Computer-Aided Design 39 (2007) 51–59www.elsevier.com/locate/cad

The application of fractal geometry to the design of grid orreticulated shell structures

M.A. Vyzantiadoua, A.V. Avdelasa,∗, S. Zafiropoulosb

a Department of Civil Engineering, Aristotle University, GR-541 24 Thessaloniki, Greeceb Department of Architecture, Aristotle University, GR-541 24 Thessaloniki, Greece

Received 14 April 2006; accepted 24 September 2006

Abstract

The paper proposes an approach where structural systems can be developed according to the mathematical theory of fractals. Architecturecan take advantage of the complexity sciences, by the use of present day computer technology, where algorithms of mathematical and geometricfunctions can produce new motifs of design. In order to demonstrate this, examples of the use of the elliptic and hyperbolic paraboloid, and thedesign of a tree on the surface of an elliptic paraboloid, are given. Further, the way to compose relevant algorithms is described.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Fractal geometry; Architectural design; Computer-aided design

1. Introduction

The evolution of non-linear dynamics and the complexitysciences have extended, in recent years, engineering thoughtbeyond the basic forms of Euclidean geometry such aslines, spheres and circles. Architecture can use the hierarchyof fractal geometry to generate new rhythms in design.At the beginning of the 20th century, new mathematicalstructures were discovered, which were at the time regarded asexceptional objects that did not fit the patterns of Euclid andNewton. These new structures were regarded as pathological,as a gallery of “monsters”. As shown by Mandelbrot, manyof these “monsters” do have, in fact, counterparts in the realworld. Mandelbrot introduced, for these shapes in 1975, theword fractals, derived from the Latin “fractus”, the adjectivalform of “frangere”, which means, “break into pieces of randomform”. Finally, they have been recognized as some of the basicstructures in the language of nature’s irregular shapes: thefractal geometry of nature [1–4].

Fractal shapes can have dimensions between the 0, 1, 2 or 3to which we are accustomed [5]. Groups of points that follow a

∗ Corresponding address: Department of Civil Engineering, Faculty ofEngineering University Campus, Aristotle University, GR-541 24 Thessaloniki,Central Macedonia, Greece. Tel.: +30 2310 995784; fax: +30 2310 995642.

E-mail address: avdelas@civil.auth.gr (A.V. Avdelas).

0010-4485/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.cad.2006.09.004

line are fractal dust, which have a dimension between 0 and 1, aterrain has dimensions greater than 2 but less than 3, and cross-sections through terrains generate fractal lines with dimensionsbetween 1 and 2 [6], but they are described in either 2-manifoldpatterns or 3-d structures. Fractals are self-similar, which meansthat each part of a structure is similar to the whole shape atmany different scales. The Koch curve, for example, is exactlyself-similar, but if a little bit of randomness is introduced, asin a coastline, the object might be statistically self-similar orself-affine (Fig. 1). Thus, seemingly, random and transformedshapes can repeat themselves across scale, and are thus fractalsin the formal sense. The self-similarity dimension D, which isgiven by the power relation between the number of pieces andthe reduction factor is:

a = 1/(s)D= (1/s)D. (1)

Thus, D = log a/ log (1/s), (2)

where a is the number of pieces and s is the reduction factor. Fornonfractal structures, the exponent D is an integer. For example,if a line is divided in 3 equal parts using a reduction factor of(1/3), then relation (1) becomes:

3 = 1/(1/3)D= 3D, and D = 1. (3)

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Fig. 1. The Koch curve and its generator.

But in a typical fractal scheme, e.g. in the Koch curve, thescaling factor is (1/3), and the number of pieces is 4 [4,7]:

Thus, D = log a/ log (1/s) = log 4/ log 3 = 1.26. (4)

Any form that is self-similar is likely to be fractal. If there isa regular motif or design, which repeats itself as the structuregrows or scales — through time or across space, then thestructure can be envisaged as a hierarchy, and thus a fractalorganization is a hierarchical organization [6]. Self-similarshapes can be found in nature, from trees to galaxies [7]. Sincesome of them are quite complex, computerized algorithms areoften used in order to render realistic models. In this context,several procedural techniques have been established. Amonga variety of methods, PL-systems (Parametric LindenmayerSystems) are capable of producing images of objects that have amore or less repetitive structure, such as plants, terrains, shells,crystals, branching structures or even buildings [2,8,9].

In the past few years, computers have helped with the appli-cation of fractals in many sciences: Physics, Biology, Mechan-ics etc. Architecture also, concerned with the control of rhythm,can benefit from the use of this relatively new mathematicaltool. The fractal dimension provides a quantifiable measure ofthe mixture of order and surprise in a rhythmic composition [7].

2. Fractal geometry in architecture

Fractal geometry in architectural composition is related tothe formal study of the progression of interesting forms, fromthe distant view of the facade to the intimate details [7].Under this concept, significant periods in architecture, suchas the Classical or Art Nouveau, tend to be explicitly fractal,since the buildings of these areas are characterised by theprogression of self-similar details from large to small scales,and by mathematical subdivisions. The houses of Frank LloydWright provide an example of this progression of detail fromthe large to the small. He often referred to the central idea thatcoordinated the design, and this idea came from nature [7].The roofline of the Robie house by Frank Lloyd Wright has

self-similar textural characteristics, though of a limited rangeof scale. It eventually flattens into straight lines, Fig. 2(a).However, there is a progression of interesting detail if oneapproaches the building–for example, in the window mullionpattern, there is a smaller detail to be seen, Fig. 2(b).

Some contemporary buildings approach the idea of fractalarchitecture by reintroducing both curvatures and subdivisionsat different scales, or a self-similar structure of the same motif,thus producing complexity and formal variety. In this way,there exist structures that do not show a self-similar repetitionof a module, as e.g. the work of Mies van der Rohe, but asubtle variation, like the petals of a fern or a snowflake. Someexamples that can be mentioned are: The Guggenheim Museumin Bilbao by Frank Gehry, where 26 self-similar petals of a‘metallic flower’ are unfolded, twisted and curved, generatinga formal and spatial geometry. The Arm’s Storey Hall inMelbourne, where “pentagon metallic patterns form a self-organising and simultaneously chaotic system in the internaland external space”, is another example Fig. 3(a) [10,11]. Thesefractal shapes are the development of the Penrose tiling patterninto an order that connects facade, floor, walls and ceiling into asingle ornamental system. Roger Penrose created an aperiodictiling system, using a fat and thin rhombus, which, however farit is extended, never results in a cyclical pattern. This unusual,“self-organizing” order was later discovered to exist in nature,in quasicrystals [12]. The Federation Square project designedby the Lab and Bates Smart Architecture Studio, where thefacade, is based on the geometry of a triangular pinwheel grid,making up a self-similar “panel” of five single triangles. Thismodular system is repeated across two scales, Fig. 3(b) [13].

Another aspect is the inspiration of forms from natureand biology as sources for creativity in design, which is alsorelated with the notion of fractals in architecture. For example,the work of Gaudi originates the organic development ofcontemporary architecture [14], and expresses a new sculpturalform consisting of continuous self-similar patterns inspiredfrom nature.

For many centuries, a variety of forms, which in many casespresent fractal geometry in their structural appearance, such astrees, cells, crystals etc., have been creatively used by architectsand engineers in projects like shells, light structures, arcs, tents,nets and bridges. Examples by Otto and Calatrava are given inFig. 4 [15–17].

Although there are several ways to connect fractal conceptswith architecture, this idea leads to the question of howcomplicated forms of design could give information forfabrication using the advantages of contemporary computertechnology. Generally, if an enclosure is to be constructed asa non-linear structure with curved lines, there are three waysfor the geometry to be chosen: (1) the sculptural, an approachbased on free-form artistic sculpting, (2) the physical, a methodbased on a physical modelling process, such as hanging cablesor sheets and (3) a procedural, generative approach, based ona composition of mathematical functions [18]. Gaudi used thesecond way of hanging models, which, when inverted, definedthe shape of masonry arches and vaults for the Colonia Guelland the Sagrada Familia. Frei Otto used also this method

M.A. Vyzantiadou et al. / Computer-Aided Design 39 (2007) 51–59 53

Fig. 2(a). Robie house by Frank Lloyd Wright.

Fig. 2(b). Robie house by Frank Lloyd Wright, the window mullion pattern.

to produce forms for fabric structures, cable nets and gridshells [18]. In the last method, the desired shapes and designscan be produced by computer programs, and the limit ofcurves and surfaces depends on the mathematical knowledgeand the imagination of the user [18,19]. The generation ofthe geometric design of the British Museum Great Court roofis a contemporary example of the use of a new algorithmbased on functions, weighted and added together (Fig. 5) [18,20]. This shell structure, constructed with a complex shapedesign, contains no two bars or meshes that are exactly alike. Itwas created thanks to the use of high-performance computers,specialized software, and to the use of sophisticated planningand technology [21]. In addition, the technique of GeometricSubstitutions can be mentioned as a geometric automaticmethod that uses simple concepts of geometric and topologicaltransformations in order to build details on an initial simpleshape, and to thus create finally complex models [22].

This paper follows the mathematical and geometrical path,by applying a set of rules of nonlinear transformations to simpleshapes in a repetitive way, in order to create models close to thefractal notion.

3. Generation of fractal, dome-shaped shells

The approach proposed here uses mathematical proceduresin the computer language C++ in order to produce elliptical,dome-like sells. The results are 3-d designs accessible to furthermanipulation. The compilation of these programs produceseach time an “.exe” file in DOS, or a file in Windows, wherethe user can choose the basic dimensions, degree of scaling,curvature, size etc. The program asks the user for:

• the number of points on the periphery• the number of rings

Fig. 3(a). The Arm’s Storey Hall in Melbourne.

Fig. 3(b). The Federation Square, Bates Smart and the Lab Architecture Studio,Melbourne.

• the number of periods of the waves along the radial distances• the width of the band of the z-level along the radial distance

and• the big Axial-A and the big Axial-B.

In this way, the automatic repetition of motives and designsin different scales is effective. The representations of the plans

54 M.A. Vyzantiadou et al. / Computer-Aided Design 39 (2007) 51–59

Fig. 4(a). The skeleton of marine protozoa (radiolarian).

Fig. 4(b). The Heart Tent, Frei Otto, Riyadh.

Fig. 4(c). Galleria and Heritage square, S. Calatrava, Toronto.

are obtained through DXF files, which can be transformed into“.dwg” files in AutoCAD, or in other drawing programs.

These drawings represent shells that are organized byan evolutionary and hierarchical process in space, which isinspired by nature, as for example in a wave periodicalmovement or a tree formation. The variety of structural systemsof steel and glass that exist nowadays give further possibilitiesfor the construction of shells. Especially the point-fixed glazingsystem, which follows a hierarchical process, is proposed hereas an ideal structural frame for fractal shapes.

Fig. 5. The glass roof over the courtyard of the British Museum in London, byFoster and Partners — Internal view.

Fig. 6(a). A Serre at La Villette by P. Rice in Paris.

Applications of point-fixed glazing systems range fromsimple structures, such as shop windows and shelters, to morecomplicated ones, such as multi-storey buildings and largeatria. Although there are different types of point-fixed glazingsystems, each one of them consists of four basic components.These main components may be defined as follows: glazingpanels, bolted fixings, glazing support attachments and themain support structure [23]. In these systems, the geometricalconfiguration and the structural performance are importantaspects in the architectural composition, and follow ahierarchical process (e.g. the Serres of the National Museumof Science, Technology and Industry at La Villette in Parisby Peter Rice), which in several cases is described from thegeometry of fractals, Fig. 6(a), [24].

Following this system, the geometry of the shells is relatedto the manufacturing and construction system, whereas the

M.A. Vyzantiadou et al. / Computer-Aided Design 39 (2007) 51–59 55

Fig. 6(b). Point-fixed glazing system: axonometric, scale 1.

Fig. 6(c). Point-fixed glazing system: axonometric, scale 2.

supporting system, the static analysis and the progressivemorphological subdivision of the structural components areimportant parameters in the generation of self-similarity. Thedimensions of the various lines, as they are computed by theprogram, can be used in practice as dimensions of the variouscomponents of the structural system.

In order to examine whether the idea of the creation ofan organized hierarchical process, by the use of the pointfixed glazing systems, is possible in practice, an algorithm hasbeen created that simulates an existing structure with a fractalhierarchy in its supporting system: the facade of a Serre atLa Villette by P. Rice in Paris. The structural system of thisbuilding can be used as a model, and be applied to the shells,which are described further, or to other similar ones, Figs. 6(b)and 6(c) [25].

Therefore, these examples exploit a new design technique,which applies the idea of fractal geometry in 3-d structures.The geometrical configuration, the structural morphology andthe hierarchical subdivision of the support system generate anew architectural expression [25–27].

Fig. 7(a). Elliptic paraboloid.

4. The elliptic paraboloid and hyperbolic paraboloidshapes

To generate computerized fractal forms, geometric shapeshave been used as a basis. Such examples are the ellipticparaboloid (e.p.) and the hyperbolic paraboloid (h.p.) shapes,Figs. 7(a) and 8(a) respectively. They are based on the followingmathematical relations.

The e.p. and h.p. for the Euclidian space E3 are respectivelydescribed by

Elliptic paraboloid :x2

a2 +y2

β2 − 2z = 0, (5)

Hyperbolic paraboloid :x2

a2 −y2

β2 − 2z = 0. (6)

The Cartesian coordinates x and y for a point on the surface ofthe e.p. or h.p., at a level

z = z0 are given as

x = α j cos(θ · i) and (7)

y = β j sin(θ · i). (8)

where i = 1, 2, . . . Num Points, and θ is the angle of rotationaround the axis zz′

[0◦, 360◦], which depends on the number of

points (Num Points) the user gives along a ring. A ring is everylevel z that subtracts, from the surface of the e.p. or h.p., a curveof the Euclidian space E3. The user gives the number of rings(Num Rounds)

θ =2π

Num Pointsor θ =

360◦

Num Points. (9)

The coefficients α j and β j represent the axes of the ellipse. Thefollowing relations give their values

α j = αj

Num Rounds(10)

β j = βj

Num Rounds, (11)

where j = 1, 2, . . . Num Roundsand α, β are the axes of the ellipse of the biggest ring, becauseforj = Num Rounds then α j ≡ α and β j ≡ β.In the program, α is the value of Big Axial-A, and β is the valueof Big Axial-B.By substituting relations (9) and (10) into relations (7) and (8),the following coordinates are obtained

x = α ·

(j

Num Rounds

)· cos (θ · i) , (12)

56 M.A. Vyzantiadou et al. / Computer-Aided Design 39 (2007) 51–59

Fig. 7(b). Elliptic paraboloid with wave.

Fig. 8(a). Hyperbolic paraboloid.

y = β ·

(j

Num Rounds

)· sin (θ · i) , (13)

z = z0. (14)

The level z is written as

z = A

(x2

a2 +y2

β2

)2

, for the e.p. and (15)

z = A

(x2

a2 −y2

β2

)2

, for the h.p. (16)

The coefficient A is given by

A = B√

a2 + β2, (17)

where B is the scale factor of the axis zz′ that produces differentlevels for the same axes α and β.

The complexity of the model can be increased if a variableequation is added in order to give a wave form along theaxis zz′. The simple e.p. and h.p., Figs. 7(a) and 8(a), aretransformed to fractal forms by the introduction of the angleof the wave. The height of the wave depends on the distance ofthe wave from the centre of the ellipse and on the number ofperiods along the axis of the ellipse, Fig. 7(b), 8(b) and 8(c).These shapes are similar to the waves created in water when astone is thrown.

In this case, Eq. (17) is written as

A = B√

a2 + β2 sin(φ · j), (18)

where j = 1, 2, . . ., Num Rounds.Angle φ depends on the number of rings (Num Rounds) and

on the number of periods (Num Periods) the user chooses to

Fig. 8(b). Hyperbolic paraboloid with wave.

Fig. 8(c). Hyperbolic paraboloid with wave — Front view.

have along the total number of rings. Above every 2-rings of theoriginal mesh, a parameterized height is added. These heightsvary according to the following function:

φ =2π · Num Periods

Num Rounds. (19)

The coordinates x , y, z are obtained by combining Eqs.(12)–(18)

x = α ·

(j

Num Rounds

)· cos (θ · i) , (20)

y = β ·

(j

Num Rounds

)· sin (θ · i) , (21)

z = B ·

√α2 + β2 ·

(j

Num Rounds

)2

·

(12

)· sin (φ · j) ,

for the e.p. (22)

or

z = B ·

√α2 + β2 ·

(j

Num Rounds

)2

·

(cos2 (θ · i) − sin2 (θ · i)

2

)· sin (φ · j) , for the h.p.,

where i = 1, 2, . . . Num Points, j = 1, 2, . . . Num Rounds.

5. A tree on the surface of an elliptic paraboloid

In this section, the structural and geometrical transformationof the surface of an elliptic paraboloid based on a tree fractalconfiguration [28], Fig. 9, is studied. The evolution of the treesystem generates two different models, depending on the rulethat is applied for the development of the model. So e.g. factor2∗ (relation (28)) and factor 2+ (relation (29)) give differentresults, as shown in Figs. 10 and 11 respectively. In orderto produce the design representations of these 3-d shells, thefollowing rules and mathematical formulas have been used:

As it has been described in the previous section, the ellipticparaboloid scheme is given by Eq. (5). On the above surface,a suitable number of points are chosen, so that a tree formis produced. To achieve this, the coordinates of the ellipticparaboloid are estimated in relation to the angle of rotationθ around the axis z, Eq. (9), and the number of levels z.

M.A. Vyzantiadou et al. / Computer-Aided Design 39 (2007) 51–59 57

Fig. 9. Generation of the tree fractal.

Fig. 10(a). Tree on the surface of an elliptic paraboloid. Model of geometricalprogression.

Fig. 10(b). Tree on the surface of an elliptic paraboloid. Model of geometricalprogression — Front view.

The Cartesian coordinate z, that represents each time one ring(level z), is parameterized according to the number of rings(Num Rounds) the user instructs the algorithm to develop, andis estimated by the paraboloid equation

z = a · x2+ b · x + c,

where x ≡ i = 1, 2, . . . , Num Rounds. (23)

The user has to select the appropriate values for the coefficientsa, b and c.

For a given value of the coordinate z (level z), which hasbeen estimated by Eq. (23), and according to the equation ofthe elliptic paraboloid, Eq. (5), the parametric equations of theCartesian coordinates x and y result in

x = αz cos(θ · j) and (24)

Fig. 11(a). Tree on the surface of an elliptic paraboloid. Model of arithmeticalprogression-Axonometric, view 1.

Fig. 11(b). Tree on the surface of an elliptic paraboloid. Model of arithmeticalprogression-Axonometric, view 2.

Fig. 11(c). Tree on the surface of an elliptic paraboloid. Model of arithmeticalprogression — View from below.

y = βz sin(θ · j), (25)

where j = 1, 2, . . ., Num Points.

58 M.A. Vyzantiadou et al. / Computer-Aided Design 39 (2007) 51–59

On both the above relations, parameter θ is given aspreviously, Eq. (9).

The coefficients αz and βz represent the axes of the ellipse.Their values are given by the following functions

αz = α ·√

2 · |z| and (26)

βz = β ·√

2 · |z|, (27)

where αz and βz are the axes of the ellipse of the ring whose|z| = 1/2, and thus αz = α and βz = β. In the program, α

is the value of Big Axial-A and β is the value of Big Axial-B,which are chosen by the user.

The user has also to select the “number of points on eachring”. This parameter is different for each ring (level) anddifferent models can describe the method of transformation. Inthe present case, the following models are chosen:

5.1. Model of geometrical progression, Figs. 10(a) and 10(b)

Num Points(i + 1) = 2∗Num Points(i), (28)

where i = 1, 2, . . ., Num Rounds, with the following initialvalues

Num Points(0) = 1 and Num Points(1) = 2.

5.2. Model of arithmetical progression, Fig. 11

Num Points(i + 1) = 2 + Num Points(i), (29)

where i = 1, 2, . . ., Num Rounds, with the following initialvalues

Num Points(0) = 1 and Num Points(1) = 2.

The coordinates x , y, z are obtained by combining Eq. (5) withEqs. (24)–(27) and the following Eq. (32)

x = α ·

√2 · |a · i2 + b · i + c| · cos

(2π

Num Points· j

), (30)

y = β ·

√2 · |a · i2 + b · i + c| · sin

(2π

Num Points· j

), (31)

z = a · i2+ b · i + c, (32)

where i = 1, 2, . . ., Num Rounds and j = 1, 2, . . .,Num Points.

Fractal geometry is observed in the tree form of the surface.The more the number of rings is increased, the more subdi-visions and points on each ring are taken (either by geomet-rical progression or by arithmetical progression). In Fig. 10,the number of points in each level is twice the numberin the previous level. The triangles are similar in eachlevel, and they can be either glass panels, or other mate-rials with a steel frame system, or they can represent areticulated supporting system as a point-fixed glazing sys-tem with cables. It must be noted that the model of geo-metrical progression is morphologically better than the modelof arithmetical progression. The complex scheme of Fig. 11could be a basis for a spiral building.

6. Conclusions

The new geometry, the geometry of Nature, has openednew routes in science, economics, urban-planning, biologyetc. This geometry has recently influenced architecture also.The proposed computational method produces algorithmsusing fractal mathematics, and can generate forms applicableto shells. Modern building technology can support suchapplications, e.g. it has been applied for the construction ofthe glass roof of the atrium in the British Museum, whereeach different node and bar was constructed according to thedimensions given by a computer program [20].

Considering the development of recent technologies indesign and building construction, and the introduction ofcomplex new forms in the architectural design, a newarchitectural mode of expression is generated.

This new morphology is produced digitally, according tothe algorithmic potential of software programs, the structuralperformance of the building materials, the support systems andusers’ demand. The applications of this method are unlimited,given the enormous capacities of computer technology, and thepossibilities for form generation are stretched far beyond thelimits of purely manual techniques. They are variable, sincethey are used either for the whole building design or for partsof the construction, e.g. the roof or the profile of a wing, theroofs of atria, the cladding of a facade, or space installations likepavilions, shelters, tents, domes etc., and in large-scale planninglike the design of a plaza or small-scale design like the patternof a pavement.

Therefore, this idea is used for further improvement of thedesign of shells and other larger scale projects that arise inarchitecture or the engineering sciences. The benefit of such anapproach is that when an algorithm is suggested for a structure,then the whole project is analysed and determined in such away that every rule it contains can be fully described — forexample, the coordinates of each node, or the dimensions ofeach bar. The result is a spatial structure providing a systemwith technological coherence and aesthetic value, rather thanmerely an interesting prototype. The challenge for the architectsand the engineers is to be inspired by the computer results,either by simplifying the undesirable complexity involved, orby using in their design small parts of the whole computerproduct, applicable in real projects, that can be realised usingnew materials and modern structural technology.

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