algorithmic form generation of a radiolarian...

677

Algorithmic FormGeneration of aRadiolarian PavilionErnesto Bueno

issue 04, volume 07international journal of architectural computing

678

Algorithmic Form Generation ofa Radiolarian PavilionErnesto Bueno

ABSTRACT

This paper describes the design stage of an on-going research projectfor the construction of a pavilion that mimics the bone structures ofRadiolarians. In the process, several constructive and generativealgorithms are developed, together with geometric and trigonometricfunctions, all implemented in RhinoScript and a math plug-in for theNURBS modeler Rhinoceros.There has been considerable emphasison the generation of the Radiolarian cell that is tessellated along astereographic surface with a honeycomb-based algorithm. It combinesdesign strategies from biomimetics, mathematical functions, generativescripting, process automation and versioning, all integrated into analgorithmic methodology for creating a non-standard structure, capableof being manufactured with CNC technology, and doing so, setting aprecedent in the academic and professional environment in which itwill be located.

679Algorithmic Form Generation of a Radiolarian Pavilion

1. Introduction

This is a work-in-progress project of a pavilion to hold a sculpture at theBarcelona campus of Universitat Internacional de Catalunya, where theSchool of Architecture is located.

The project is conceived and directed by Alberto T. Estevez, head ofthe Genetic Architectures research group. It consists of an ovoid-shapedstructure designed as an interpretation of bone structures of Radiolarians(Figure 1). Its design develops a series of irregular hexagonal cells that arerepeated in a tessellation of a surface and generate volumes that, followingthe logic of bone structures, are thickened at nodes forming spikes andleaving rounded holes.

This complex structure is one of the latest projects developed by theGenetic Architecture research group as a real application in architectureand construction of the concepts developed there, that are feasible thanksto emerging strategies such as algorithmic design and digital fabrication.

The non-standard geometry outcomes from programming and modelingmethods using RhinoScript, a dialect of Visual Basic Scripting Edition(VBScript) for Rhinoceros, a 3-D NURBS modeler; and also a Math plug-inthat generates NURBS surfaces from mathematical expressions.

! Figure 1.The Radiolarian script anda visualization of the resulting model.

This paper is focused on the first design stage of the project in whichthe author developed implementations of constructive and generativealgorithms, and combined them with the use of some complementarymathematical expressions to create a pavilion of sophisticated biomimeticaesthetics that can be seen as architecture taking advantage of thealgorithmic beauty of life forms.

1.1 Algorithmic strategies for biomimetics

It has been said that biological forms have existed in architecture since itsbeginning.As argued by Estevez [1], this “biomorphic architecture” hasexisted mainly in ornamental elements, always limited by the constructiontechnology available. Only recently, technology has allowed us to reachdeeper into life forms to understand their logic, abstract them as algorithmsand implement them by means of computation, creating designs that areable to deliver biomimetic solutions more efficiently.

Nowadays there are a variety of examples of biomimetic projectsdeveloped with these strategies. Especially after the publication of majorreference books such as Prusinkiewicz and Lindenmayer’s AlgorithmicBeauty of Plants [2], some architects and designers have implementedalgorithmic design strategies for the generation of form that can berelated to biomimetic solutions such as Dollens [3] or Oxman [4],among others.

With the belief that especially the combination of algorithmic techniques(like scripting) with NURBS modeling strategies (with tools like Rhinoceros)is where lies the true potential of current design technologies for theconception of form [5], the use of RhinoScript was considered the basetechnique for the development of this project. In relation to that, the casesof Loukissas and Sass [6], and Dritsas’ MiranScript [7] need to be cited asprecedent.

However, the present project should be viewed as a continuation ofanother project in our research group which, although for another clientand in different circumstances, shares the idea of Radiolarian structuresgenerated by a script.The previous project an interior design for a therapyand treatment clinic in Barcelona.

2. Metodology

Differently from the previous project, a more generative approach was usedthis time, in which almost all forms were generated by the script from thebeginning. Instead of modeling each hexagon and circle and then input themone by one to the script, only the base surface and a couple of numericalvalues were needed as input data and the rest was generated by the script.A special algorithm was implemented to set the general configuration of thestructure: the honeycomb algorithm, with which a difficult and inaccuratemodeling technique was avoided.

680 Ernesto Bueno

2.1 Stereographic sphere generation

Since the sphere and all the spheroids that can be optained from it did notproduce a correct construction of the base surface, a different approachwas required, one in which the isoparametric curves were not converginganywhere within the domain of the surface.The draft of some complexcurves was considered for the construction of a lofted surface, but it wasnot sufficiently accurate.Then, the solution was achieved through the use ofMath plug-in for Rhinoceros, developed by Rhino3DE [8], with which, amongother things, a stereographic sphere can be plotted.

! Figure 2. Stereographic projectionof a Cartesian plane. Illustration by J.R. Davis [9].


This surface is a stereographic projection of a planar grid on a sphere(Figure 2), commonly used in optics and cartography, expressed in Cartesiancoordinates as:

(1)

To calculate a point on the surface, we would have to take this Eqn (1)and extract its coordinated parameters in functions of u and v, as surfacedirections:

(2)

(3)

(4)z

u vu v

= + !+ +

2 2

2 2

11

yv

u v=

+ +2

12 2

xu

u v=

+ +2

12 2

x y zx

x yy

x yx yx y

, , , ,{ } =+ + + +

+ !+ +

" 21

21

112 2 2 2

2 2

2 2##$

%&'

In order to have the array of points to construct a NURBS surface, theMath plug-in calculates these Eqns (2-4) iteratively when they are inputtedalong with the data for u and v: minimum and maximum values to set thebounds for the iteration; and a point count for each direction, acting as thenumber of iterations.

For this project, the outputted stereographic sphere is then non-regularly scaled and rotated to reconstruct the shape of the requiredspheroid.

2.2 Hexagonal tessellation algorithm

While computing a set of points on the given surface with a common anduseful domain division algorithm, an instruction was inserted in the nestedincremental loop to collect each point into a two-dimensional array (alsoknown as a matrix array), in order to have a correct topologic relationshipbetween points in terms of u and v as parametric dimensions of the surface.

" Figure 3. Construction of thehexagonal tessellation from thedivision points of a spheroid (left) andof the transformed stereographicsphere (right).

682 Ernesto Bueno

Then, this matrix was used to trace a hexagonal tile able to repeatitself along the division points of the surface domain (Figure 3).Asimplified version of the Andrew Kudless’ Honeycomb algorithm [10] wasused to iteratively compose the hexagonal tile out of six relative points asfollows:

(5)

q

p u v

p u v

u v

p u v°

( +( )( +( )( +( )( + +(

0

1

2

3

1

1

2

3 1

,

,

,

,

p

))( + +( )( + +( )(

)

*

++++++++++

p u v

p u v

p p

4

5

6 0

2 2

1 2

,

,++

,

-

.

.

.

.

.

.

.

.

.

..

Where q is the hexagonal tile composed of p0 . . . p5, which are thehexagon vertices.A seventh point (p6) was assigned with the same values asthe first one in the script to construct a periodic (or closed) polyline thatdefines this tile. In this case, u and v, in addition of being surface directionsare also used as iteration variables to reference coordinates in the matrix,given that this is also in an incremental nested loop. Since the tile needs tobe repeated in a hexagonal fashion, the secondary loop iterates at a step of4 and the primary loop at a step of 2.

When an integer higher than 1 is assigned to the step of incrementalloops, some iterations are skipped and, in tessellations, that involves skippinga few ‘rows’ or ‘columns’ of tiles. In this particular case, a single execution ofthis subroutine would leave intermittent empty columns that would need tobe filled by a second execution with an offset of the iteration start.This wasdone using another variable for the starting value of the iteration counter,assigned in the argument of the tessellation subroutine.Thus, the subroutineis being called twice, each time with different values assigned to thatvariable.As a result, the use of conditional statements and a repetition ofvery similar loops in the code scope were avoided.

Since it is not used to create honeycomb-like cells, in a broader senseit can be called a hexagonal tessellation algorithm, which returns theconstruction points to input to the next subroutine that creates the finalstructure.

2.3 Radiolarian component algorithm

! Figure 4.Visualization of a singleRadiolarian component generatedfrom the six vertices of a unit of thetessellation.


This is the algorithm that constructs each component of the Radiolarianstructure. Its implementation, the Radiolarian Component subroutine, is beingcalled from within Hexagonal Tessellation loops, so it takes the six vertices ofeach tile to construct a single complex surface, whose partial outputresembles a doughnut trimmed at its boundaries, leaving lifted corners and arounded hole at the center (Figure 4).When components get tiled, thecorners form the spikes.The algorithm is explained in the following step list:

1. Find midpoints between hexagon vertices.2. Set an array of all twelve points.3. For each point, calculate a pair of offset points as outer points:

i. If point count is even, offset with a distance for peak points.ii. If point count is odd, offset with a distance for valley points.

4. Set a construction hole-curve:i. Create a curve using vertices as control points.ii. Scale it to its center with a given factor.

5. For each point in the array of twelve, generate a transversal curve:i. Set a hole-point finding the closest point in the hole-curve.ii. Calculate a pair of offset points from the hole-point as inner

points.iii. Interpolate first outer point, first inner point, second inner point

and second outer point with a transversal curve.6. Generate a ring-surface lofting through all transversal curves.7. Set a trimmer polysurface:

i. Calculate a wider pair of offset points from hexagon vertices.ii. Construct lines joining each pair of offset points.iii. Generate a polysurface lofting construction lines.

8. Trim the ring-surface with the trimmer polysurface.9. Output the resulting surface.

Although this algorithm may look as a process automation routine atfirst sight, it is actually more than that, considering operations like the offsetof points, which are made with a custom function that finds the normalvector of the base surface at a given point and scale it to a specific distance.It acts in isoparametric terms of the surface, without being limited by anytransformation or deformation the surface may suffer. In other words, itoperates in topological terms and returns a geometrical object, somethingthat would not only be time-consuming, but also would require anenormous intellectual effort in vector calculations and NURBS modelingeven for the most advanced draftsperson.

The outer peaks and valley points produce wavy borders in the ring-surface which exhibits a third of a spike for each vertex when it is trimmedby the polysurface, which is a hexagonal prism.The sliced spikes eventuallyget together when the subroutine runs again, called by other iterations ofthe previous procedure, to form the neighbor components.Although it maynot be the most accurate method of forming spikes from a hexagonal grid,

684 Ernesto Bueno

this is a practical and efficient method for generating smooth rounded holesand forming the rest, all in one continuous surface per component.

2.4 Sine wave scaling pattern

Due to the regularity in the size of the holes generated so far, a scalingpattern function was added to correct the lack of material thickness in thelower part of the structure (that bears more loads than the upper part).From a simple sine wave expression sin(x) + sin(y), some parameters wereadded to adapt the results to a particular range:

(6)

Where a and b are wavelength variables, c is a factor for amplitudereduction and h modifies the frequency, or in geometric terms, thetranslation of the function in plot dimension (z). Since a and b are dividers,the amplitude increases as these values grow.

f x y

xa

yb

ch( , )

sin sin=

)*+

,-.+

)*+

,-. +

! Figure 5. Script-generated surfacesas three-dimensional plots of the initialsine wave function (left) and theadapted one (right).


In the RhinoScript implementation, a series of NURBS surfaces weregenerated as plots of Eqn (6) and, by trial and error, adequate values (7.0,6.1 and 5.0 for a, b and c respectively) where assigned to these variables, sothat the wave fits only one peak at the center of the studied range(Figure 5).The wave surface was moved upward with the value 0.5 on h, justto have the entire wave in the Z+ space, hence, to have all positive values.

Since this was inserted inside the Hexagonal Tessellation loops,sequential u and v iteration variables were set for x and y, to get thefunction output, not as a dimension value z, but as a numeric value w.Thiswas used as an ever-changing factor for scaling the holes of the Radiolariancells following this sine wave pattern (Figure 6).

3. Process summary

Having the opportunity to develop scripts form the beginning, an efficientfeedback process was had in terms of time spent. In a period of two weeksof work, a big amount of temporary output structures were generated, also

producing lots of interesting blocks of code that can be useful for futureprojects.

On Figure 7, a series of these temporary structures are shown. Step 4exhibits cells formed from longitudinal section curves; the sharp borderssuggest ‘primitive’ spikes. On step 8, border tangencies are met, withoutspikes. Step 13 implements the current construction of spikes, the sine wavepattern for holes scaling and a random component. On step 15, randomizingwas aborted. In turn, some execution efficiency issues were met.

Some issues will require to be addressed in the next stages of theproject, which include the digital fabrication of the pavilion. Further stepsincorporate smoothing of spikes for better piece assembly; and substitutionof the base surface by another one that, following the same stereographicsurface configuration, could end horizontally to the ground for a continuousstructure foundation instead of just four supporting axes.

4. Conclusion

Although not taking into consideration some topics that should be relevantin architectural design projects (e.g., usability, sustainability), the process herepresented is an important step forward in the realization and application of

" Figure 6.The complete Radiolarianscript applied on a flat surface tovalidate the sine wave pattern forscaling holes.

686 Ernesto Bueno

concepts studied and developed in the Genetic Architectures research group.It combines design strategies from biomimetics, mathematical functions,generative scripting, process automation and versioning, all in an integratedalgorithmic methodology to create a unique architectural piece. Even though,it could be said that the versioning process is here closer to softwaredevelopment than to a form-finding design strategy.

From the beginning, this was a very different design approach, incomparison to the research projects developed so far. I believe it is settingprecedents for a prolific line of design research.

Its dual characteristic of being an academic research project, but also areal work at the same time is important as well, since it is an opportunity topotentially reach interested clients that could support these applications onour professional practice.

Acknowledgements

The author wants to thank Universitat Internacional de Catalunya forsupporting this research and acknowledge the initiative of Alberto T. Estévez.Thanks also to Pablo Baquero and Daniel Wunsch for their pertinentobservations.

! Figure 7. Shape outputs fromdifferent script versions.


References1. Estévez,A.T., Biomorphic Architecture, in: Arquitecturas Genéticas II: Medios

digitales y formas orgánicas, SITES Books/ESARQ-UIC, Barcelona, 2005, 18-53.

2. Prusinkiewicz, P. and Lindenmayer,A., The Algorithmic Beauty of Plants, 2nd ed.,Springer-Verlag, New York, 1996.

3. Dollens, D., Digital-Botanic Architecture, SITES Books, Santa Fe NM, 2005.

4. Oxman, N., ‘Get Real:Towards Performance Driven Computational Geometry’,International Journal of Architectural Computing, 2007, 4(5), 663-684.

5. Bueno, E., ‘Consideraciones y recursos para la concepción de la forma en laarquitectura de la era digital’, Pesquisa em Arquitetura e Construçao, 2008, 1(3).

6. Loukissas,Y. and Sass, L., ‘Rulebuilding: a generative approach to modelingarchitecture using 3D printers’, in: Fabrication: Examining the Digital Practice ofArchitecture: Proceedings of the 23rd Annual Conference of the Association forComputer Aided Design in Architecture, Cambridge, 2004, 176-185.

7. Dritsas, S., ‘MiranScript, Intuitive Calculations’, in: Digital Design:The Quest for NewParadigms: Proceedings of the Education and Research in Computer Aided ArchitecturalDesign in Europe Conference, Lisbon, 2005, 705-712.

8. http://www.rhino3.de/_develop/__v3_plugins/math/ [19-02-2009].

9. http://en.wikipedia.org/wiki/File:CartesianStereoProj.png [18-06-2009].

10. http://www.materialsystems.org/?page_id=384 [03-12-2008].

688 Ernesto Bueno

Ernesto BuenoUniversitat Internacional de CatalunyaDepartment of ArchitectureC. Immaculada, 22, 08017 Barcelona, Spain

[email protected]

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