FLOW ALONG SURFACE VARIATIONS
3D PRINT TESTS
MODELLING THE INFLORESCENCE
PAPER TO PLY
METHOD OF EXTENSION
The inspiration for this body of work is the complex formations of tree branching. It is fascinating to comprehend the geometry of the branches and the reasoning for this, from aerodynamics to solar gain. The branching can be seen as fractal, with each branch migrating into two or more as it moves through space, also the thickness of the trunks is mathematically interesting, with Leonardo Da Vinci’s deduction of the sum of branch thickness moving up the tree, at a certain height is equal to the thickness of the initial tree trunk. This analysis has been taken forward throughout these studies, creating more structurally sound physical and digital models. Through studying these geometries the work aims to produce architecture which is thoughtfully designed, and can be adapted based on its environment.
Point Cloud Branching
Travelling through the point cloud the branching lines are created, although to produce a smoother geometry the initial branch lines are reformed with smooth curvature. Then, a pipe structure is applied to the branch lines, with varying thickness moving from top to bottom, as seen in the drawing above.
To form the point cloud branching system, first the path the branches aim to take must be formed, in the below case a cone shape was chosen to closely mimic the geometry naturally formed by the dragon-blood tree. Next a bounding box is created to inform the population of points. Once the points are established within the bounding box they can be joined to its nearest counter points, thus forming a point cloud.
The voronoi geometry is offset, with the distance halved each iteration.
The outer most geometry is divided into points which meet on the offset geometry.
The process of meeting the points continues.
Finally the points are all joined at a central point.
To form the 3D printed models the below file was used, is it comprised of 10 pyramid like branching structures, perfectly joined. Therefore the 3D printed model to the right could form the below geometry, if all of its counterparts were also printed.
Curved Plywood Branching
The initial test model using 0.8mm plywood utilises an initial layer of 8 segments. Once built the model had little tension, therefore needed string to create the dome like form. Although, even with the tension string the resulting form was flatter than anticipated.
Proceeding the first test, the second aimed to produce a more substantial dome geometry. To do this 5 rather than 8 initial segments where used.
When the tension string was weaved the dome form was created, giving the model an ability to stand upright, as shown in the image above.
The final test, aimed to decreased the percentage of material necessary to construct the model. This lead to cutting parts to make them thinner, causing breakage. Through the breakage, it was found that the dome like form could be created without the use of tension string.
Curved Paper Void Branching
Initially points are set using populate tool in grasshopper, then the points are plugged into the voronoi tool, giving the geometry of meeting circular entities.
From the voronoi geometry, the curves are filleted, then offset, at an increment doubling each time.
On the outer curve points are set equidistant apart, from these points a mid point is set along the inner curve, then creating a line to the closest point on the inner curve.
The task of joining the points at their midpoint, along the closest point on the adjacent curve, is repeated. Finally the points all join together in the centre of the final curve.
Once all the branches meet at the central point of the final curve the offset curves can be removed to reveal the branching geometry.
Finally, tracing over the straight branching geometry with the curve tool. Revealing petal like closed curves filling the voids between the straight branches.
Thread Voronoi Branching
Dragon Blood Tree Analysis
Through extensive research into the construction of grid shells, as well as differential geometry, I present a design solution for a complex grid structure inspired by the highly symmetrical and optimised physical properties of a triply periodic minimal surface. The proposal implements the asymptotic design method of Eike Schling and his team at Technical University of Munich.
‘Minimal Matters’ utilises the several geometric benefits of an asymptotic curve network to optimise cost and fabrication. From differential geometry, it is determined asymptotic curves are not curved in the surface normal direction. As opposed to traditional gridshells, this means they can be formed from straight, planar strips perpendicular to the surface. In combination with 90° intersections that appear on all minimal surfaces (soap films) this method offers a simple and affordable construction method. Asymptotic curves have a vanishing normal curvature, and thus only exist on anticlastic surface-regions.
Asymptotic curves can be plotted on any anticlastic surface using differential geometry.
On minimal surfaces, the deviation angle α is always 45 (due to the bisecting property of asymptotic curves and principle curvature lines). Both principle curvature networks and asymptotic curve networks consist of two families of curves that follow a direction field. The designer can only pick a starting point, but cannot alter their path.
(a) Planes of principle curvature are where the curvature takes its maximum and minimum values. They are always perpendicular, and intersect the tangent plane.
(b) Surface geometry at a generic point on a minimal surface. At any point there are two orthogonal principal directions (Blue), along which the curves on the surface are most convex and concave.
Their curvature is quantified by the inverse of the radii (R1 and R2) of circles fitted to the sectional curves along these directions. Exactly between these principal directions are the asymptotic directions (orange), along which the surface curves least.
(c) The direction and magnitude for these directions vary between points on a surface.
(d) Starting from point, lines can be drawn to connect points along the paths of principal and asymptotic directions on the respective surface.
The next step is to create the asymptotic curve network for the Gyroid minimal surface; chosen from my research into Triply Periodic Minimal Surfaces.
As the designer, I can merely pick a starting point on an anticlastic surface from which two asymptotic paths will originate. It is crucial to understand the behaviour of asymptotic curves and its dependency on the Gaussian curvature of the surface.
Through rotational symmetry, it is resolved to only require six unique strips for the complete grid structure (Seven including the repeated perimeter piece).
The node to node distance, measured along the asymptotic curves, is the only variable information needed to draw the flat and straight strips. They are then cut flat and bent and twisted into an asymptotic support structure.
Eight fundamental units complete the cubit unit cell of a Gyroid surface. Due to the scale of the proposal, I have introduced two layers of lamellas. This is to ensure each layer is sufficiently slender to be easily bent and twisted into its target geometry, whilst providing enough stiffness to resist buckling under compression loads.
‘Minimal Matters’ aims to create an explorative, meditative and interactive experience for visitors. It is a strained grid shell utilising the geometrical benefits of an asymptotic curve network; digitally designed via algorithmic rules to minimise material, cost, and construction time.
Inspired by the highly symmetrical and optimised physical properties of a triply periodic minimal surface, ‘Minimal Matters’ aims to create an explorative, meditative and interactive experience for visitors. It is a strained grid shell utilising the geometrical benefits of an asymptotic curve network; digitally designed via algorithmic rules to minimise material, cost, and construction time.
The proposal takes the form of a crystalline structure found in nature, interpreted through parametric design into a timber grid art piece. In the sense of repeating themselves in three dimensions, a gyroid is an infinitely connected triply periodic minimal surface. A minimal surface is a single surface articulation which minimises that amount of surface needed to occupy space. The proposal represents restoring a balance in energy, taking only that of the earth’s resources required to fulfil the form. Our inability to distinguish our needs from our greeds leads to excessive desires for life’s commodities. The efficiency of the design complements the beauty of rotational symmetry of a single node.
The lattice structure will create foot and hand holds to help climbers onto the series of sloping platforms; allowing users to survey the desert camp from different perspectives.
More than just a climbable structure, Minimal Matters is to be a resting place for festival-goers and a shelter from the strong sun of the site. The layers of grids cast shadows of varied patterns throughout the day. At night, LED lighting along the lamellas will celebrate it’s form and illuminate the playa.
Inspired by nature, the proposal brings a parametrically designed structure into the realm of physical interaction. The piece is a culmination of thorough research and physical exploration of timber’s potential. The combination of conceptual bravery matched with architectural reality seeks an architecture of playfulness and beauty which will respond to the inclusive environment of Burning Man. It will celebrate a new design method for timber grid construction, and symbolise the harmony between nature and computational design.
A minimal surface is the surface of minimal area between any given boundaries. In nature such shapes result from an equilibrium of homogeneous tension, e.g. in a soap film.
Minimal surfaces have a constant mean curvature of zero, i.e. the sum of the principal curvatures at each point is zero. Particularly fascinating are minimal surfaces that have a crystalline structure, in the sense of repeating themselves in three dimensions, in other words being triply periodic.
Many triply periodic minimal surfaces are known. 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.
My research into grid structures with the goal of simplifying fabrication through repetitive elements prompted an exploration of TPMS. The highly symmetrical and optimised physical properties of a TPMS, in particular the Gyroid surface, inspired my studio proposal, Minimal Matters.
The gyroid is an infinitely connected periodic minimal surface discovered by Schoen in 1970. It has three-fold rotational symmetry but no embedded straight lines or mirror symmetries.
The boundary of the surface patch is based on the six faces of a cube. Eight of the surface patch forms the cubic unit cell of a Gyroid.
For every patch formed by the six edges, only three of them is connected with the surrounding patches.
Note that the cube faces are not symmetry planes. There is a C3 symmetry axis along the cube diagonal from the upper right corner when repeating the cubic unit cell.
Curiously, like some other triply periodic minimal surfaces, the gyroid surface can be trigonometrically approximated by a short equation:
Using Grasshopper and the ‘Iso Surface’ component of Millipede, many TPMS can be generated by finding the result of it’s implicit equation.
Standard F(x,y,z) functions of minimal surfaces are defined to determine the shapes within a bounding box. The resulting points form a mesh that describes the geometry.
- A cube of points are constructed via a domain and fed into a function. Inputs of standard minimal surfaces are used as the equation.
- The resulting function values are plugged into Millipede’s Isosurface component.
- The bounding box sets up the restrictions for the geometry.
- Xres, Yres, Zres [Integer]: The resolution of the three dimensional grid.
- Isovalue: The ‘IsoValue’ input generates the surface in shells, with zero being the outermost shell, and moving inward.
- Merge: If true the resulting mesh will have its coinciding vertices fused and will look smoother (continuous, not faceted)
The above diagrams show Triply Periodic Minimal surfaces generated from their implicit mathematical equations. The functions are plotted with a domain of negative and positive Pi. By adjusting the domain to 0.5, the surface patch can be generated.
Many TPMS can best be understood and constructed in terms of fundamental regions (or surface patches) bounded by mirror symmetry planes. For example, the fundamental region formed in the kaleidoscopic cell of a Schwarz P surface is a quadrilateral in a tetrahedron, which 1 /48 of a cube (shown below left). Four of which create the surface patch. The right image shows a cubic unit cell, comprising eight of the surface patch.
Schoen’s batwing surface has the quadrilateral tetrahedron (1/48 of a cube) as it’s kaleidoscopic cell, with a C2 symmetry axis. As shown in the evolution diagram below, the appearance of two fundamental regions is the source of the name ‘batwing’. Twelve of the fundamental regions form the cubic unit cell; however this is still only 1/8 of the complete minimal surface lattice cell.
‘Growth From The Ger’ seeks to analyse the vernacular structure of the traditional nomad home and use parametric thinking to create a deployable structure that can grow by modular.
‘Ger’ meaning ‘home’ is a Mongolian word which describes the portable dwelling. Commonly known as a ‘yurt’, a Turkish word, the yurt offered a sustainable lifestyle for the nomadic tribes of the steppes of Central Asia. It allowed nomads to migrate seasonally, catering to their livestock, water access and in relation to the status of wars/conflicts. An ancient structure, it has developed in material and joinery, however the concept prominently remaining the same.
Growing up in London, I fell in love with the transportable home when I first visited Mongolia at the age of 17. The symmetrical framework and circulating walls create a calm and peaceful environment. In the winter it keeps the cold out and in the summer keeps the heat out. The traditional understanding of placement and ways of living within it, which seems similar to a place of worship, builds upon the concept of respect towards life and its offerings.
Understanding the beauty of the lifestyle, I also understand the struggles that come with it and with these in mind, I wanted to explore ways of solving it whilst keeping the positives of the lifestyle it offers.
Pros: Deployable, transportable, timber, vernacular, can be assembled and dissembled by one family, can vary in size/easily scaleable depending on user, low maintenance, sustainable, autonomous.
Cons: Difficult to sustain singularly, not water proof, no privacy, no separation of space, low ceiling height, can’t attach gers together, low levels of security.
Lattice Analysis and Testing
To understand the possibilities of the lattice wall, I created a 1:20 plywood model using 1mm fishing wire as the joinery. This created various circular spirals and curves. The loose fit of the wire within the holes of timber pieces allowed such curves to happen and created an expanding body. The expansion and flexible joinery allows it to cover a wider space in relation to the amount of material used.
I created the same latticework at 1:2 scale to see if the same curvature was created.
Locking the curve to create a habitable space. I did this by changing the types of joints in different parts of the structure.
To create a smoother and more beautiful curve I change the baton to a dowel and densify the structure.
To lock the lattice curve in expansion I extrude legs that meet the ground and tie together.
Manufacturing and assembly
The model made from sheet plywood cost approximately £30 and took one working day to make for one person. However, a more sustainable material and process needed to be considered as the process of making plywood contradicted this.
This model can be made by one person with the use of a wood workshop. The timber pieces were bought at 18mm x 95mm x 4200mm, 13 pieces of these were enough to make three modules, roughly costing £170 in total. Each module takes approximately 5 hours to construct, this involves the tying of the measured length twine joints. The structure is lightweight and each module is easily transportable by one person.