The Nature of Gridshell Form Finding









7.0 Introduction to Shells




7.1 – Single Curved Surfaces


Uniclastic: Barrel Vault (Cylindrical paraboloid)



Uniclastic: Conoid (Conical paraboloid)


7.2 – Double Curved Surfaces


Synclastic: Dome (Elliptic paraboloid)



Anticlastic: Saddle (Hyperbolic paraboloid)
Kellogg’s potato and wheat-based stackable snack
11 Tensile and Compressive behaviour of shells.jpg

7.3 – Translation vs Revolution

Hyperbolic Paraboloid: Straight line sweep variation


Ruled Surface & Surface of Revolution (Circular Hyperboloid)
Hyperboloid Generation [Source:Wikipedia]


These towers are pretty cool acoustically as well:


8.0 Geodesic Curves


Geodesic Curves - Ribon.jpg
Conic Plank Lines (Source: The Geometry of Bending)




8.1 – Basic Grid Setup


Basic geodesic curves on a plane


Initial set of curves


Addition of secondary set of curves


Grid with independent sets


8.2 – Basic Gridshells


Uniclastic: Barrel Vault Geodesic Gridshell


Uniclastic: Conoid Geodesic Gridshell


Synclastic: Dome Geodesic Gridshell


Anticlastic: Saddle Geodesic Gridshell


On another note:


6 - Alternate Grid


Biosphere Environment Museum - Canada
Biosphere Environment Museum – Canada


Centre Pompidou-Metz - France
Centre Pompidou-Metz – France


8.3 – Gridshell Construction


O - Saddle 2
Hyperbolic Paraboloid: Straight Line Sweep Variation With Rotating Plank Line Grid


Hyperbolic Paraboloid: Flattened Plank Lines With Junctions





9.0 Form Finding


9.0 – Minimal Surfaces


hyperbolic paraboloid soap bubble


Kangaroo2 Physics: Surface Tension Simulation



The Eden Project – United Kingdom




Kangaroo2 Physics: Catenary Model Simulation




9.3 – Biomimicry in Architecture

Example of Biomimicry








Triply periodic minimal surfaces _topology


Frequently occuring in nature, minimal surfaces are defined as surfaces with zero mean curvature.  These surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame.

The thin membrane that spans the wire boundary is a minimal surface of all possible surfaces that span the boundary, it is the one with minimal energy. One way to think of this “minimal energy” is that to imagine the surface as an elastic rubber membrane: the minimal shape is the one that in which the rubber membrane is the most relaxed.


A minimal surface parametrized as x=(u,v,h(u,v)) therefore satisfies Lagrange`s equation


(Gray 1997, p.399)

This year`s research focuses on triply periodic minimal surfaces (TPMS). A TPMS is a type of minimal surface which is invariant under a rank-3 lattice of translations. In other words, a TPMS is a surfaces which, through mirroring and rotating in 3D space, can form an infinite labyrinth. TPMS are of particular relevance in natural sciences, having been observed in observed as biological membranes, as block copolymers, equipotential surfaces in crystals, etc.

From a mathematical standpoint, a TPMS is the most interesting type of surface, as all connected RPMS have genus >=3, and in every lattice there exist orientable embedded TPMS of every genus >=3. Embedded TPMS are orientable and divide space into disjoint sub-volumes. If they are congruent the surface is said to be a balance surface.

The first examples of TPMS were the surfaces described by Schwarz in 1865, followed by a surface described by his student Neovius in 1883. In 1970 Alan Schoen, a then NASA scientist, described 12 more TPMS, and in 1989 H. Karcher proved their existence.

The first part of my research focuses on understanding TPMS geometry using a generation method that uses a marching cubes algorithm to find the results of the implicit equtions describing each particular type of TMPS. The resulting points form a mesh that describes the geometry.

Schwartz_P surface

schwartz_p_formation   Schwartz_p

Neovius surface

Neovius_formation neovius

Gyroid surface

gyroid_formation gyroid

Generated from mathematical equations, these diagrams show the plotting of functions with different domains. Above, the diagrams on the left illustrate the process of forming a closed TMPS, starting from a domain of 0.5, which generates an elementary cell, which is mirrored and rotate 7 times to form a closed TPMS. A closed TMPS can also be approximated by changing the domain of the function to 1.

The diagrams below show some examples generating a TMPS from a function with a domain of 2. The views are front, top and axonometric.

FRD surface

dd = 8 * Math.Cos(px) * Math.Cos(py) * Math.Cos(pz) + Math.Cos(2 * px) * Math.Cos(2 * py) * Math.Cos(2 * pz) – Math.Cos(2 * px) * Math.Cos(2 * py) – Math.Cos(2 * py) * Math.Cos(2 * pz) – Math.Cos(2 * pz) * Math.Cos(2 * px)


D Prime surface

dd = 0.5 * (Math.Sin(px) * Math.Sin(py) * Math.Sin(pz) + Math.Cos(px) * Math.Cos(py) * Math.Cos(pz)) – 0.5 * (Math.Cos(2 * px) * Math.Cos(2 * py) + Math.Cos(2 * py) * Math.Cos(2 * pz) + Math.Cos(2 * pz) * Math.Cos(2 * px)) – 0.2


FRD Prime surface

dd = 4 * Math.Cos(px) * Math.Cos(py) * Math.Cos(pz) – Math.Cos(2 * px) * Math.Cos(2 * py) – Math.Cos(2 * pz) * Math.Cos(2 * py) – Math.Cos(2 * px) * Math.Cos(2 * pz)


Double Gyroid surface

dd = 2.75 * (Math.Sin(2 * px) * Math.Sin(pz) * Math.Cos(py) + Math.Sin(2 * py) * Math.Sin(px) * Math.Cos(pz) + Math.Sin(2 * pz) * Math.Sin(py) * Math.Cos(px)) – 1 * (Math.Cos(2 * px) * Math.Cos(2 * py) + Math.Cos(2 * py) * Math.Cos(2 * pz) + Math.Cos(2 * pz) * Math.Cos(2 * px))


Gyroid surface

dd = Math.Cos(px) * Math.Sin(py) + Math.Cos(py) * Math.Sin(pz) + Math.Cos(pz) * Math.Sin(px)


This method of approximating a TPMS is high versatile, useful in understanding the geometry, offsetting the surfaces and changing the bounding box of the lattice in which the surface is generated. In other words, trimming the surface and isolating parts of the surface. However, the resulting topology is unsuitable for fabrication purposes, as the generated mesh is unclean, being composed of irregular polygons consisting of triangulations, quads and hexagons.

The following diagrams show the mesh topology for a Gyroid surface, offset studies and trimming studies.



4  23

For fabrication purposes, my proposed method for computationally simulating a TPMS is derived from discrete differential geometry, relying on the use of Kangaroo Physics, a Grasshopper plugin for modeling tensile membranes. Bearing in mind that a TPMS has 6 edge conditions, a planar hexagonal mesh is placed within the space defined by a certain TPMS`s edge conditions. The edge conditions are interpreted as Nurbs curves. Constructed from 6 predefined faces, the initial planar hexagonal mesh, together with the curves defining the surface boundaries are split into the same number of subdivisions. The subdivision algorithm used on the mesh is WeaveBird`s triangular subdivision. The points resulted from the curve division are ordered so that they match the subdivided mesh`s edges, or its naked vertices. The naked vertices are then moved in the corresponding points on the curve, resulting in a new mesh describing a triply periodic surface, but not a minimal one. From this point, Kangaroo Physics is used to find the minimal surface for the given mesh parameters, resulting in a TPMS.

Sequential diagram showing the generation of a Schwartz_P surfaces using the above method.


A Gyroid surface approximated with the above method

gyroid_full  8

This approach towards approximating a TPMS leads to a study in the change of boundary conditions, gaining control over the geometry. The examples below present various gyroid distorsions generated by changing the boundary conditions.

6  7

5  4

Being able to control the boundary conditions defining a gyroid, or any TPMS, opens up to form optimization through genetic algorithms. Here, various curvatures for the edge conditions have been tested with regards to solar gain, using Galapagos for Grasshopper.

1_1                2_1

3_1                 4_1

The following examples show some patterns generated by different topologies of the starting mesh.





















Tensegrity structure or also known as tensional integrity, a portmanteau term that was coined by Buckminster Fuller in the 1960s. It is a structural principle which is based on the use of detached components in compression inside a net of continuous tension.It is also known as “floating compression”, a term that was promoted by Kenneth Snelson. Each compressed members such as struts or bars do not touch among another and the prestressed tensioned members , tendons or cables for example, define the system spatially. Snelson defines tensegrity as a closed structural system composed of a set of three or more elongate compression struts within a network of tension tendons, the combined parts mutually supportive in such a way that the struts do not touch one another, but press outwardly against nodal points in the tension network to form a firm, triangulated, prestressed, tension and compression unit. Triangulated network are stronger and even more firm, if compared to non-triangulated network. 

Basic tensegrity structure made up a tower. Photo courtesy to LiftArchitects
Basic tensegrity structure made up a tower. Photo courtesy to LiftArchitects

Precedent project  by students of Ball State University
Precedent project by students of Ball State University


Biotensegrity, is a term coined by Dr. Stephen Levin, is the application of tensegrity principles to biologic structures.Harvard physicians and scientist Donald Ingber has developed a theory of tensegrity in molecular biology to explain cellular structure.The shape of cells are all can be mathematically modelled if a tensegrity model is used for cell’s cytoskeleton. Cytoskeleton is a network of fivers composed of proteins contained within a cell’s cytoplasm, which is dynamic structure, parts of which are constantly destroyed, renewed or newly constructed.



The study of tensegrity structure started through geometric remodeling the tensegrity model using digital softwares, to understand the deployability of the system . The basic form of tensegrity is being explored through addition of struts in the cell modules . The basic cell module is then are combined according to geometry tessellation . They were applied on regular surface and also irregular surface ,  while maintaining the structural frequency and mesh tension .



By using basic cell module of 3 struts, the tensional components were explored both using cables and fabric . The fabric helps in creating enclosures and more aesthetically pleasing in installation. It is still in the beginning of the design by studying the behavior of the system, with the goal to expand the possibilities of design in the future , either for Burning Man or Buro Happold.





For the next project, the possibility of the tensegrity structures will be explored more and the design will be highlighted on the playful intervention of tensegrity structure and the advantages of this super lightweight system for both proposal for Burning Man and Buro Happold .

Lotus Hypars

Lotus Hypars – A study of hyperbolic bamboo structures

The Lotus Hypars symbolise the “Caravansary” trading centre. The structure is assembled as the centre for exchange after journeying across land and water to a resting point, Burning Man. Hammocks offer a space for the festivals unique style of trading to be discussed and carried out. The tangible nature of the Lotus also creates a playfulness in an otherwise formal system of resources exchange. The lightweight structure evolves from the horizontal lines of the desert and forms a hyperbolic shelter. The user can inhabit not only underneath the structure, but also the petal shaped hammocks. Here, individuals can exchange stories, supplies and treasures.


In Buddhism, the Lotus flower is symbolic of fortune. It grows in muddy water, and it is this environment that gives forth the flower’s first and most literal meaning: rising and blooming above the murk to achieve enlightenment. The Lotus Hypar story has evolved from the same principles. In the harsh desert environment, man can create beauty. The folded geometries are playfully excited by human participation. A twist, a fold and a push.


The structure is assembled using bamboo sticks that are arranged in a reciprocal formation. These canes are then bound using high strength elastic bands. This allows for the flat cells to twist and take on new shapes. The Lotus Hypar is formed by a repetitive series of folds and the result forms petals. These are symbolic of the Lotus flower. The cells are covered with a white semi-elastic membrane that adds to the strength of the structure and the petal geometries become more visible. These are also the hammocks that can be inhabited by the Burning Man users.


In order to test the structural performance of the proposal, I constructed a series of 1:1 scale models. This was done using 6m and 3m bamboo canes (35mm diameter). By testing a small segment of the full proposal, it is easier to determine the success of the final proposal.

LH.4 LH.5

Black Light Tower by Luka Kreze

Midnight Butterfly Release

My design for this year’s Burning Man Festival is titled ‘Black Light Tower.

Black Light Tower is a membrane tensegrity structure which is constructed out of 38 metal rods and a transparent structural membrane. One of the most important characteristics of the Black Light Tower tower is also one of the most important principles of Burning Man Festival – Self Reliance. Just like people of Burning Man are encouraged to rely solely on themselves and their own resources this structure is fascinating mainly because it does not use any additional support such as columns or cables, but relies solely on its own structure. The membrane almost miraculously holds the structure up and in tension. To be more precise, instead of the structure holding up the cover, cover is holding up the structure.

Tensegrity structure of this kind seems to be a perfect allegory of the Burning Man principles and a symbol of self reliance.
Black Light Tower will provide a unique experience to its users.

Its interior space is essentially a ladder which leads towards the top of the structure. From the day one the space is filled with hundreds and hundreds of butterflies which cannot escape the tower.

Butterflies then fill the tower storey by storey as the days go by. Every day the net which separates one storey from the other gets removed and butterflies can fly higher into the structure. Visitors can climb only as high as the butterflies go.

On the final day magic happens as the butterflies are symbolically released in the air, at the same time as the Burning Man Figure is set on fire. Butterflies are lit with UV light which allows the visitors to see them in their true colours, in the same way as butterflies see each other, while creating a spectacular event for the viewers’ eyes.

Immediacy is probably one of the most important values of the Burning Man culture. As written in the ten principles immediate experience means to seek to overcome barriers that stand between us and a recognition of our inner selves, the reality that surrounds us and contact with a natural world exceeding human powers.
The experience of climbing the tower will definitely make the climbers fell more alive aware of their inner selves. The night view of the Black Light Tower should be spectacular as all of the rods will be equipped with UV light which will reveal not only the magic of the floating structure under the membrane but ever changing projection of butterflies in flight under the Black Light. One of the most thrilling experiences, however, will most probably be the sublime view of the Playa from the top of the tower on the last day of the festival, that seems like its going to collapse any minute.

Elevation View of the Black Light Tower

Precedent Study

Large Scale Test Model

Timeline Drawing of the Butterfly Release

Section Drawing

MOOM Tensegritic Membrane Structure (Noda) by Kazuhiro Kojima

Architectural students at the Tokyo University of Science developed an experimental, extremely lightweight, load-bearing structure for a temporary pavilion. The 26-metre-long, up to 7,5-metre-wide and 4,25-metre-high volume is self-supporting and comprises only two kinds of component: the metal bearing elements and a delicate space-enclosing skin consisting of an 0,7-mm membrane of elastic polyester fabric. The membrane is drawn over metal tubes that create a tensegrity system and forms the tension element. The 131 compression bars are 25-mm diameter aluminum tubes of various lengths and there is no contact between them; instead, they are connected to the skin by sliding the ends into sheaths sewn on. The membrane is anchored at the base like a conventional tent with pegs consisting of aluminum tubes with tips pressed together to form a point. The compression members are pushed into these pegs and fixed in position by means of steel pins. With a weight of only 600 kg, this airy structure covers a ground area of 146 square meters. The pavilion was erected by 70 students in a single day. Initially, they laid out the ready tailored skin, then slid the tubular members into the sheaths. The overall structure was tensioned on all sides, pushed upwards at the same time in the interior and finally fixed to the ground. The convex and concave forms resulting from this create an animated surface and a lively interplay of light and shade. Since the membrane screens off 80% of the UV radiation, but allows 50% of the daylight to pass through, the softly filtered light creates a fascinating spatial impression internally. When illuminated, the translucent pavilion has the appearance of a lighted sculpture.

Via Archetipo


“The Water Cathedral is a large, horizontal urban nave for public use, discreetly exposed so that enigmatic and semi-dark atmospheres can be made out through its topographic lines of floor and ceiling. Both topographies are surface systems shaped by numerous slender, vertical components, which hang or rise like stalactites and stalagmites in a cave, varying their heights and concentrations. A physical organization with regimes of proliferation and differentiation; a series of columns, platforms, arches, curtains, domes and caverns emerge, qualifying the project spatially and atmospherically.Water embedding / Dripping, duration, delay by integrating water within its spatial and material logics, the project promotes a form of architecture that systematizes dripping pulses and speeds, consistently exacerbating the water drop as primal atmospheric matter.”

“This is accomplished through the hanging components of the roof topography, constituted geometrically and materially like textile prisms placed upside-down with a partial granular substratum filling. They are fed by a hydraulic network that continuously irrigates using measured doses. When filled with small amounts of water, these elements act as regulators or atmospheric interfaces, experiencing a phenomenon of absorption and release where, through capillarity, water drops gradually flow against gravity along the prism edges, slowing down the time of concentration and fall. This delay process that consequently produces multiple dripping rhythms over the ground topography, formally demonstrates management of duration. It materializes an architectural form in which water can be particularized in a system that has extensive effects over an area. The delay principle, established through mediation by the prisms, is translated into the possibility to define the layout and use the water as it drops.Form Principles as an abstraction and geometrical transposition of the stalactite, a three-dimensional spatial component, is developed. A three-phase prism with an equilateral base that varies in height and area. These elements proliferate in a grid system, forming curves of different lengths and intensities. This variation capacity component, allows to effectively producing diversity of spaces, textures, patterns and interactions with environmental dynamics based in a principle of material abundance.Geometrical studies from two dimensional curve variation and three dimensional arrangements constrained to equilateral modules. These forms of proliferation deliver rich arch formations, defining interiorities, structural points and modular connectivity.”




via WATER CATHEDRAL | GUN Architects.