Dissertation - Fractal Content Khajuraho Temples

_________________________________________________________________________

DISSERTATION

FRACTAL CONTENT OF THE SURFACE OF ARCHITECTURAL COMPOSITION THE TEMPLES OF KHAJURAHO DEMIS ROUSSOS BHARGAVA (9604) Guide: MR. RAJAT RAY

JULY 2001

TULSI VIDYA BHARTI SCHOOL OF HABITAT STUDIES Vasant Kunj, New Delhi DISSERTATION Title: FRACTAL CONTENT OF THE SURFACE ARCHITECTURAL COMPOSITION – THE TEMPLES OF KHAJURAHO

OF

CONTENTS Acknowledgements Intro Hypothesis Methodology Scope and Limitations ____________________________________________________________________________________________________________

I: Information Content Painting Space Layering Ordering Elements of Composition Composite Layers, Discrete Elements Coding Space Coding Styles Genes and Systems Generation II: Fractal Geometry Infinite Complexity: The Fractured Surface of Space Variables Quantifying Complexity Case Study: Khajuraho Evolution of a Style: Deconstructing Surface Coded Layers, or The Spatial Representation Of Ordering Principles The Complex Surface of Sacred Space Fractal Dimensions: Quantification of Layered Complexity Results Conclusion

ACKNOWLEDGEMENTS

Mr. Rajat Ray, for being patient whenever I went off on one of my more ambitious – or just 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 and the 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 territory or dominance, but for self-enlargement. Let something come from nothing. …. Texas Suite: Stan Rice

Very little is more worth our time than understanding the 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 it through its surface. The information encoded by the surface determines our correspondence with the space defined. The wall becomes a surface of interaction, rather than a passive element of delineation. While information encoded may not be designed, 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 alphanumeric characters. The form of information storage, transmission or retrieval – whether digital or analog, 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. The complexity of this surface, and its relationship with the ordering principles organizing the space within, is dependent on the resultant geometry of composite layers. Nowhere is this complexity more evident than in the temples of Khajuraho. The objective of this study 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 content of the composite layers constituting its surface, caused by isometric transformations and addition of sub-elements within the overall shape pattern schema SCOPE AND LIMITATIONS OF THE STUDY !

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

!

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

!

Comparisons between different objects within and without architecture highlight differences in complexity based solely on the variables chosen. They are not comments on the validity of the generating systems of thought.

METHODOLOGY !

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

!

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

!

Applying this method as a means of deconstructing the evolution of the temple style at Khajuraho

I: information content

If we fully apprehend the pattern of things of the world will it not be found that every thing must have a reason why it is as it is? …a rule [of co-existence with all things] to which it cannot 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 can See the world outside… Lemon: U2

PAINTING SPACE The wall forms the patina of the spatial painting. This patina or surface acts as a screen that transmits information to observers [users]. Information here refers to an organizing mechanism that allows the user to deal with the environment defined by the wall. Information

has

conventionally

been

classified

according to its mod of transmission: cultural, genetic and exosomatic. These units exist through time as lineages of information, in a manner similar to genealogical communication. Conservative structures are passed down different channels of transmission such as books or CDs.1 Information transmitted by the surface links the user to the building system in a nonlinear manner. Here information flow is not a unidirectional movement through time: it is two-way traffic taking place in real-time. Architecture, then, is not a collection of noninteracting forms and voids, abstractly represented through lines on paper.2 It is a complex 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 property arises which may have been completely unanticipated.

3

This means that complex

systems are irreducible, a conclusion that goes against the assumption of 19th century mechanistic 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. Pollock replaced brush-strokes with trajectories: his paintings were created by dripping a continuous flow of paint from a can suspended over canvas laid flat on the ground. The process mimics the creation of frescoes, with discrete layers being formed over a period of months at a time. Following the establishment of an anchor or base layer, subsequent layers would be added immediately after the preceding layer had dried and set. The process has been described as ‘chaotic layering in an ordered manner’.4 The difference arises with the replacement of broken lines by a continuous trajectory. 1

The Evolution of Information - Susantha Goonatilake

2

A Pattern measure – Nikos A. Salingaros

3

A Pattern measure – Nikos A. Salingaros

4

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

To prove the use of chaotic flow systems by Pollock, scientists from the University of New South Wales recreated the process. A pendulum was hung over a canvas and its normal periodic motion modified using electromagnetic coils. The resultant motion was recorded 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 Fractals are complex geometrical objects that will be discussed in more detail later. They show statistical self-similarity (SSS), rather than exact self-similarity (ESS). This means that patterns observed at different magnifications may not be identical but they can be described by the same statistics. (a) ESS in geometry: the Koch Snowflake (b)

ESS

in

physics:

Sinai

billiard

magnetoresistance (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 metaphor for

the

surface

of

space.

Discrete

architectural elements combine at small length scales to form higher scales with emergent properties. Fractal patterns are built up over time. In Pollock’s paintings, different colors are introduced sequentially, with the same color deposited during the same period in the painting evolution. Taylor and his team electronically deconstructed the paintings into their constituent colored layers and calculated each layer’s fractal content. Fractal content is given by the value of the fractal dimension – a property explained in II: Fractal Geometry. The higher the fractal dimension, the higher the coverage of the canvas surface area. Composition with Pouring II

1943

1.0

Number 14

1948

1.45

Autumn Rhythm

1950

1.67

Blue Poles

1952

1.72

Within the overall composition, each layer consists of a uniform fractal pattern. As each of the patterns is incorporated to build up the complete pattern, the fractal dimension of the overall composition increases. Thus the combined pattern of many layers has a higher fractal dimension than those of individual layer contributions. The first layer acts as an anchor layer for subsequent layers that then fine-tune the high fractal dimension of the anchor layer. The anchor layer of ‘Autumn Rhythm’ occupies 32% of the canvas surface area, with the complete pattern occupying 47%. The anchor layer is thus designed 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 to an architectural style. Pollock’s individual paintings evolved through the addition of discrete layers, while in the larger context of style evolution, fractal content was increased. Initial paintings occupy 20% of the 0.35m2 canvas area while later multilayered paintings occupy 90% of the 9.96m2 area.

ORDERING ELEMENTS OF COMPOSITION Building up the surface of architectural space therefore involves layering of discrete elements. The fractured surfaces that results from a high degree of layering encodes organized complexity. Information and Detailing in the Horizontal Plane

Vertical facets and flutes

Amphitheatres

Colonnades

Columns and pilasters

Courtyards

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

Similarity symmetry

Translation

Chiral 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 geometry arising from the interaction of discrete elements within the whole. Since the fractal is governed by its own peculiar geometry, the level of information encoded by it should be dependent on its geometrical properties.

CODING SPACE Discrete elements interacting at larger scales change the total subtended angle for which each solution works. To ensure an averaged equivalence of signal transmission to observers at different locations with respect to a surface, the overall piecewise concavity shows spatial differentiation at the smaller and intermediate scales. With enough segmentation, it shows different substructures. These sub-structures are organized and ordered through compositional rules that can be coded using shape data schema. Developed by Myung Yeol Cha and John S. Gero, shape data schema describe patterns based on visual organization and the recognition factors of typicality, similarity, frequency, dominance and multiplicity. Conceptual shape descriptions are constructed in a hierarchical tree structure using pre-defined shape knowledge. Shape pattern schemas are generalized from a set of multiple representations for a single object or a set of representations for a class of shape objects using inductive generalization.6 Put simply, transformations applied to an object to create a new one are mathematically represented. The initial object e1 and the resultants of the operation k repeated n times designated 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 has a 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 apply to operations 2 to 4. The operation is described in further detail using a nesting operator i= 1Π

x

. The nesting operator denotes x recursive applications of isometric transformation k

to shape elements ei with transformation arguments ak. The resultant shape S is described by

S = i= 1Πx k { ei, ak }

For complex compositions, the description involves multiple transformations coded in a hierarchical 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 of rotation is represented by e2 = 2 { e1, (a2, a5) }, where a2 and a5 are the angle and centre of 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 different sub-shapes. More complex relationships are represented through shape pattern schema where shape elements

and

lower

level

relationship

elements or schemas are considered as variables. The composition on the left, for example, is the result of two operations, translation and rotation, 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 object translated in the direction defined by a3, at intervals of a1.

6

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

CODING STYLE Shape pattern schema can be used to represent properties that characterize a particular style. Schapiro defines style as constant forms and qualities, particularly with regard to replication of shape qualities. Cha and Gero represent style through a basic schema Style (N) = {(UM), (UF) } where N, M and F are the name, members and form elements respectively For example, the Gothic style can be described by Style(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 the style 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 an element being rotated by 90 four times, the second has one element being rotated by 45 eight times. The two buildings therefore belong to the same shape schema,

with

Sa

nested

within

Sb.

Similarly, the members of a style can be confirmed as such by verifying a common shape data schema, or genetic code. As a genetic code, shape data schema code, among others, fractured surfaces. Organisms have a multiplicity of intermediate scales in the various functional systems of the human body: circulatory, respiratory, neural and locomotory. This large hierarchy of structural and functional levels has a high 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 of molecules as well as at the level of cells, organs, individuals, social groups or ecosystems. The growth of most organisms is dependent on density: as soon as the distance between two neighboring relevant levels gets sufficiently large, a new intermediate level emerges. In living organisms, DNA translation produces proteins that constitute fractal networks within the larger organism – composite layers and discrete elements. In the human body, the fractal nature of systems allows for the occupation of a large area within a restricted volume. The lungs have a large surface area for air exchange due to the fractured surface of its constituent bronchioles. Constituent systems maximize surface area within a 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 the computer microscope, reveal new molecules, each resembling the main set and yet not quite the 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, never identical, fulfilling some mandate of infinite variety, a miracle of miniaturization in which every detail was sure to be a universe of its own, diverse and entire. …Their [Peitgen and Richter] pictures of such [fractal basin] boundaries displayed the peculiarly beautiful complexity that was coming to seem so natural, cauliflower shapes with progressively more tangled knobs and furrows. As they varied the parameters and increased their 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 another signpost 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 towards the overall fractal content. The layers and the way they aggregate is dependent on compositional rules laid down by the architect, which can be represented through shape data schema. Within the shape data schema, the fractal dimension variable is of significance. Either in terms of a resultant value describing the geometry of the composition, or as a generating code describing the creation of that geometry, the fractal dimension gives the level of space 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 the result that each iteration adds a little area to the inside of the preceding figure, but the total area remains finite, since the figure produced is bounded by the area of the original figure. However, the length of the figure produced is infinitely long. The end result is that infinite 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 conceive space 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 to infinity in all three directions.

The concept of a dimension 1. 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 pieces is 1/nth the size of the whole line and each piece, if magnified n times, would look exactly the 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 figure is 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 the number 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 using the Box Counting Method. The Box Counting Method (The BCM) In the BCM, a square mesh of various sizes is laid over the image (containing the object). The number of meshes N(r) that contain part of the image is counted. The slope of the linear portion of a log [N(r)] vs. log (1/r) graph gives D the fractal dimension. The graphed value of N(r) is usually the average of N(r) from the different mesh origins. The limited resolution of most data renders the estimation of D sensitive to the range of box lengths ∆ used. In the fractal analysis software used, the range of error caused by low resolution of the image is negated. The limited resolution of digitized images results in an underestimation of counts for smaller boxes, resulting in a convex log-log plot (and an underestimate 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 and minimally similar to members of other categories.

9

Commonalities characterize style:

similarity in materials, shapes, and space organization. Shape pattern schemas can be used as rules for shape generation, and learned shapes and shape patterns can be initial shape elements for shape grammar generation. Base shapes for shape generation can be constructed from the combination of properties of family style. The initial application therefore involves constructing a set of preliminary shape pattern schema tracing the transformations applied in the evolution of temple style at Khajuraho. These give the process of addition of sub-shapes to the overall composition: our concern is more with the resultant fractal contents. Increasing fractal content in the course of evolution and identification of the anchor layer would prove the relationship between subshape 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

• 9

The Cave Mountain and Shelter 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 bamboo construction, with the base derived from the Vedic sacrificial altar and the spire from the tabernacle formed by tying bent bamboo at their apex. This combination of altar and spire gave shape to the Nagara style of temple architecture in North India. The configuration of vertical axis, square altar and

enclosure

persisted

in

Indian

architecture

to

‘demonstrate the participation of each monument in the cosmogonic process’.10 The temple form evolved from a centralized, bilaterally symmetrical structure to one with a defined longitudinal axis to aid access and approach. The early wooden construction gave way to stone as a building material, with the basic formal composition being retained. Stone construction in temple architecture was taken to its pinnacle by the Chandella dynasty at Khajuraho, in the period 950 – 1050 A.D. The temples demonstrate a unified style that differs only

in

detailed

surface

expression

though

belonging to three sects: Shaiva, Vaishnava and Jaina. The basic code of elevated porch, linear axis culminating in the garbhgrha, capped by the shikhara persists throughout. Refinements in the temple

structure

were

made

mainly

to

the

superstructure 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 the following classes by A.G. Krishna Menon and S. Punja in their study of the evolution of temples 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 of its members. It comprises three temples: Lakshman, Visvanath and the Kandariya Mahadeo, in increasing order of complexity. The isometric operations applied successively to the skin can be coded by shape data schema. Divided into discrete elements and composite layers, the fractal content of the overall composition coded can be 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-elements B. Primary transformations applied to sub-elements within the overall composition C. Isometric transformation applied to the overall composition D. Shape pattern schema of Khajuraho temple style, using inductive generalization E. 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

Scaling Gradation of translations described by i= 1Π

n

4 { 1[j= 1Πn 1 (ej.i, (a1,a3)), (a1,a3)], a4 }

a1 = distance, a3 = axis of translation, a4 = scale factor 1 and 4 are isometric transformations translating and scaling respectively.

Reflection Mirroring of parts within the whole, described by i= 1Π

n

3 { 1[j= 1Πn 1 (ej.i, (a1,a3)), (a1,a3)] }

a1 = distance, a3 = axis of translation 1 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, isometric transformations 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 subelements, therefore two different shape pattern descriptions in different domains can be generalized by the turning constants into variables rule. D. Shape pattern schema describing each stage of temple style evolution N changes based on the number of sub-elements, but the equations remain embedded in the schema. If these shapes or patterns are members of a class that are linked to a style, then the embedded shapes or patterns characterize the style by the dropping condition rule of inductive generalization [see Style Learning: Inductive Generalization of Architectural Shape Patterns by Myung Yeol Cha And John S. Gero] If Sa and Sb : : > [Khajuraho Style] <

Sa & Sb <

i= 1Π

n

1 { 4 [xe, xa4] (xa1, xa3) } : : > [Khajuraho Style]

::> is the implication linking a concept description with a concept name and< is the generalization

Since the two shape patterns characterize the Khajuraho style, the conjunction of two shapes 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 multicentric form, patterns of expansion, self-similar iteration and radiation from a core organize its information field. The plan achieves complexity through self-similar iteration in a diminution scale. The offset projections continue as vertical latas. In the shikhara, this results in diminutive multiples of its shape in relief. In the aedicules, quarter shikharas at the corners arise from the half-shikharas on the sides. In a multipartite shikhara, several sub-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 contributes towards the overall fractal content. The resultant fractal content can be calculated using the fractal analysis software described earlier.

L1 = 1.121490

L2 = 1.148719

L3 = 1.182849

∆ L2-1 = 0.027229

∆ L3-2 = 0.03413

∆ L4-3 = 0.129715

L4 = 1.312564

Ln = Fractal dimension of each layer

With the addition of scaled shape elements, the fractal content increases gradually, with the 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 average value of: D = 1.684 +/- 1.2%. _____________________________

lakshman temple: front elevation

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: side 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 average value of: D = 1.71 +/- 1.2%.

______________________________

lakshman temple: side elevation

Computer generated mesh overlaid over image

Magnified view of self-similar components composing the temple superstructure

Gradient image created to facilitate fractal dimension calculation _____________________________

visvanath 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 average value of: D = 1.694 +/- 1.2%.

____________________________

visvanath temple: front elevation

Computer generated mesh overlaid over image

Magnified view of self-similar components composing the temple superstructure

Gradient image created to facilitate fractal dimension calculation

_____________________________

visvanath temple: side 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 average value of: D = 1.755 +/- 1.2%.

_____________________________

visvanath temple: side elevation

Computer generated mesh overlaid over image

Magnified view of self-similar components composing the temple superstructure

Gradient image created to facilitate fractal dimension calculation ________________

kandariya mahadeo 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 average value of: D = 1.731 +/- 1.2%.

________________

kandariya mahadeo temple: front elevation

Computer generated mesh overlaid over image

Magnified view of self-similar components composing the temple superstructure

Gradient image created to facilitate fractal dimension calculation

________________

kandariya mahadeo temple: side 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 average value of: D = 1.780 +/- 1.2%.

_________________

kandariya mahadeo temple: side elevation

lakshman temple

.

visvanath temple

Fractal Dimension: 1.750

kandariya mahadeo temple

Fractal Dimension: 1.773

Fractal Dimension: 1.776

_________________

fractal dimensions for perspective views

chaturbhuja temple

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

jagadambi devi temple

Fractal Dimension: 1.560

Fractal Dimension: 1.567 Results 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 elevation

FDs

= Fractal Dimension of the side elevation

FDp

= Fractal Dimension of the perspective view

FDav

= Average fractal dimension

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. As coded by shape pattern schema, sub-shapes added to the anchor layer to increase the occupation 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 through Ultrasonic Sound: Hive

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

This exploration attempts to describe a generic shape pattern schema for the fractured surface of the Hindu temple. The codes themselves are rudimentary: variables are left as unsolved. The main objective has been to quantify one of the variables

–

fractal

dimensions

–

as

an

evolutionary development characteristic of the temple style at Khajuraho. The fractal dimensions measure the fractal content of the temple surface, indicating how sub-shapes are added to the anchor layer devised initially. The schema codes the addition of these sub-shapes and transformations applied to them. Over a period of time, the fractal content increases steadily. The anchor layer contributes the most to the overall fractal content of the composition. This results from the desire to create a dominant background of color and texture against which additional elements are added. Fine-tuning leads to increasing occupation of the canvas surface area. The architectural object as painted space is a recurring motif. Architectural destiny has always been guided by changes in ‘outside’ fields: metaphysics, programming, technology, genetics, beliefs, art movements et al. Architecture depends on disciplines as varied as science and religion: the approach here has been to understand the architect’s role as ‘an artist and a poet’, and as ‘a scientist and a technologist’.11 Above all 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 you crossed to find me, and imagine a trillion universes encapsulated in each grain of that desert; and within each universe an infinity of others. We tower over these universes from our pitiful grass vantage point; with one swing of your boot you may knock a billion billion worlds flying off into darkness, in a chain never to be completed. Size, gunslinger...Size... Yet suppose further. Suppose that all worlds, all universes, met in a single nexus, a single pylon, a Tower. A stairway, perhaps, to the Godhead itself. Would you dare gunslinger? Could it be that somewhere above all of endless reality, there exists a Room...? You dare not.’12 11 12

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

Bibliography 1. The Shape of Space

Graham Nierlich

2. Space is the machine

Bill Hillier

3. The Architect’s Eye

Tom Porter

4. Architectural Morphology

J.P. Steadman

5. Does God Play Dice

Ian Stewart

6. Superforce

Paul Davies

7. Origins Rediscovered

Richard Leakey

8. The Sleepwalkers

Arthur Koestler

9. The Architecture of the Jumping Universe

Charles Jencks

10. The Blind Watchmaker

Richard Dawkins

11. Nature in Question

J.J. Clarke

12. A New Model of the Universe

P.D. Ouspensky

13. Chaos

James Gleick

14. About Time

Paul Davies

15. The Evolution of Information

Susantha Goonatilake

16. Imagenation: Popular Images of Genetics

Jose Van Dijck

17. Ecology and the fractal Mind

Victor Padron and Nikos A. Salingaros

18. Chaos, Fractals and Self-Organization

Arvind Kumar

19. A Text Book of Biology

P.S. Dhami

20. Nature’ s Numbers

Ian Stewart

21. Cybertrends

David Brown

22. The Theory Of Architecture 23. Architecture in the 20th Century

Udo Kultermann

24. Fractal Expressionism

Richard Taylor, Adam Micolich and David Jonas (Physics World Vol.12 No. 10 October 1999)

25. The Hindu Temple

Stella Kramrisch

26. Patterns of Transformation

Adam Hardy

27. Concept of Space

IGNCA Publication

28. Architecture, Time and Eternity

Adrian Snodgrass

29. Living Architecture

Andreas Volwahsen

30. Form, Transformation and Meaning

Adam Hardy

31. The Hindu Temple: Axis of Access

Michael W. Meister

32. Architecture of the World: India

Andreas Volwahsen

33. Indian Architecture (Buddhist and Hindu)

Percy Brown

34. The Legacy of Khajuraho

A.G. Krishna Menon

35. Dissertation

Geetanjali Chordia

36. Dissertation

Harsha Vishwakarma

37. Dissertation

Rishi Dev

38. Hindu Temples: Models of a Fractal Universe

Kirti Trivedi The Visual Computer (1989)5

39. Jurassic Park

Michael Crichton

40. The Lost World

Michael Crichton

41. Timeline

Michael Crichton

42. The Dark Tower I : The Gunslinger

Stephen King

43. The Dark Tower II: The Drawing of the Three

Stephen King

44. The Dark Tower III: The Wastelands

Stephen King

45. The Dark Tower IV: Wizard and Glass

Stephen King

46. Style Learning: Inductive Generalization of

Myung Yeol Cha And John S.

Architectural Shape Patterns 47. Attack of the Deranged Mutant Killer Monster Snow

Gero Bill Watterson

Goons 48. Interrogating Modern Indian Architecture

A.G. Krishna Menon (Architecture + Design Vol. XVII No.6 NovemberDecember 2000)

49. Bionic Vertical Space

Javier Pioz, Rosa Cervera and Eloy Celaya (Architecture + Design Vol. 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