Instructional Objectives (SIO)
specific measurable objectives to be achieved at the end of a lesson.
Spiral
a circular motion of Arabesque
Symmetry
correspondence in size, shape, and relative position of parts on opposite sides
of a dividing line or medium plane or about a centre or axis (Hens,
2004:pp.46).
JAN2011/MTH/015
Page 117 of 117
Symmetry group
a group of symmetries for a given pattern is called a symmetry group (Asche
and Holroyd, 1994:pp.10).
Tessellate
covers without gaps or overlaps by congruent plane figures of one type or a
few types (Hens, 2004:pp.46).
Transformation
rotation, reflection, glide-reflection, translation or dilation (Majewski and
Wang, 2009:pp.1)
Translation
direct isometry without fixed points (Blanco and Harris, 2011:pp.35)
Wallpaper group
a group of plane isometric movement, which is the symmetry group of
wallpaper patterns (Asche and Holroyd, 1994:pp.10).

classification of wallpaper group in islamic arts

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