Fractal geometry applications in landscape

FRACTAL GEOMETRY AND ITS APPLICATIONS IN LANDSCAPE DESIGN DISSERTATION REPORT APOORVA TIWARI 133701128 VII D

INTRODUCTION WHAT ARE FRACTALS • A natural phenomenon or mathematical set that exhibits a repeating pattern that displays at every scale.

FRACTALS IN NATURE

• This self-similar object undergoes modification such that dimensions of the structure are all modified by the same scaling factor.

• The shape may change in size or orientation but the shape remains same.

FRACTALS IN ARCHITECTURE

• These shapes are seen in nature, architecture and design, mathematics, ecology etc.

SOME IDENTIFIED FRACTAL FORMS

FORMATION • A fractal is generally composed of two parts : the initiator and generator.

• The formation can be understood by taking the example of ‘the Koch snowflake’.

GENERATOR

1 ITERATION

INITIATOR

2 ITERATIONS

3 ITERATIONS

AIM •

This study aims to review and understand fractal geometry, evaluate landscape preferences based on fractal character of the environment and explore how these concepts maybe applied in landscape design.

OBJECTIVES • To understand fractal geometry- its meaning and underlying principles.

• To understand how it has been used in architecture, specifically landscape design.

• To study the significance- why and how it has been used and how it can be beneficial.

• To explore and identify how a few fractal forms may be used in landscape design.

SCOPE AND LIMITATIONS • It has been studied that fractal character is associated with restorative properties, thus there is a scope of it’s application in landscape.

• This study focuses on the application of fractal geometry in landscape design and discusses only a few of the many possible applications.

• However, it does not take into account the fractal dimension.

(The complexity of a fractal structure is quantified by it’s fractal dimension D)

RELEVANCE OF STUDY • It has been established that fractals are associated with restorative character.

• They are being studied and applied in the fields of visual arts, natural processes, music, mathematics etc. In architecture, however the use of fractals is still generally limited to motifs and decorations.

• Therefore there is a scope of study on fractals in architecture and design.

OUTCOME The following outcome is expected by the end of this study: • Understand the meaning and principles of fractal geometry • The possible applications in the field of architecture • Evaluate if fractal scenes have a higher preference associated with them • Compare with landscaping components and principles and analyze how it may be applied to landscape design.

METHODOLOGY

AIM (Study of fractals and applications in landscape)

REVIEW AND UNDERSTANDING OF FRACTAL GEOMETRY

SECONDARY LITERATURE STUDY

OBSERVATIONS- USES IN ARCHITECTURE

STUDY OF LANDSCAPE PREFERENCES

PRIMARY FIELD RESEARCH

SECONDARY LITERATURE STUDY

SURVEY THROUGH QUESTIONNAIRE

EVALUATION OF PREFERENCES BASED ON RESULTS

CONCLUSION IN FORM OF SUGGESTIONS AND POSSIBLE APPLICATIONS

EXPLORATION OF SCOPE OF FRACTAL GEOMETRY IN LANDSCAPE AND DESIGN

PRIMARY FIELD RESEARCH

SECONDARY LITERATURE STUDY

CASE STUDIES AND ANALYSIS

OBSERVATIONS AND ANALYSIS

DATA COLLECTION • CASE STUDIES : - Ryoanji Temple, Kyoto Japan - IUCAA, Pune

• LIVE SURVEY: Online survey to study and compare preferences between different scenes with varying fractal characteristics.

ROCK GARDEN AT RYOANJI TEMPLE, KYOTO • Dry landscape garden created during the Muromachi era (AD 1333- 1573) • An abstract composition of rocks and moss on an empty rectangle of raked gravel.

• The rocks are arranged so as to incorporate two opposing elements as a manifestation of the forces of nature.

• The principal viewing area of the garden is located on a balcony which provides the full view of the garden.

Ryoanji Temple

SECTION OF THE TEMPLE

VIEWING VERANDAH ROCK GARDEN

The garden is to be viewed while seated (meditating) on the slightly raised verandah.

ROCK GARDEN

VIEWING VERANDAH PLAN OF THE TEMPLE

uidaho.edu

BRANCHING

PLAN OF THE GARDEN SHOWING POSITION OF ROCKS

TRUNK

• In a study conducted by Nitschke, Blum, Van Tonder, of Kyoto University, the spatial structure of the garden was examined using a technique called “medial axis transformation”.

VIEWING VERANDAH

• The overall structure is a simple, dichotomously branched tree that converges on the principal garden-viewing area on the balcony.

• The connectivity pattern of the tree is self-similar, with the mean branch length decreasing monotonically from the trunk to the tertiary level.

The features are reminiscent of actual trees.

ANALYSIS USING ‘MEDIAL AXIS TRANSFORMATION’ (Tonder et al. 2002) The dark lines show the structure of a tree suggested by the location of the rocks with respect to the viewing area (abstract representation)

• Viewed from the right position, this empty space created the image of a tree in the subconscious mind.

• The layout of the garden uses “suggestive symmetry” to make the brain visualize a tree by connecting the empty space between the rocks.

• The study also found that by changing the number of rocks or their spatial arrangement the self-similar structure and the convergence on the main viewing area were lost.

• It may be concluded that the unconscious perception of this pattern contributes to the enigmatic appeal of the garden and abstract fractal structure of the garden is linked to its aesthetic appreciation.

ANALYSIS BASED ON LANDSCAPE ELEMENTS Figure showing different elements of landscape in the garden

IUCAA, Pune • The Inter University Centre for Astronomy and Astrophysics, designed by Charles Correa has three principal courtyards.

• The courtyard in the residential block houses what is known as the Sierpinski gasket.

• The triangle has been used as a layout for different types of vegetation detail, creating a pattern and the different textures are created through different types of detail. The edges have low walls in order to create a distinct segregation between the different spaces.

IUCAA, Pune

COURTYARD HOUSING THE SIERPINSKI TRIANGLE

archnet.org

SITE PLAN OF THE UNIVERSITY

SIERPINSKI TRIANGLE • The triangle has been used as a layout for different types of vegetation detail, creating a pattern and the different textures are created through different types of detail.

• The edges have low walls in order to create a distinct segregation between the different spaces.

• The figure shows the different components of landscape.

SURVEY • The aim of the survey was to compare preference between natural scenes with a fractal character, computer generated or non-natural fractal scenes and geometric or abstract scenes with a non-fractal character.

Method • The participants were asked to rate different scenes based on their individual preferences on a scale of 1 to 5; where 1 was the most preferred scene and 5 –the least. A Sample size of 55 participants was considered. The scenes were of the following categories:

Natural scenery

Forms in architecture

The following results were observed:

Patterns/ designs

Art-work

Landscaping layouts

5 4.5

`

4

3.5

3 2.5 2 1.5 1 0.5 0 Natural scenery

Forms in architecture

Patterns/ Designs

Artwork

Landscaping layouts

Natural fractal scene

Dome with fractal character

Natural fractal pattern

Fractal artwork

Fractal layout

Computer generated fractal scene

Organic architecture with fractal character

Computer generated fractal pattern

Abstract artwork

Organic layout

Painting

Minimalistic architecture

Geometric pattern

Geometric artwork

Geometric layout

Based on this, it can be concluded that there is an inherent preference for fractal scenes.

ANALYSIS AND CONCLUSION • The results of the survey indicate that individuals have an inherent preference for fractal geometry or fractal patterns.

•

These preferences vary with the fractal dimension associated with the particular pattern or scene, however that is beyond the scope of this study.

• Thus a relation can be drawn between the visual and aesthetic appeal and the restorative effect of the scene.

IMPLICATIONS & APPLICATIONS THE VON KOCH SNOWFLAKE AT LEVEL 4 • The patterns of the snowflake can influence the form of layout created for a particular space. For example, it may be used as a plan form for a garden or could define the edge of a flower bed.

• However, at this level of iteration, the increasing number of projections reduces its feasibility as an option for seating detail.

•

It would also be lesser suited to be used as a pathway due to a high number of turns; although a lower level of iteration maybe used for the same purpose.

LOWER ITERATIONS OF THE FORM MAYBE USED FOR PLAN, LAYOUT, SEATING OR PATH DETAIL. HOWEVER THE FEASIBILITY OF PATH AND SEATING DEATAIL REDUCES WITH INCREASING ITERATIONS.

18-SEGMENT QUADRIC VON-KOCH CURVE AT LEVEL TWO AND FASS CURVES • The forms of the quadric von-Koch curve and the FASS curves give a layout that would be feasible for a pathway, especially an exploratory path. FASS curves, specifically maybe an option for designing the layout of a maze.

• The square formed at the center of the 18 segment quadric von-Koch curve could function as a node while creating a central focus. The enclosures formed within the arms of the shape can be thresholds that may be used for various purposes. For example, they could have vegetation detail or built detail creating points of interest.

• This particular layout may also have the potential of being used as a seating detail due to the smoothness of the edges.

THE GOSPER CURVE AND PEANO CURVE • The Gosper curve is a variation of the von-Koch curve. Due to the rough edges, the interlocking of the form to another of its kind is possible. This particular form may find application in detail- surface texture, pattern etc.

1 UNIT OF THE GOSPER CURVE

TILING PATTERN

• The Peano curve may also find a similar application especially in tiling where the number of iterations and the type of unit can change giving various options for textures and patterns

FRACTALS WITH CIRCLES • The Apollonian packing of circles gives a possible arrangement for vegetation detail.

• The same may also be applicable for spaces or a mixture of different components.

• It may help in creating edges that define the thresholds, nodes, foci and other areas or points of interest.

•

By increasing the number of iterations, the number of spaces or the complexity of the layout can be increased.

1 UNIT OF THE PEANO CURVE

TILING PATTERN

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