Dissertation - Fractal Content Khajuraho Temples

_________________________________________________________________________ DISSERTATION

FRACTAL CONTENT OF THE SURFACE OF ARCHITECTURAL COMPOSITIONTHE TEMPLES OF KHAJURAHO

DEMIS ROUSSOS BHARGAVA (9604) JULY 2001

Guide: MR. RAJAT RAY

TULSI VIDYA BHARTI SCHOOL OF HABITAT STUDIESTULSI VIDYA BHARTI SCHOOL OF HABITAT STUDIESTULSI VIDYA BHARTI SCHOOL OF HABITAT STUDIESTULSI VIDYA BHARTI SCHOOL OF HABITAT STUDIES

Vasant Kunj, New Delhi

DISSERTATION Title:DISSERTATION Title:DISSERTATION Title:DISSERTATION Title: FRACTAL CONTENT OF THE SURFACE OFARCHITECTURAL COMPOSITION – THE TEMPLES OF KHAJURAHO

CONTENTS

AcknowledgementsIntroHypothesisMethodologyScope and Limitations____________________________________________________________________________________________________________

I: Information ContentPainting SpaceLayeringOrdering Elements of CompositionComposite Layers, Discrete ElementsCoding SpaceCoding StylesGenes and Systems Generation

II: Fractal GeometryInfinite Complexity: The Fractured Surface of SpaceVariablesQuantifying Complexity

Case Study: KhajurahoEvolution of a Style: Deconstructing SurfaceCoded Layers, or The Spatial Representation Of Ordering PrinciplesThe Complex Surface of Sacred SpaceFractal Dimensions: Quantification of Layered Complexity

ResultsConclusion

ACKNOWLEDGEMENTS

Mr. Rajat Ray, for being patient whenever I went off on one of my more ambitious – orjust plain weird – tangents (and for being a closet rock fan),Dr. K.L. Nadir, for guiding this particular sheep when he needed the help,Mr. A.B. Lall, for the confidence,Mr. A.G.K. Menon, for quietly encouraging me to speak up,Mrs. Madhu Pandit, for the words of encouragement,Mr. Anand Bhatt, for putting things into perspective and making life a lot more interesting,Mr. Nikos A. Salingaros, for the prompt and valuable correspondence,Martin Nezadal and Oldrich Zmeskal, for the HarFA program,Anvita for her valuable advice,The rest of the ‘Pandavas’,Vishal, for the friendship (and the C&Hs),Jaspreet, for listening,Anyone else from the class or elsewhere I may have forgotten (please don’t sue me!),Limp Bizkit, Korn, U2 and all the other bands, for the company during the long days andthe even longer nights,Stephen King, for the Dark Tower,Dana and Dev, for the uninterrupted access to the computer (more or less, anyway!),All my esteemed colleagues at the Academy and

My parents, for everything.

intro

No one is listening.Now you may sing the selfsong,as the bird does, not for territoryor dominance,but for self-enlargement.Let somethingcome from nothing.….Texas Suite: Stan Rice

Very little ismore worth our timethan understandingthe talent of Substance.…A bee, a living bee,at the windowglass, trying to get out, doomed,it can’t understand.Untitled: Stan Rice

The architectural object is painted space. We relate to the space and respond to itthrough its surface. The information encoded by the surface determines ourcorrespondence with the space defined. The wall becomes a surface of interaction,rather than a passive element of delineation. While information encoded may not bedesigned, modes of transfer and representation are within the scope of the designer.

Information can be defined as an abstraction from any meaning a message might have.It can be represented as a sequence of bits 0 1 1 0, or as a sequence of alphanumericcharacters. The form of information storage, transmission or retrieval – whether digital oranalog, binary or decimal – is irrelevant to the issue of conveying meaning to people.Information stored in architectural composition is encoded in and by its very geometry.Discrete elements and compositional layers combine to form the surface of space. Thecomplexity of this surface, and its relationship with the ordering principles organizing thespace within, is dependent on the resultant geometry of composite layers. Nowhere isthis complexity more evident than in the temples of Khajuraho. The objective of thisstudy would therefore be to study the temples of Khajuraho, and test the hypothesis that:

Evolution of the temple style at Khajuraho is characterized by increasing fractal contentof the composite layers constituting its surface, caused by isometric transformations andaddition of sub-elements within the overall shape pattern schema

SCOPE AND LIMITATIONS OF THE STUDY

! Metaphysical rituals and belief systems influencing temple form are brieflyintroduced. The primary focus is on the physical resultant of these ‘genes’.

! The changing phenotype of the temple form is quantified by fixing variables within thespatial representation of the generating code. Other variables and the causes forthese changes are not discussed in detail as being beyond the scope of this study.

! Comparisons between different objects within and without architecture highlightdifferences in complexity based solely on the variables chosen. They are notcomments on the validity of the generating systems of thought.

METHODOLOGY

! Understanding compositional operations that order discrete elements into thecomposite layering of the surface of space.

! Spatially representing the generation of this surface and deriving a method forquantifying variables within the representation, namely geometrical properties

! Applying this method as a means of deconstructing the evolution of the temple styleat Khajuraho

I: information content

If we fully apprehend the pattern of things of the world will it not be found that every thingmust have a reason why it is as it is? …a rule [of co-existence with all things] to which itcannot but conform? Is not this just what is meant by Pattern?Hsu Heng

A man builds a city,With banks and cathedrals,A man melts the sand so he canSee the world outside…Lemon: U2

PAINTING SPACE

The wall forms the patina of the spatial painting. Thispatina or surface acts as a screen that transmitsinformation to observers [users]. Information hererefers to an organizing mechanism that allows the userto deal with the environment defined by the wall.Information has conventionally been classifiedaccording to its mod of transmission: cultural, geneticand exosomatic. These units exist through time aslineages of information, in a manner similar togenealogical communication. Conservative structuresare passed down different channels of transmissionsuch as books or CDs.1

Information transmitted by the surface links the user to the building system in a non-linear manner. Here information flow is not a unidirectional movement through time: it istwo-way traffic taking place in real-time. Architecture, then, is not a collection of non-interacting forms and voids, abstractly represented through lines on paper.2 It is acomplex system tied together by both static and dynamic linkages.

This complex system represents something not found in isolated, discrete elements.When elements at one scale combine to form a higher scale, an emergent propertyarises which may have been completely unanticipated. 3 This means that complexsystems are irreducible, a conclusion that goes against the assumption of 19th centurymechanistic physics that a complex system can never be more than the sum of its parts.

The work of the mid 20th century artist Jackson Pollock offers valuable insight. Pollockreplaced brush-strokes with trajectories: his paintings were created by dripping acontinuous flow of paint from a can suspended over canvas laid flat on the ground. Theprocess mimics the creation of frescoes, with discrete layers being formed over a periodof months at a time. Following the establishment of an anchor or base layer, subsequentlayers would be added immediately after the preceding layer had dried and set. Theprocess has been described as ‘chaotic layering in an ordered manner’.4 The differencearises with the replacement of broken lines by a continuous trajectory.

1 The Evolution of Information - Susantha Goonatilake2 A Pattern measure – Nikos A. Salingaros3 A Pattern measure – Nikos A. Salingaros4 Fractal Expressionism – Richard Taylor, Adam P. Micholich and David Jonas

To prove the use of chaotic flow systems by Pollock, scientists from the University ofNew South Wales recreated the process. A pendulum was hung over a canvas and itsnormal periodic motion modified using electromagnetic coils. The resultant motion wasrecorded on the canvas below by paint dripping from the pendulum. The chaotic patterns‘painted’ on the canvas were compared with examples of Pollock’s work.

Normal motion Chaotic motion Painting by Jackson Pollock

Natural chaotic systems form fractals in the patterns that record the process. 5 Fractalsare complex geometrical objects that will be discussed in more detail later. They showstatistical self-similarity (SSS), rather than exact self-similarity (ESS). This means thatpatterns observed at different magnifications may not be identical but they can bedescribed by the same statistics.

(a) ESS in geometry: the Koch Snowflake(b) ESS in physics: Sinai billiardmagnetoresistance(c) SSS in nature: Coastlines

Fractal patterns can be inferred from the following visual clues:

" Fractal scaling: Difficulty in judging the object’s magnification and the length scale" Fractal displacement: The possibility of describing the pattern by the same

statistics at different spatial locations.

5 Fractal Expressionism – Richard Taylor, Adam P. Micholich and David Jonas

LAYERING

Pollock’s paintings serve as a metaphorfor the surface of space. Discretearchitectural elements combine at smalllength scales to form higher scales withemergent properties. Fractal patterns are

built up over time. In Pollock’s paintings, different colors are introduced sequentially, withthe same color deposited during the same period in the painting evolution. Taylor and histeam electronically deconstructed the paintings into their constituent colored layers andcalculated each layer’s fractal content. Fractal content is given by the value of the fractaldimension – a property explained in II: Fractal Geometry. The higher the fractaldimension, the higher the coverage of the canvas surface area.

Composition with Pouring II 1943 1.0Number 14 1948 1.45Autumn Rhythm 1950 1.67Blue Poles 1952 1.72

Within the overall composition, each layer consists of a uniform fractal pattern. As eachof the patterns is incorporated to build up the complete pattern, the fractal dimension ofthe overall composition increases. Thus the combined pattern of many layers has ahigher fractal dimension than those of individual layer contributions. The first layer actsas an anchor layer for subsequent layers that then fine-tune the high fractal dimension ofthe anchor layer. The anchor layer of ‘Autumn Rhythm’ occupies 32% of the canvassurface area, with the complete pattern occupying 47%. The anchor layer is thusdesigned to dominate the composition.

" Complex surfaces are composed of discrete layers of architectural elements" Each layer contributes towards building up the fractal content of the overall

composition" The anchor layer dominates with subsequent layers increasing the fractal

dimension slightly

These principles are valid for the evolution of one building or for buildings belonging toan architectural style. Pollock’s individual paintings evolved through the addition ofdiscrete layers, while in the larger context of style evolution, fractal content wasincreased. Initial paintings occupy 20% of the 0.35m2 canvas area while later multi-layered paintings occupy 90% of the 9.96m2 area.

ORDERING ELEMENTS OF COMPOSITION

Building up the surface of architectural space therefore involves layering of discreteelements. The fractured surfaces that results from a high degree of layering encodesorganized complexity.

Information and Detailing in the Horizontal Plane

Vertical facets and flutes Amphitheatres Courtyards

Colonnades Columns and pilasters Fluted columns

Information and Detailing in the Vertical Plane

Facets Roof edges Roof corners Arches

COMPOSITE LAYERS, DISCRETE ELEMENTS

These elements are organized and ordered through compositional rules.

Bilateral symmetry Translation Chiral symmetry

Similarity symmetry Helical symmetry Multiple symmetry

The perception of architectural forms can therefore be divided into three aspects:(i) The information content depends on the design and geometry of discrete

elements and their subdivisions(ii) Information access is governed by the orientation of surfaces, their differentiation

on the smallest scale, and the microstructure in the materials(iii) Interactions between discrete elements create fractured surfaces

Emergent properties arising from these fractured surfaces depend on the geometryarising from the interaction of discrete elements within the whole. Since the fractal isgoverned by its own peculiar geometry, the level of information encoded by it should bedependent on its geometrical properties.

CODING SPACE

Discrete elements interacting at larger scales change the total subtended angle for whicheach solution works. To ensure an averaged equivalence of signal transmission toobservers at different locations with respect to a surface, the overall piecewise concavityshows spatial differentiation at the smaller and intermediate scales. With enoughsegmentation, it shows different substructures. These sub-structures are organized andordered through compositional rules that can be coded using shape data schema.

Developed by Myung Yeol Cha and John S. Gero, shape data schema describe patternsbased on visual organization and the recognition factors of typicality, similarity,frequency, dominance and multiplicity. Conceptual shape descriptions are constructed ina hierarchical tree structure using pre-defined shape knowledge. Shape pattern schemasare generalized from a set of multiple representations for a single object or a set ofrepresentations for a class of shape objects using inductive generalization.6

Put simply, transformations applied to an object to create a new one are mathematicallyrepresented. The initial object e1 and the resultants of the operation k repeated n timesdesignated en. Special conditions that guide the operation are termed as arguments an.

The diagram above shows the four basic possible operations translation, rotation,reflection and scaling, represented by k = 1, 2, 3 and 4 respectively. Each operation hasa special argument an that dictates the direction of the operation. For example,translation (1) of object e1 by distance a1 creates object e2. Similarly, arguments a2-5 applyto operations 2 to 4. The operation is described in further detail using a nesting operator

i= 1Πx. The nesting operator denotes x recursive applications of isometric transformation kto shape elements ei with transformation arguments ak. The resultant shape S isdescribed by

S = i= 1Πx k { ei, ak }

For complex compositions, the description involves multiple transformations coded in ahierarchical manner.

For example, the above diagram shows two seemingly different compositions Sa and Sb.Sa is formed by rotating (k = 2) an oval (ei) through 90 (a2) four (x) times. The process ofrotation is represented by e2 = 2 { e1, (a2, a5) } , where a2 and a5 are the angle and centreof rotation respectively. Therefore,

Sa = i= 1Π4 2 { Ovali, (90, a5) }Similarly, Sb can be represented as

Sb = i= 1Π4 2 { Trianglei, (90, a5) }

The two group shapes are therefore structurally similar though composed of differentsub-shapes.

More complex relationships are represented through shape pattern schema where shapeelements and lower level relationshipelements or schemas are considered asvariables.The composition on the left, for example, isthe result of two operations, translation androtation, described by

S = j=1Π3 1 { i=1Π4 2[ei.j, (90, a5)], (a1, a3) }

The sub-shapes are rotated about a5 through 90 four times, and the overall objecttranslated in the direction defined by a3, at intervals of a1.

6 Style Learning: Inductive Generalization of Architectural Shape Patterns - Myung Yeol Cha andJohn S. Gero

CODING STYLE

Shape pattern schema can be used to represent properties that characterize a particularstyle. Schapiro defines style as constant forms and qualities, particularly with regard toreplication of shape qualities.Cha and Gero represent style through a basic schemaStyle (N) = { (UM), (UF) }where N, M and F are the name, members and form elements respectivelyFor example, the Gothic style can be described byStyle(Gothic) = { (Paris Cathedral, Laon Cathedral, Rheims Cathedral, Nayon Cathedral)

(Pointed arches, flying buttresses, ribbed vaults, stained glass)}The basic schema is then elaborated to describe the shape pattern schema, so that thestyle is characterized by a numerical representation of its distinctive formal qualities.

If Style(Gaudi) = { (Casa Batlo, gratings, windows, Casa Mila roof), (reflection, gradation,

translation) } , the diagram above can be represented through shape data schema as:

7

GENES, CODES AND SYSTEMS GENERATION

Shape data schema can serve two primary functions:" Code the formation of the composition from its sub-shapes" Group objects into styles based on compositional rules

While the first building on the left has anelement being rotated by 90 four times,the second has one element being rotatedby 45 eight times. The two buildingstherefore belong to the same shapeschema, with Sa nested within Sb.Similarly, the members of a style can beconfirmed as such by verifying a commonshape data schema, or genetic code.

As a genetic code, shape data schema code, among others,fractured surfaces. Organisms have a multiplicity ofintermediate scales in the various functional systems of thehuman body: circulatory, respiratory, neural and locomotory.This large hierarchy of structural and functional levels has ahigh value of relevance over a continuum of scales, and

7 Image source: Style Learning - Myung Yeol Cha and John S. Gero

mutually interacts. Each of these aspects of living organisms takes place at the level ofmolecules as well as at the level of cells, organs, individuals, social groups orecosystems. The growth of most organisms is dependent on density: as soon as thedistance between two neighboring relevant levels gets sufficiently large, a newintermediate level emerges.

In living organisms, DNA translation producesproteins that constitute fractal networks within thelarger organism – composite layers and discreteelements. In the human body, the fractal nature ofsystems allows for the occupation of a large areawithin a restricted volume. The lungs have a largesurface area for air exchange due to the fractured

surface of its constituent bronchioles. Constituent systems maximize surface area withina fixed volume to maximize efficiency of the overall system.

II: fractal geometry

It is the perfect law of Unreason.F. Galton

A strange place this dirt ball is…Dirt Ball: Insane Clown Posse with Twiztid

INFINITE COMPLEXITY: THE FRACTURED SURFACE OF SPACE

Any segment – no matter where, and no matter how small – would, when blown up by thecomputer microscope, reveal new molecules, each resembling the main set and yet not quitethe same. Every new molecule would be surrounded by its own spirals and flame-like

projections, and those, inevitably, would reveal molecules tinier still, always similar, neveridentical, fulfilling some mandate of infinite variety, a miracle of miniaturization in which everydetail was sure to be a universe of its own, diverse and entire.

…Their [Peitgen and Richter] pictures of such [fractal basin] boundaries displayed thepeculiarly beautiful complexity that was coming to seem so natural, cauliflower shapes withprogressively more tangled knobs and furrows. As they varied the parameters and increasedtheir magnification of details, one picture seemed more and more random, until suddenly,unexpectedly, deep in the heart of a bewildering region, appeared a familiar oblate form,studded with buds: the Mandelbrot set, every tendril and every atom in place. It was anothersignpost of universality. “Perhaps we should believe in magic,” they wrote.

Extract from ‘Chaos’ by James Gleick

VARIABLES

In the mind’s eye, a fractal is a way of seeing infinity.8

Fractured surfaces are therefore composed of individual layers that contribute towardsthe overall fractal content. The layers and the way they aggregate is dependent oncompositional rules laid down by the architect, which can be represented through shapedata schema.

Within the shape data schema, the fractal dimension variable is of significance. Either interms of a resultant value describing the geometry of the composition, or as a generatingcode describing the creation of that geometry, the fractal dimension gives the level ofspace occupied by the surface. The higher the value – as seen in Pollock’s paintings –the more space occupied and, by corollary, the more there is for the user to react to.

A fractal is produced by iterating (repeating) a basic function onto an object, with theresult that each iteration adds a little area to the inside of the preceding figure, but thetotal area remains finite, since the figure produced is bounded by the area of the originalfigure. However, the length of the figure produced is infinitely long. The end result is thatinfinite length exists within a finite area. Fractals therefore occupy fractional dimensions.

8 Chaos – James Gleick

As a non-fractal object is magnified, no new features are revealed

As a fractal object is magnified, ever finer new features are revealed

QUANTIFYING COMPLEXITY

The concept of a fractional dimension is difficult to grasp intuitively, since we conceivespace as existing in three dimensions, moving through the fourth dimension of time.Mathematically, it can be described simply:

A point has no dimensions - no length, no width, no height.

A line has one dimension - length. It has no width and no height, but infinite length.

A plane has two dimensions - length and width, no depth.

Space, a huge empty box, has three dimensions, length, width, and depth, extending toinfinity in all three directions.

The concept of a dimension1. Take a self-similar figure like a line segment, and double its length.

Doubling the length gives two copies of the original segment.2. Take another self-similar figure, this time a square 1 unit by 1 unit. Now multiply the

length and width by 2.

Doubling the sides gives four copies.

3. Take a 1 by 1 by 1 cube and double its length, width, and height.

Doubling the side gives eight copies.

The dimension is the exponent. So when we double the sides and get a similar figure,we write the number of copies as a power of 2 and the exponent will be the dimension.

Figure Dimension No. Of Copies

Line 1 2=21

Square 2 4=22

Cube 3 8=23

Doubling similarity d N=2d

This means that a line can be divided into n = n1 separate pieces. Each of these piecesis 1/nth the size of the whole line and each piece, if magnified n times, would look exactlythe same as the original. In the case of the square, the value 2 signifies that it can be

divided into n2 pieces, and the cube is composed of n3 pieces. This means that if a figureis divided into pieces, magnifying these pieces by a factor of n reveals the original figure.As a result, the dimension of the figure can be calculated by dividing the logarithm of thenumber of divisions by the logarithm of the magnification factor 1/n. For fractal objects,this value would be fractional. This fractional value can be calculated for buildings usingthe Box Counting Method.

The Box Counting Method (The BCM)In the BCM, a square mesh of various sizes is laid over the image (containing theobject). The number of meshes N(r) that contain part of the image is counted. The slopeof the linear portion of a log [N(r)] vs. log (1/r) graph gives D the fractal dimension. Thegraphed value of N(r) is usually the average of N(r) from the different mesh origins. Thelimited resolution of most data renders the estimation of D sensitive to the range of boxlengths ∆ used. In the fractal analysis software used, the range of error caused by lowresolution of the image is negated. The limited resolution of digitized images results in anunderestimation of counts for smaller boxes, resulting in a convex log-log plot (and anunderestimate of D). Probabilities are assigned using a binomial model and solving for p.

case study: khajuraho

If you don’t know history, you don’t know anything.Edward Johnston

God is in the TV.Marilyn Manson

EVOLUTION OF A STYLE: DECONSTRUCTING SURFACE

Good examples for a style are maximally similar to members of their category andminimally similar to members of other categories. 9 Commonalities characterize style:similarity in materials, shapes, and space organization. Shape pattern schemas can beused as rules for shape generation, and learned shapes and shape patterns can beinitial shape elements for shape grammar generation. Base shapes for shape generationcan be constructed from the combination of properties of family style. The initialapplication therefore involves constructing a set of preliminary shape pattern schematracing the transformations applied in the evolution of temple style at Khajuraho. Thesegive the process of addition of sub-shapes to the overall composition: our concern ismore with the resultant fractal contents. Increasing fractal content in the course ofevolution and identification of the anchor layer would prove the relationship between sub-shape transformations and the changing fractal content of the overall canvas.

In general, temple evolution has been driven by the need to represent:• The Vastupurushmandala, a square diagram on which the temples are founded, in

the centre of which is the place for Brahman, the formless, ultimate superior reality• The Cave Mountain and Shelter

9 Style Learning - Myung Yeol Cha and John S. Gero

• The Sanctum as womb/cave• The Temple as mountain

The north Indian temple had its origins in bambooconstruction, with the base derived from the Vedic sacrificialaltar and the spire from the tabernacle formed by tying bentbamboo at their apex. This combination of altar and spiregave shape to the Nagara style of temple architecture inNorth India. The configuration of vertical axis, square altarand enclosure persisted in Indian architecture to‘demonstrate the participation of each monument in thecosmogonic process’.10 The temple form evolved from acentralized, bilaterally symmetrical structure to one with adefined longitudinal axis to aid access and approach. The

early wooden construction gave way to stone as a building material, with the basic formalcomposition being retained. Stone construction in temple architecture was taken to itspinnacle by the Chandella dynasty at Khajuraho, in the period 950 – 1050 A.D.The temples demonstrate a unified style that differsonly in detailed surface expression thoughbelonging to three sects: Shaiva, Vaishnava andJaina. The basic code of elevated porch, linear axisculminating in the garbhgrha, capped by theshikhara persists throughout. Refinements in thetemple structure were made mainly to thesuperstructure and the surface treatment.

10 The Hindu Temple: Axis of Access – Michael W. Meister

Based on parameters of design and form, the temples at Khajuraho were divided into thefollowing classes by A.G. Krishna Menon and S. Punja in their study of the evolution oftemples at Khajuraho, ‘The Legacy of Khajuraho’:

" Lalguan Mahadeo type" Varaha type" Brahma type" Chaturbhuja type" Javari type" Devi Jagadambi type" Duladeo type" Lakshman type

The Lakshman group is the highly developed in terms of the complexity of the surface ofits members. It comprises three temples: Lakshman, Visvanath and the KandariyaMahadeo, in increasing order of complexity. The isometric operations appliedsuccessively to the skin can be coded by shape data schema. Divided into discreteelements and composite layers, the fractal content of the overall composition coded canbe calculated. The changing values act as variables describing each schema.CODED LAYERS

Shape pattern schema when used to describe the Khajuraho style:A. Basic deconstruction of layering of sub-elementsB. Primary transformations applied to sub-elements within the overall compositionC. Isometric transformation applied to the overall compositionD. Shape pattern schema of Khajuraho temple style, using inductive generalizationE. Fractal content of individual and composite layers coded

A. Layering of sub-elements

Lakshman Temple

Visvanath Temple

Kandariya Mahadeo Temple

B. Sub-Elements and Primary Isometric Transformations

ScalingGradation of translations described by

i= 1Πn 4 { 1[j= 1Πn 1 (ej.i, (a1,a3)), (a1,a3)], a4 }a1 = distance, a3 = axis of translation, a4 = scale factor1 and 4 are isometric transformations translating and scaling respectively.

ReflectionMirroring of parts within the whole, described by

i= 1Πn 3 { 1[j= 1Πn 1 (ej.i, (a1,a3)), (a1,a3)] }a1 = distance, a3 = axis of translation1 and 3 are isometric transformations translating and reflecting respectively.The reflection description acts independently as an alternative schema

Layering adds discrete sub-shapes to the overall composition. Here, isometrictransformations applied to elements build up fractal content through successive layering.C. Overall Composition and Primary Isometric Transformation

Sa: Scaling in the XY plane translated along the z-axis

Sb: Scaling in the YZ plane translated along the x-axis

Sa = i= 1Πn 1 { 4 [eai, aa4] (aa1, aa3) } Sb = i= 1Πn 1 { 4 [ebi, ab4] (ab1, ab3) }

Shape pattern description Sa and Sb have the same predicates, translation axes and sub-elements, therefore two different shape pattern descriptions in different domains can begeneralized by the turning constants into variables rule.

D. Shape pattern schema describing each stage of temple style evolutionN changes based on the number of sub-elements, but the equations remain embeddedin the schema. If these shapes or patterns are members of a class that are linked to astyle, then the embedded shapes or patterns characterize the style by the droppingcondition rule of inductive generalization [see Style Learning: Inductive Generalizationof Architectural Shape Patterns by Myung Yeol Cha And John S. Gero]

If Sa and Sb : : > [Khajuraho Style] <

Sa & Sb < i= 1Πn 1 { 4 [xe, xa4] (xa1, xa3) } : : > [Khajuraho Style]::> is the implication linking a concept description with a concept name and < is thegeneralization

Since the two shape patterns characterize the Khajuraho style, the conjunction of twoshapes that is the scaling of sub-elements characterizes this aspect of Khajuraho style.

THE COMPLEX SURFACE OF SACRED SPACE

The temple is a symbol of the manifestation of a dynamic continuum. In its multicentricform, patterns of expansion, self-similar iteration and radiation from a core organize itsinformation field. The plan achieves complexity through self-similar iteration in adiminution scale. The offset projections continue as vertical latas. In the shikhara, thisresults in diminutive multiples of its shape in relief. In the aedicules, quarter shikharas atthe corners arise from the half-shikharas on the sides. In a multipartite shikhara, severalsub-spires are attached in a proportionate order, giving sub-scale to the shikhara form.

The shape pattern schema derived for the temple codes the layering that contributestowards the overall fractal content. The resultant fractal content can be calculated usingthe fractal analysis software described earlier.

L1 = 1.121490 L2 = 1.148719 L3 = 1.182849 L4 = 1.312564 ∆ L2-1 = 0.027229 ∆ L3-2 = 0.03413 ∆ L4-3 = 0.129715Ln = Fractal dimension of each layer

With the addition of scaled shape elements, the fractal content increases gradually, withthe anchor layer contributing the most towards the overall fractal content of the schema.

FRACTAL DIMENSIONS: QUANTIFICATION OF LAYERED COMPLEXITY

Computer generated mesh overlaid over image

Magnified view of self-similar components composing the temple superstructure

Gradient image created to facilitate fractal dimension calculation_____________________________ lakshman temple: front elevation

Lines plotted for DB, DBW and DW

Slope analysis of the three lines (from the equation of a line being y = mx + c,where m is the slope of the line that gives the fractal dimension) gives an averagevalue of:D = 1.684 +/- 1.2%.

_____________________________ lakshman temple: front elevation



Gradient image created to facilitate fractal dimension calculation

_____________________________ lakshman temple: side elevation



______________________________ lakshman temple: side elevation




_____________________________ visvanath temple: front elevation



____________________________ visvanath temple: front elevation




_____________________________ visvanath temple: side elevation




________________ kandariya mahadeo temple: front elevation




________________ kandariya mahadeo temple: side elevation


Slope analysis of the three lines (from the equation of a line being y = mx + c, where m isthe slope of the line that gives the fractal dimension) gives an average value of:D = 1.780 +/- 1.2%.

_________________ kandariya mahadeo temple: side elevation

.Fractal Dimension: 1.750

Fractal Dimension: 1.773


_________________ fractal dimensions for perspective views

laks

hman

tem

ple

visv

anat

h te

mpl

eka

ndar

iya

mah

adeo

tem

ple

To serve as a test group, two examples have been taken at different length scales


Fractal Dimension: 1.567Results

TEMPLE FDfr FDs FDp FDav

Chaturbhuja - - - 1.567

Jagadambi Devi - - - 1.560

Lakshman 1.684 1.710 1.750 1.714

Visvanath 1.694 1.755 1.773 1.740

Kandariya 1.731 1.780 1.776 1.762

FDfr = Fractal Dimension of the front elevationFDs = Fractal Dimension of the side elevationFDp = Fractal Dimension of the perspective viewFDav = Average fractal dimension

jaga

dam

bi d

evi t

empl

ech

atur

bhuj

a te

mpl

e

In a larger context:

No. Object FD No. Object FD

1 Hong Kong Bank 1.200 11 Eiffel Tower 1.598

2 Villa Savoy 1.200 12 Barcelona Pavilion 1.599

3 Hawa Mahal 1.300 13 Houses at Amasya 1.600

4 Robie House 1.352 14 Mt. Kailash 1.693

5 Protein (sample) 1.410 15 Taj Mahal 1.695

6 Sagrada Familia 1.493 16 Fern (sample) 1.698

7 Apartments by Gaudi 1.520 17 Sydney Opera House 1.712

8 Mt. Meru 1.520 18 Cathedral 1.730

9 Unity Temple 1.538 19 Lightning (sample) 1.734

10 Building by Gehry 1.584 20 Kandariya Mahadeo Temple 1.780

Fractal content increases steadily in the evolution of the temple style at Khajuraho. Ascoded by shape pattern schema, sub-shapes added to the anchor layer to increase theoccupation of the canvas that is the temple surface. The temple surface as an ‘interface’between the devotee and God is thus designed with increasing efficiency.

conclusion

There are doors I haven’t opened,Even doors I’ve yet to look throughUltrasonic Sound: Hive

You stand on the edge, of a silver futureSilver Future: Monster Magnet

This exploration attempts to describe a genericshape pattern schema for the fractured surface ofthe Hindu temple. The codes themselves arerudimentary: variables are left as unsolved. Themain objective has been to quantify one of thevariables – fractal dimensions – as anevolutionary development characteristic of thetemple style at Khajuraho. The fractal dimensionsmeasure the fractal content of the temple surface,

indicating how sub-shapes are added to the anchor layer devised initially. The schemacodes the addition of these sub-shapes and transformations applied to them. Over aperiod of time, the fractal content increases steadily. The anchor layer contributes themost to the overall fractal content of the composition. This results from the desire tocreate a dominant background of color and texture against which additional elements areadded. Fine-tuning leads to increasing occupation of the canvas surface area.

The architectural object as painted space is a recurring motif. Architectural destiny hasalways been guided by changes in ‘outside’ fields: metaphysics, programming,technology, genetics, beliefs, art movements et al. Architecture depends on disciplinesas varied as science and religion: the approach here has been to understand thearchitect’s role as ‘an artist and a poet’, and as ‘a scientist and a technologist’.11 Aboveall of this there is pure architecture – elements that build the surface of space.

Fiction, Metaphysics and Painted Space: To Sum‘Imagine the sand of the Mohaine Desert, which youcrossed to find me, and imagine a trillion universesencapsulated in each grain of that desert; and withineach universe an infinity of others. We tower overthese universes from our pitiful grass vantage point;with one swing of your boot you may knock a billionbillion worlds flying off into darkness, in a chain neverto be completed. Size, gunslinger...Size... Yet supposefurther. Suppose that all worlds, all universes, met in asingle nexus, a single pylon, a Tower. A stairway,perhaps, to the Godhead itself. Would you dare gunslinger? Could it be that somewhereabove all of endless reality, there exists a Room...?You dare not.’12

11 The Theory of Architecture – Paul Allan - Johnson12 The Dark Tower I : The Gunslinger – Stephen King

Bibliography1. The Shape of Space Graham Nierlich2. Space is the machine Bill Hillier3. The Architect’s Eye Tom Porter4. Architectural Morphology J.P. Steadman5. Does God Play Dice Ian Stewart6. Superforce Paul Davies7. Origins Rediscovered Richard Leakey8. The Sleepwalkers Arthur Koestler9. The Architecture of the Jumping Universe Charles Jencks10. The Blind Watchmaker Richard Dawkins11. Nature in Question J.J. Clarke12. A New Model of the Universe P.D. Ouspensky13. Chaos James Gleick14. About Time Paul Davies15. The Evolution of Information Susantha Goonatilake16. Imagenation: Popular Images of Genetics Jose Van Dijck17. Ecology and the fractal Mind Victor Padron and Nikos

A. Salingaros18. Chaos, Fractals and Self-Organization Arvind Kumar19. A Text Book of Biology P.S. Dhami20. Nature’ s Numbers Ian Stewart21. Cybertrends David Brown22. The Theory Of Architecture23. Architecture in the 20th Century Udo Kultermann24. Fractal Expressionism Richard Taylor, Adam

Micolich and David Jonas(Physics World Vol.12 No. 10October 1999)

25. The Hindu Temple Stella Kramrisch26. Patterns of Transformation Adam Hardy27. Concept of Space IGNCA Publication28. Architecture, Time and Eternity Adrian Snodgrass29. Living Architecture Andreas Volwahsen30. Form, Transformation and Meaning Adam Hardy31. The Hindu Temple: Axis of Access Michael W. Meister32. Architecture of the World: India Andreas Volwahsen33. Indian Architecture (Buddhist and Hindu) Percy Brown34. The Legacy of Khajuraho A.G. Krishna Menon

35. Dissertation Geetanjali Chordia36. Dissertation Harsha Vishwakarma37. Dissertation Rishi Dev38. Hindu Temples: Models of a Fractal Universe Kirti Trivedi

The Visual Computer (1989)539. Jurassic Park Michael Crichton40. The Lost World Michael Crichton41. Timeline Michael Crichton42. The Dark Tower I : The Gunslinger Stephen King43. The Dark Tower II: The Drawing of the Three Stephen King44. The Dark Tower III: The Wastelands Stephen King45. The Dark Tower IV: Wizard and Glass Stephen King46. Style Learning: Inductive Generalization of Myung Yeol Cha And John S.

Architectural Shape Patterns Gero47. Attack of the Deranged Mutant Killer Monster Snow Bill Watterson

Goons48. Interrogating Modern Indian Architecture A.G. Krishna Menon

(Architecture + Design Vol. XVII No.6 November-December 2000)

49. Bionic Vertical Space Javier Pioz, Rosa Cervera and Eloy Celaya(Architecture + DesignVol. XVII No.5 September –October 2000)

50. Time Magazine Special: The Age of Discovery

! Yahoo image gallery! www.webshots.com! www.digitalblasphemy.com! www.ucomics.calvinandhobbes.com! www.ccat.sas.upenn.edu! www.visualparadox.com! www.fch.vutbr.cz! www.library.upenn.edu! www.ultrafractal.com! www.math.utsa.edu! www.swin.edu.au

Dissertation - Fractal Content Khajuraho Temples

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