CHAPTER 1
INTRODUCTION
1.1
Brief Overview
This chapter opens the discussion on the subject under study. It defines the
phenomenon, its history, and a brief overview of the extent of academic research on
the matter. It then moves on to provide the roundup of the rationale and challenges
of the thesis as a prologue to the contents of the present work. The different
subchapters presented here are:
1. Definition
2. Backgrounds
3. Preceding Research
4. Research Gaps
5. Research Objectives
6. Problem Statement
7. Scopes and Limitations
8. Methods
9. Structure of the Thesis
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1.2
Definition
Rasmi consists of the vertical projection of a star polygon onto the surface
of a rotating solid (Hashemi Nik, 2008). The definition of star polygon on the other
hand given by Wolfram Math World is “A star polygon
, with
positive
integers, is a figure formed by connecting with straight lines every th point out
of
regularly spaced points lying on a circumference”1. This definition has been
modified by the author into a more comprehensible one as “an enclosed polygon
with alternately concave angles derived from the connection of each vertex to an
alternate one on a base convex polygon”.
It is deduced from the definitions above on rasmi and star polygon that the
former derives from the latter, since it is a reflection of the latter over a spherical
surface. However because the spatial field by virtue of reason contains in itself the
plain fields as well, then it will be conclusive that although rasmi itself is derived
from the star polygon, yet it includes the latter in itself turning the regular star
polygon into the two dimensional attribute of rasmi.
As realized by the author, there are three facets to the phenomenon of rasmi.
First there is the logos, or the theoretical concept. This aspect deals with the
definition of rasmi and the understanding of it. This is followed by the
manifestation of rasmi. Manifestations are what materialize in the visual sense.
They may be built, hence real, or graphical, hence virtual; they may also be spatial
or planar. What relates the concept to the manifestations is the third facet of rasmi,
which is the science that deals with its ratios and the mechanics that interpret the
concept and definition into manifestations. This last aspect has been least
understood and explored in science. It is therefore this aspect of rasmi that the
current research intends to examine.
1
http://mathworld.wolfram.com/StarPolygon.html
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1.3
Backgrounds
Expressions of art and the deliberate recreation of forms are the only
workable evidences to a conscious appreciation of star shapes by man. Investigating
this claim is not up to the current research, but evidence at hand attributes the first
mathematical understanding of such shapes to classical Greece. Proclus (412 – 485
A.D.) credits the study and discovery of platonic solids to Pythagoras (570 – 495
B.C.) (Rosán, 1949). However, scholars speculate Pythagoras to have only known
three such solids, namely cube, tetrahedron, and dodecahedron, and that the other
two solids, namely octahedron and icosahedron, and the realization of the five
shapes as the only possibilities in regular solids is credited to Theaetetus (417 – 369
B.C.) (Nails, 2002). Theaetetus was a contemporary of Plato and the nomination of
the indicated solids as Platonic suggests that if not a preoccupation of the general
population of philosophers/mathematicians of the time, it at least was that of the
two individuals above.
This much tangible information on the solids and their discoverers is
available, however the issue of stellation as the subject of this research is only
vaguely understood to have been covered as a subsidiary topic under polytopes –
covering polyhedra, hence the Platonic solids, but also including polygons- by the
latter scholars. To continue with this topic would be an encroachment into the
review of literature on the subject that is to follow in its own pertinent chapter. It is
therefore appropriate to focus more on the history of the material manifestations of
stars as a preview to the findings of the preceding research.
1.3.1
Architectural Background
Attention to the existence of the phenomenon in science was drawn through
the study of a structural technique in architecture that applies the material
manifestation of rasmi in construction. The architectural manifestation has been
known and in practice since 961 A.D. where findings by Hashemi Nik (2008)
attribute the first application of the form to a number of cloister vaults at the
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Mosque of Cordova, Spain (10th century). It is from here that the technique spreads
respectively into the Iberian Peninsula and Northwest Africa, followed by Greater
Iran (Persia and Central Asia-12th century), Britain (14th century), India (16th
century) and Italy (17th century); and with international modernism of the 20th
century, in sporadic manifestations around the globe.
In its Moorish sphere of prevalence that encompasses Northwest Africa and
the Iberian Peninsula, rasmi follows a narrow line of evolution that experiments
only with numbers but not expressions; leaving a spectrum of works from 8 to 32,
all in the primeval expression cheshmé (Ibid, 2008). The same is true about almost
all other manifestations outside the Indo-Iranian range, to include the few samples
of medieval England, Renaissance Italy, and the sporadic modern samples. Indeed
the range of these latter works is also more limited than the Moorish ones, all
invariably cheshmés of 5, 6, and 8. This leaves the credit for the greatest range of
experiments to the Indo-Iranian sphere, where as early as its first introduction in the
12th century, experiment with the phenomenon takes on to new levels, leading to a
spectrum of definitions and classifications. An inquisitive aesthetic tradition,
established with the very first manifestations of rasmi in this new sphere lead to the
discovery of the many revelations and the evolution of entirely new genera of forms
out of the rasmi core, either composite i.e. kasébandi and naxlbandi, or
deconstructed and symmetrically rearranged i.e. yazdi. The history of rasmi in
Indo-Iran begins with its three extant Seljuk manifestations, hence two cloister
vaults at the Jame’ of Isfahan (Figures 3.13 and 3.14), and the cupola over the tomb
of Sultan Sanjar, at Marv, Xorassan (Fig3.24). The samples are few and far apart,
causing debate over the issues of origin and discontinuity of tradition over the next
two centuries. Yet their impact is undeniable considering that –albeit with a two
hundred year lapse- they eventually set the trend for propagation in the new sphere.
Further, it is understood from these very first precedents that from the moment of
touch-base at its new oriental home, rasmi is on a scientific expedition of
mechanical and aesthetic boundaries and challenges.
The greatest diversity of experiments with rasmi commence at the
architectural renaissance of the 15th century in Xorassan, expanding over the next
5
centuries into Persia and India. It is here and in this phase that amputation and
segmentation lead to the many different revelations of rasmi. Here on, mutations
enter, leading to the complete evolution of rasmi into new genera. However, the
core of this entire family of forms remains as rasmi in the simple sense. It is for this
reason that the mathematical questions raised by the current study are answered
solely through the examination of rasmi, needless of encroachment into any of the
three other evolved genera of kaarbandi. Indeed one may argue that most if not all
the geometrical rules governing the other three, are inherent of rasmi, and their
discovery in the latter will pave the way for the understanding of the prior.
1.3.2
Mathematical Background
Citations on Abolwafaa Mohammad bin Mohammad Al-Bouzjani reveal his
familiarity with manifestations of star polygons in the 10th century (Jazbi 2, 1997).
However, the first conscious attempt in the study and taxonomy of the forms appears in
the writings of Thomas Bradwardine in the 14th century (Molland, 1978). Bouzjani on
the other hand hints to forms with star patterns while discussing the topic of gereh. The
latter is a practical mathematician and his treatises largely cater to industrial
applications of math and geometry. In discussing gereh therefore, Bouzjani delves into
the calculation of the areas of the component elements of quasi-crystal, only some of
which possess star shapes. His work focuses on the subject of tessellation and not
exclusively the star polygon.
The first scientific attempt to introduce and independently examine the chapter
of star polygons takes place in the 14th century through the English scholar Thomas
Bradwardine, archbishop of Canterbury. The original writings of Bradwardine have not
been accessible to the author, leaving the accurate delineation of his discoveries
obscure as of this writing. However, an overview of the dominant trends in the
successive study of star polygons leaves a clear historical picture on the areas of
scientific interest and expansion. The second scholar known to have extended the
research on star polygons is Johannes Kepler (17th century). At this point, an
understanding of the definition and scope of polytope is necessary since it encompasses
parallel issues in planar and spatial geometries. The manifestation of a polytope on the
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Euclidean plain is the polygon. Its manifestation in space, which automatically
becomes non-Euclidean, is the polyhedron. Extending this definition into the stellar
manifestations in both geometries the self-intersecting concave polygon that emerges
from the alternate connection of vertices on a regular polygon on the Euclidean plain
becomes the star polygon, and the similarly drawn self-intersecting or stellated
polyhedron becomes a star polyhedron. From Kepler on, the study of star polytopes
immediately focuses on star polyhedra, and all future mathematicians follow by
focusing the greatest portion of their research on the analysis of these volumes. Kepler
was an astronomer, and it is through the demands of astronomy that an analysis of the
spatial geometry of polytopes deems more vital than the Euclidean manifestation, the
star polygon. An overview of existing mathematical sources on star polygons renders a
picture that focuses on the more general frame of star polytopes, with little distinction
between the planar and spatial research. It has been the trend therefore to start
discussions with a brief overview of the subject in the planar form, general conclusions
of which prepare the ground for excursion into polyhedra and then exclusively focusing
on the latter. This in the opinion of the author has left many pristine areas of research
to explore in the topic of star polygons, most especially on areas concerning the study
of rasmi, hence the focus of this research.
1.4
Preceding Research
In spite of the familiarity of construction guilds with rasmi, the technique
wasn’t documented until the 20th century. Even then, as argued in the previous
research, the architectural historians who noticed and photographed the
phenomenon didn’t delve into its scientific analysis. Through Lorzaadé, the late
royal architect of the Pahlavis, himself the product of a pre-academic tradition in
architecture, the technique was documented in design manuals in the 1970s.
Following the late architect, a number of colleagues and disciples produced
manuals on design and construction with conscious knowledge of the phenomenon.
Yet none of these sources attempted a documentation of the history, evolution, and
taxonomy of the phenomenon in architecture. Further, although the phenomenon of
star polygons was first identified by Thomas Bradwardine of 14th century England
as a mathematical chapter the connection between the practice in architecture and
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its mathematical paradigm was never made. This was the task carried out by the
present author under the preceding research.
As argued in the previous research, English citations of the phenomenon
were equally inadequate in its clear explanation. The work under discussion
therefore initiated the research by establishing its industrial and artistic vocabulary
of terms. Subsequently, this vocabulary was compiled together with the
mathematical one. The second portion of the previous study then attempted a
chronology of forms exposing a number of sequential patterns that were
subsequently processed into historical and evolutionary frames documenting the
expansions and transformations of the phenomenon. Finally, an arrangement and
regrouping of all extant possibilities, i.e. manifestations, mutations and revelations,
led to a taxonomical framework for a more technical acquaintance with the family
of forms. The former study then identified a number of areas wherein research
could expand into the more extensive understanding of rasmi or kaarbandi. A
number of suggestions dealt with the historical and archaeological analysis of
buildings, while a suggestion was made on the expansion of knowledge into the
mathematical aspects of rasmi. In this regard, the previous research only laid the
foundations by establishing a basic vocabulary of mathematical terms for the study.
However, the current research, intends to continue into the deeper mathematical
understandings; a long anticipated area of study.
1.5
Research Gaps
The areas unchartered under this new chapter in science are vast and cover a
number of disciplines. These range in addition to such aspects of the science as the
documentation of precedents and a detailed scrutiny of the evolution of
architectural manifestations through the reevaluation of historic models and
archaeological analysis of extant samples, to structural analysis and load
management in rasmi, to establishing the mathematical rules and definitions of this
chapter in geometry.
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1.6
Research Objectives
To elaborate the last issue, it must be added that rasmi and star polygon as
mutual attributes of one another fall into the realm of geometry, since in the purely
theoretical sense, they are mathematical logoi that find material manifestations in
the built world. Further, since regularity is a prerequisite of rasmi, it implies order,
which then alludes to ratios and consequently mathematical formulae. It is here and
up to this research to discover and establish these rules for the advancement of
knowledge.
1.7
Problem Statement
This research intends to discover -and establish the findings of this
discovery- in the most basic areas of mathematical inquiry into the subject of rasmi.
By the definitions given over the first subchapter, rasmi and star polygon become
interchangeable at least in the planar sense. For this reason, the questions delved
into by the research will partially apply to the universal premise of star polygon,
and partially to those areas that find significance from the view point of rasmi and
its pertinent situations.
Three general areas of enquiry have been perceived by the author as the
most basic in the reconnaissance work at hand namely a reevaluation of
Bradwardine’s findings on the sum of internal angles, the method of prediction of
the number of sequels per star polygon, and an enquiry into the correlations
between rasmis and their inscriptions into regular and semi-regular polygons, with
the last category only limited to rectangular and trapezoidal inscriptions.
The questions of the research have been formalized and listed down into
seven items at the end of Chapter 3, under the subtitle 3.4 Statement of the
Problem.
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1.8
Scopes and Limitations
Because this work is the first of its kind, it is more important to establish the
most basic scientific notions as the stepping stones or building blocks of the new
area of science at hand. For this the work draws a premeditated range of
explorations for itself. This predefined range in turn explains the limitations of the
work in a complimentary manner. It is understood that the spatial geometry of
rasmi would be more complex than the planar one. As a first step it then becomes
necessary to establish the planar discussions before moving on to the spatial
analysis. This work will focus on that as its scope.
The limitations of the work are internal; meaning that they are not imposed
by external factors, but are rather inherent of the expedition. It is understood that as
a logical and mathematical realm, there will be no limits to the possible
explorations, and therefore it will be up to the work to draw a demarcation point
beyond which the work may extend indefinitely and run out of the prescribed
timeframe. Time could therefore be considered a limitation here. Another limitation
would be on the background of the researcher who is conducting this work as a
pioneering research presented to an architectural faculty. For this the lack of
discreet mathematical knowledge is considered an inherent limitation. The scopes
and limitations are explained in full detail under Chapter 5.
1.9
Research Approach
This research is on the science of mathematics, albeit approached by the
discipline of architecture. Being of geometrical nature -and hence theoreticalimplies that the data is logical, meaning that it could be generated in the mind as
opposed to tangible material data gathered on the field. And yet because it is in the
premise of geometry, it deals with shapes, which can only be examined empirically
as drawings. A manual execution and compilation of the data proved cumbersome.
AutoCAD design software was utilized for this purpose. The advantages of this
software include in addition to accelerating the design production process, with due
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consideration of amputations and inscriptions that will eventually influence the
results, such additional functions as the measurement of angles, superimposition for
comparative analysis, and the ability to design spatially should the need arise. In the
buildup of knowledge it was realized that star polygons start with the pentagram
and continue indefinitely with the number of vertices. For this the maximum limit
of data generation was defined as the last permutation of the 48-sided polygram
hence R48.
The next step was to refine the data according to research question. For each
question a smaller sample group was taken from the data pool, rearranged for
pattern observation, and if the question pertained to the issues of amputation and
inscription, the necessary processes were implemented over the sample units before
a rearrangement by pattern group. Visible patterns were examined by mathematical
and rational notions in an attempt to formulate or at least document them
scientifically. It must be indicated here that trial and error played an important role
in rearranging the samples in such a way that would reveal a pattern, and in
examining their correlations and attempting to formulate these mathematically.
The work therefore relies largely on the intuition and mathematical capacity
of the examiner. In this sense it is similar to the empirical sciences where the
discovery of a phenomenon depends on the background knowledge of the
researcher with considerable influence from external factors, much in the same way
that natural selection acts upon speciation in the science of biology. The procedures
dictating the examination process have been laid out in full detail over Chapter 6,
Methodology, and followed categorically under Chapter 7. These are the
mathematical and logical procedures of the research and because of their solid and
tangible nature as the scientifically reliable methods they are vital to the structure of
the study, however the trial and error aspects which rely largely on intuition are a
soft mind process that cannot be formulated and categorized and have therefore
done away with minute documentation. Indeed the documentation of the softer
mental processes will be an impossible task and may compromise the structuralism
of this or any logical research.
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1.10
Structure of the Thesis
The present work comprises of eight chapters. However to prepare the
reader for the topics involved and equip him with the technical vocabulary, the
Glossary is supplied at the preliminary stage enlisting the technical terms and their
essential definition, equipped with illustrations where a deeper understanding is
needed. To enrich this work, a complete list of proposed and established scientific
symbols is also presented at the preliminary stage that should be accepted as the
convention for a better scientific representation of the phenomena under discussion.
The main course of the text follows with Chapter 1 which attempts to
present an overview of the problem, highlight the gaps and questions in the
scientific study of the phenomenon, and present a flow of the backbone of the work
comprising of aims of the research, existing literature, data and technical methods.
However because the introductory chapter is not enough to comprehensively
illuminate this entirely new topic in science, a number of its contents are elaborated
in further detail as subsequent independent chapters. For this the second and third
chapter work exclusively to buildup the knowledge of the reader on the importance
of the topic of rasmi to science as a whole and the importance of this research in
particular. The second chapter intends to illustrate the significance of the research
by pointing to the fact that rasmi is not merely the structure visible to the eye but is
in fact a manifestation of a deeper mathematical reality that has yet been awaiting
discovery and exploration. The third chapter presents a pictorial list of the existing
samples in architecture and on drawing, categorizes them by revelation and in the
same manner that the processed data of the research will be regrouped for pattern
observation under Chapter 7, here the forms are regrouped to encourage the reader
into a pattern observation and trigger curiosity on the subject, and then tutor that
curiosity into a directed rational format that is formulated into the seven questions
of the problem statement at the end of the chapter. Chapter 4 then takes a turn from
this buildup of knowledge and curiosity, to delve into the question of existing
knowledge and literature in the world. For this and based on the backgrounds given
both under the first and second chapters, as well as throughout the previous
research by the author, it is understood that knowledge rests with the two unrelated
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groups of traditional builders and mathematical scholars. These are discussed in full
detail throughout the chapter highlighting the gaps of scholarship which in turn
shed more light on the significance of the research. Chapter 5 focuses solely on the
elaboration of scopes and limitations in the work. Chapter 6 presents the research
design and scientific methods applied both in data generation as well as analysis.
Chapter 7 applies this structure to the gathered data and documents all the processes
and outcomes from observation to the extraction of scientific axioms and formulae
as the answers to the seven questions raised at the end of Chapter 3. Chapter 8 lists
down these findings in summary, and then proceeds to discuss the need that gave
rise to the research and -building on which- projects the anticipated course of future
scholarship into the subject.
The body of the research is followed by a compilation of all pertinent
literature that assisted the author in the preparation of the work. Those that were
directly cited throughout the text are listed under References, as textual and
pictorial sources; and those that merely assisted in the buildup of knowledge and
would be useful to the reader who intends to develop similar knowledge
independently are listed down by category as Bibliography under Appendices. The
second part of the Appendix comprises of the data of the research, hence the
AutoCAD designs of the star polygons under discussion, categorized based on
research question.