Square Tessellation Patterns on Double Layer Minimal Surface Structures Geometric Investigation and Design Algorithms

Katherine Liapi1, Andreana Papantoniou2 1,2University of Patras 1,2{kliapi|apapantoniou}@upatras.gr

Minimal surfaces, defined as surfaces of the smallest area spanned by a given boundary present advantages for architectural applications in terms of their structural and material performance. Therefore, the investigation of their properties including their geometric ones deserve special attention. In this regard, methods for tessellating minimal surfaces need to be studied. In this paper, patterns that consist of four squares with partly overlapping sides have been considered. A constrain in this study was the square tiles maintained their planarity. Three different types of surfaces have been considered, namely the helicoid, catenoid and Enneper's surface. Design algorithms that generate tiling patterns in all three minimal surface types have been developed and are presented in the paper. The geometric investigation of the application of the developed methods to double layer structures has also been examined and discussed in the paper. Finally, the accuracy of the developed algorithms has been tested through the construction of a physical model.

Keywords: minimal surfaces, double layer, square tessellation

DOUBLY CURVED SURFACE TESSELLA- a spherical or an elliptical surface. In the process of TION PROBLEMS: EARLIER WORK determining the most appropriate approach for this Within the general frame of surface tessellation prob- study, all methods that generated significantly dis- lems, recent research was focused on the tessella- torted or non-planar square shapes on these surfaces tion of curved surfaces with flat square tiles, where were excluded. For spherical surfaces several existing the tessellation pattern consists of four squares with projection methods were examined. With regard to partly overlapping sides [Liapi et al.2017] the elliptical surface, among the methods that were Specifically, a method for the square tessellations examined, an approach that is based on the Merca- of several of doubly curved surfaces has been devel- tor projection [Osborne, 2013] was given special con- oped. The main objective in this study was to de- sideration and an analytical solution that utilizes the velop a parametric method that allows for the uni- inverse equation of the Mercator projection was de- form distribution of the square pattern throughout veloped. In order to achieve highly accurate results,

PARAMETRIC MODELLING | Concepts - Volume 2 - eCAADe 36 | 385 approximation techniques were also needed. lem [Osserman, 2013]. In mathematical terms mini- In both cases by following this method a net- mal surfaces can be described as surfaces with zero work of equidistant points on the studied surfaces is mean curvature. Specifically the principal curvatures placed first. Then a four square tessellation pattern of minimal surfaces K1 and K2 are equal and reverse on the surface is constructed by the rotation of all K1=-K2, and thus their Gaussian curvature is negative the flat square tiles by the same angle, around an axis K=K1K2<0 at any point of the surface. Phased differ- that passes through their centers and is perpendic- ently in these surfaces for the value H of their princi- ular to their plane. The value of the rotation angle pal curvature we got H=0, where H=1/2(K1+K2). determines the overlap ratio of the adjacent sides of An additional feature of minimal surfaces is that the four square tiles that form the pattern and their their parametric curves (u and v) intersect orthogo- dimensions. nally. This means that the angle of the intersected In the case of minimal surface tessellation, the lines maintains a 90o value (φ=90o) and therefore a properties of minimal surfaces need to be taken into conformal parameterization is possible. Using this account and a method based on these properties will property, an orthogonal grid of intersected curves be developed as described in the following sections. can be created. The intersection points consist the centre points of each square or otherwise tile of the MINIMAL SURFACES TESSELLATION: tessellation. MAIN FEATURES AND GEOMETRIC Accordingly, the (circular) helicoid and catenoid geometries have been considered first. The catenary PARAMETRIZATION surface is a rotational surface generated by the ro- The use of minimal surfaces in architecture has been tation of the catenary curve γ : {cosh u, y=0, z=0} associated with the design of fabric tensile structures around the axis Oz. The helicoid is the minimal and eventually with a great variety of curved config- surface having a helix as its boundary and it is the urations. As minimal surfaces, defined as surfaces of only ruled minimal surface other than the plane (do the smallest area spanned by a given boundary [2], Carmo). present advantages in terms of their structural and For the development of the Enneper’s surface its material performance, the investigation of their ge- parametric equations have been applied: ometric properties for architectural applications de- 1 serves special attention. x = u − u3 + uv2, (1) Covering minimal surface with square tile pat- 3 1 terns is a geometric problem that falls within the do- y = −v − u2v + v3, (2) main of surface tiling or tessellation problems. Tessel- 3 lating flat surfaces with regular or semi-regular pat- z = u2 − v2 (3) terns constitutes a geometric problem that can be These transformation equations have been applied in easily solved and parametrized. However, tessella- a code in order to create the network of points uni- tions of minimal surfaces in which several constrains formly distributed on the surface. As in the square may apply in terms of the pattern or the planarity of tessellation method of spherical and elliptical sur- the composing tiles is not a problem addressed in re- faces, a four square tessellation pattern on the min- cent bibliography. imal surface is constructed by the rotation of all the Minimal surfaces and the area minimizing prob- flat square tiles by the same angle, around an axis lem was first formulated by J.L. Lagrange in 1760 and that passes through their centers and is perpendic- later addressed by Plateau (1849) who experimented ular to their plane. The value of the rotation angle with physical models to find a minimum area with a determines the overlap ratio of the adjacent sides of given boundary, a problem known as Plataeu’s prob-

386 | eCAADe 36 - PARAMETRIC MODELLING | Concepts - Volume 2 the four square tiles that form the pattern and their It is also interesting to note that the helicoid can be dimensions. continuously deformed into a catenoid by the trans- formation: Figure 1 Intermediate stages x = cos asinhvsin u + sin acoshvcos u, (4) of the transformation y = −cos asinhvcos u + sin acoshvsin u, (5) process from a catenoid to a z = ucos a + vsin a. (6) helicoid for where α=0 corresponds to a helicoid and α =π/2 to a different values of catenoid [1]. the variable α As indicated in figure 1, by giving different values before and after the to the variable α we can take intermediate stages of rotation of the the transformation process. square tiles The code permits the application of other mini- mal surfaces equations, generating new networks of points and double square tessellations on different surfaces. Also, another value that is affect the configura- tion of minimal surfaces and respectively the tessel- lation patterns on them are related to the values of the domain ranges for u and v, which are related to different values on the axes x and y. The x - y ratio of each case is proportional to the values u and v. Accordingly for the catenoid the following do- main values provide significantly different results as follows: For a complete circle, the variables u=2π and v=π and the number of rectangles on axis x=30 and on axis y =15, for three quarters of the circle, the vari- ables u=3π/2 and v=π and the number of rectangles on axis x= 20 on axis y =13, while for the half of a circle, the variables u=π and v=π and the number of rectangles on axis x=15 on axis y =15. Finally, for one quarter of the circle, the variables u=π/2 and v=π and the number of rectangles on axis x=10 on axis y=20. In figure 2 three different catenoid surfaces that derive from the application of different domain val- ues are shown.

PARAMETRIC MODELLING | Concepts - Volume 2 - eCAADe 36 | 387 Figure 23 A. catenoid for various domain values, Figure 3 Enneper’s surface for various domain values

Similarly, for Enneper’s surface for a complete circle, While for the half of a circle, we define the variables the variables are defined as u=2π and v=π and the u=π and v=π and the number of rectangles on axis x= number of rectangles on axis x= 30 and on axis y =15. 20 on axis y =20.

388 | eCAADe 36 - PARAMETRIC MODELLING | Concepts - Volume 2 In figure 3 three different Enneper’s surfaces that de- square, that represent the bases on the units, on the rive from the application of different domain values two layers of the network, need to fall on the same are shown. axis that is perpendicular to their plane. So first a method for creating a pattern of squares on the sec- TESSELLATION OF DOUBLE LAYER MINI- ond layer was developed. Subsequently, taking into MAL SURFACE STRUCTURES: THE CASE OF account that the bases of a tensegrity unit of pris- matic form are rotated against each other by a 45 de- TENSEGRITY NETWORKS grees angle, the squares on the second layer were ro- The applicability of the minimal surface tessellation tated by a 45 degree angle and scaled until the adja- method has also been examined in double layer min- cent squares were connected properly. imal surface structures. Tensegrity networks com- All three different types of minimal surface con- posed of square base prismatic units that form dou- figurations that were discussed in the previous sec- ble layer structures [Hanaor, 1992 & 1998] provide tion, namely a catenoid, a helicoid and Enneper’s sur- an ideal testbed for experimentation with this con- face, have been addressed. cept. Structural performance optimization parame- An algorithm that generates the results in a ters, such as minimum weight and amount of mate- graphical environment was then developed and the rial, as well as the possibility of using uniformly ten- Grasshopper visual programming language has been sioned membranes, further justify the investigation used for the parametric description of the two layers minimal surface double layer tensegrity configura- of the tensegrity networks (Figure 4). tions. Taking into account that the method for the gen- eration of the square pattern of flat tiles on a minimal TESTING THE METHOD WITH PHYSICAL surface can be applied for the construction of the ge- MODELS. ometry of the first layer of a tensegrity network [Li- The developed algorithm has been used for the study api et al. 2017], the question at hand was the de- of an actual structure that was planned to be built in velopment of methods that make possible the con- the context of a two day-hand on workshop with un- struction of the second layer of a tensegrity network dergraduate students. The structure would be con- composed of square base prismatic units. In other structed from wood dowels and elastic string and words, assuming that the arrangement of the upper had to be placed as an overhead structure. The de- bases on the composing units of a tensegrity network veloped algorithm has permitted the generation of is a solved problem, a method for determining the ar- several models of minimal surface tensegrity struc- rangement of the lower bases of the tensegrity units tures and the comparison of the models against each that form the second layer of the network, had to be other. A tensegrity structure of helicoidal shape developed. was chosen as the preferred configuration in this in- Accordingly, for the development of the second stance. Specifically, a section of the network that cor- layer of the tensegrity network, a vertical projection responds to the first quadrant of the helicoid was se- of the squares of the first layer of the structure on lected as the basis for exploration on the form of the a surface placed parallel to the previous was per- accrual structure. formed. The requirements that had to be fulfilled, as determined by the shape of the tensegrity units and their method of assembly, are: a) the centers of the squares of the second layer need to lie on a surface that is placed parallel to the first minimal surface of the network, and b) the centroids of the composing

PARAMETRIC MODELLING | Concepts - Volume 2 - eCAADe 36 | 389 for simplicity purposes, the thickness of all members Figure 4 were kept the same and the joints were kept simple. Minimal surface The algorithm was used to generate the dimensions tensegrity networks of each one of the units and the dimensions of the of a) catenoid, b) overlapping sections of adjacent unit bases. subse- helicoid and c) quently, the dimensions of all tension and compres- Enneper’s geometry sion members were derived. Some of the challenges encountered in this stage of the process were due to the materials that were used., i.e. the elastic string, was probably not tight enough. The construction of the model has also indicated some minor problems of the developed algorithm that need to be fixed. Specifically, in some areas of the digital model, the square bases did not perfectly meet. Overall the construction of the physical model has proven the appropriateness, accuracy and appli- cability of the developed methods.

REFERENCES Do Carmo, M. P 1986, ’The Helicoid’, in Fischer, G. (eds) 1986, Mathematical Models from the Collections of Universities and Museums, Vieweg, Braunschweig, Germany, pp. 44-45 Hanaor, A 1992, ’Aspects of Design of Double-Layer Tensegrity Domes’, International Journal of Space Structures, 7(2), pp. 101-103 Hanaor, A 1998, TensegritytheoryandApplication, Beyond the cube, J. Francois Gabriel, John Wiley & Sons, Inc., New York, pp 385-408 Liapi, K, Papantoniou, A and Nousias, Chr 2017a ’Mor- phological exploration of curved tensegrity net- works: Towards minimal surface double-layer con- figurations’, IASS, Hamburg, Germany Liapi, K, Papantoniou, A and Nousias, Chr 2017b ’Square For the development of the virtual models several tessellation patterns on curved surfaces’, Proceed- other constraints and parameters had to be taken ings of the 35th eCAADe Conference - Volume 2, Rome, into account, i.e. various parameters related to the to- pp. 371-378 Osborne, P 2013, The Mercator Projections; The Normal tal number of units to be used, the total length of the and the Mercator Projections on the Sphere and the El- linear compression members, the amount of cable lipsoid with Full Derivations of all Formulae, Zenobo, overlap between adjacent units, the number of joints Edinburgh etc. As the time was limited and the participating Osserman, R 1986, Plateau, Dover Books on Mathemat- students had no experience with tensegrity model ics, New York, Dover building, solutions with a smaller number of units for [1] http://mathworld.wolfram.com/Helicoid.html [2] https://www.encyclopediaofmath.org/index.php/Mi the same surface area had to be considered. Also, nimal_surface

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