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.




















All living organisms are composed of cells, and cells are fluid-filled spaces surrounded by an envelope of little material- cell membrane. Frei Otto described this kind of structure as pneus.

From first order,  peripheral conditions or the packing configuration spatially give rise to specific shapes we see on the second  and third order.

This applies to most biological instances.  On a larger scale, the formation of beehives is a translated example of the different orders of ‘pneu’.

Interested to see the impact of lattice configuration on the forms, I moved on to digital physics simulation with Kangaroo 2 (based on a script by David Stasiuk). The key parameters involved for each lattice configuration are:

Inflation pressure in spheres
Collision force between the spheres
Collision force of spheres and bounding box
Surface tension of spheres


Physical exploration is also done to understand pneumatic behaviors and their parameters.

This followed by 3D pneumatic space packing. Spheres in different lattice configuration is inflated, and then taken apart to examine the deformation within. This process can be thought of as the growing process of seeds or pips in fruits such as pomegranates and citrus under hydrostatic pressure within its skin; and dissections of these fruits.

As the spheres take the peripheral conditions, the middles ones which are surrounded by spheres transformed into Rhombic dodecahedron, Trapezoid Rhombic dodecahedron and diamond respectively in Hex Grid, FCC Grid,  and Square Grid. The spheres at the boundary take the shape of the bounding box hence they are more fully inflated(there are more spaces in between spheres and bounding box for expansion).


Physical experimentation has been done on inflatables structures. The following shows some of the outcome on my own and during an Air workshop in conjunction with Playweek led by Will Mclean and Laylac Shahed.

To summarize, pneumatic structures are forms wholly or mainly stabalised by either
– Pressurised difference in gas. Eg. Air structure or aerated foam structures
– liquid/hydrostatic pressure. Eg. Plant cells
– Forces between materials in bulk. Eg. Beehive, Fruits seeds/pips

There is a distinct quality of unpredictability and playfulness that pneumatic structures could offer. The jiggly nature of inflatables, the unpredictability resulted from deformation by compression and its lightweightness are intriguing. I will call them as pneumatic behaviour. I will continually explore what pneumatic materials and assembly of them could offer spatially in Brief 02. Digital simulations proved to be helpful in expressing the dynamic behaviours of pneumatic structures too, which I intend to continue.

I have been researching Miura pattern origami as a structural solution for rapidly deployable structures. Miura ori are interesting as structures due to their ability to develop from a flat surface to a 3D form, and become fully rigid, with no degrees of freedom, once constrained at certain points. 141110_Year 2 working folio2 Physical and digital experiments with Miura Ori have taught me that certain topographies can be generated by developing a modified Miura pattern. With the help of Tomohiro Tachi’s excellent research on the subject of curved Miura ori, including his Freeform Origami simulator ( I have learned that Miura ori surfaces that curve in the X and Y axes can be generated by modifying the tessellating components, however these modifications require some flexibility in the material, or looseness of the hinges. 141110_Year 2 working folio6 As a system for a rapidly deployable structure, I am most interested in the potential for the modified Miura ori to work as a structure built with cheap, readily available sheet materials which are generally planar, so I will continue to develop this system as a rigid panel system with loose hinges that can be tightened after the structure is deployed. 141110_Year 2 working folio4 In order to test the crease pattern’s ability to form a curved surface, I have defined a component within the Miura pattern that can tessellate with itself. The radius of this component’s developed surface is measured as it is gradually altered.

With the objective being to develop a system for the construction of a rapidly deployable structure, I have also been interested in understanding the Miura ori’s characteristics as it is developed from flat. Physical and digital tests were performed to determine the system’s willingness to take on a curve as its crease angles decrease from flat sheet to fully developed. I found the tightest radius was achieved rapidly as the sheet was folded, with the radius angle reaching a plateau. This is interesting from the perspective of one with the desire to create a structure that has a predictable surface topography, as well as from a material optimisation standpoint; the target topography can be achieved without the wasteful deep creases of an almost fully developed Miura ori. 141110_Year 2 working folio5 With the learnings of the modified Miura ori tests in mind, a simple loose hinged cylinder is simulated. As the pattern returns on itself and is fastened, the degrees of freedom are removed and the structure is fully rigid. 141110_Year 2 working folio A physical model of the system was constructed with rigidly planar MDF panels and fabric hinges. The hinges were flexible enough to allow the hinge movement necessary in developing this particular modified Miura ori, however some of the panels’ corners peeled away from the fabric backing as the system was developed from flat. A subsequent test will seek to refine this hinge detail, with a view to creating a scalable construction detail that will allow sufficient flexibility during folding, as well as strength once in final position. 141110_Year 2 working folio3

John Konings

The Cloud at Burning Man

The Cloud at Burning Man

So easily can fun and playfulness be neglected within Architecture. My proposal stands as an embodiment of these aspects, creating an area of inclusive participation, a space that can be explored and is only complete when occupied.

Fallen from the sky and tied down in the middle of Black Rock City ‘The Cloud’ stands as a mirage for weary-eyed travellers from far and wide, a beacon of sanctuary that creates spaces that provide respite from the harsh conditions of the desert using permeable fabric to create a cool atmosphere diffusing light within daylight and emitting a soft glow from within in the evening.

Principle Stress Analysis

Principle Stress Analysis

Walking through the dessert after a long journey along the silk road ‘The Cloud’ emerges as a whimsical mirage. Mimicking the form of a cloud the easily recognisable form is transformed into Architecture; a sinuous billowing form allowing us to fulfil a childhood dream, walking on clouds.

The principle structure of the cloud is composed of hollow rolled steel tubes ,sandwiched between thick perforated fabric, strategically placed to withstand the extreme wind conditions as well as human interaction. Elevated from the floor these tubes are secured to the ground using the kandy kane re-bar method.
Keeping the form soft and playful so that not only is the installation safe but also malleable, responding to people climbing and walking it, bungee rope is securely looped over the steel tubes and threaded through the ‘ground’ fabric to hold it up, as illustrated in the accompanying drawing.

Structural Breakdown

Structural Breakdown

The Cloud Perspective

The Cloud Perspective

Orthographic Cut

Orthographic Cut

Interactivity is an integral part of the installation. Bringing to life the stranded cloud people are encouraged to explore the piece climbing in, over and around it, finding intricate crevasses that provide discreet hidden entrances to the inner cloud where an intimate social environment softly illuminated by the diffused daylight, providing an area of solace.

Physical Model 1:5

Physical Model 1:5

Evening View of The Cloud

Evening View of The Cloud

DIMENSIONS // 5000mm(l) x 3100mm(w) x 4100mm(h)

After our post on Jake Hebbert‘s tutorial, here are some great Grasshopper tricks to create boids or fractals by Kristof Crolla, Architect at LEAD and teacher at Honk Kong University on his vimeo channel:

Below: Boids behaviour with Hoopsnake


Below: Fractals using Hoopsnake:


Below: Catenary Network on Kangaroo:


Below: Explaining the path mapper:


Below: Organizing hexagons on flat list




Below are two videos made from the exercises shown at the DS10/Inter9 Grasshopper class at the Architectural Association.

The first video shows the trail left by points constrained by springs, end points and gravity. The Arch moves up and to the side, leaving a beautiful trace which reminded me of the pictures of Edouard Muybridge. It was done with Grasshopper and the free Kangaroo plugin by Daniel Piker.

The second video is a very simple example of recursion using Hoopsnake byVolatile Prototype for Grasshopper: A line rotates on another line and this new line becomes the currrent one on which the rotation is done and so one and so forth. Depending on the angle of the rotation and its location on the curve, these amazing patterns get created.

It has almost been already a year that Toby and I started tutoring DS10 at Westminster. One of our main ambitions was to link physical and digital experiments so that one feeds the other.

Physical reality is much more than surfaces on a screen therefore students created complex parametric models working as systems linked to many forces (gravity, environment, structure…etc…) and not just finished objects. These very precise digital models allow students to implement what they learn from their physical models, to simulate even more design options and further understand the rules behind them.

To do so, they used Grasshopper and its numerous plugins provided by generous developers. Grasshopper is a graphical algorithm editor integrated with Rhinoceros 3D modelling tool and a 18,000 strong community exchanging ideas and helping each other on the forum.

Below are most of the printscreens that I used to help the students with their journey into parametric modelling which is based on help that I also received previously. I hope that this will help others to design amazing things! If you have any questions on one of the images, please do not hesitate to ask.

Below is my favourite image: packing balloons on a surface using Kangaroo (with Emma Whitehead)