In mathematics, a Scherk surface (named after Heinrich Scherk in 1834) is an example of a minimal surface. A minimal surface is a surface that locally minimizes its area (or having a mean curvature of zero). The classical minimal surfaces of H.F. Scherk were initially an attempt to solve Gergonne’s problem, a boundary value problem in the cube.

The term ‘minimal surface’ is used because 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. However the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, minimal surface of revolution, Saddle Towers etc.).

Scherk’s minimal surface arises from the solution to a differential equation that describes a minimal monge patch (a patch that maps [u, v] to [u, v, f(u, v)]). The full surface is obtained by putting a large number the small units next to each other in a chessboard pattern. The plots were made by plotting the implicit definition of the surface.

An implicit formula for the Scherk tower is:

sin(x) · sin(z) = sin(y),

where x, y and z denote the usual coordinates of R3.

Scherk’s second surface can be written parametrically as:

x = ln((1+r²+2rcosθ)/(1+r²-2rcosθ))

y = ((1+r²-2rsinθ)/(1+r²+2rsinθ)) 

z = 2tan-1[(2r²sin(2θ))/(r-1)]      

for θ in [0,2), and r in (0,1).

Scherk described two complete embedded minimal surfaces in 1834; his first surface is a doubly periodic surface, his second surface is singly periodic. They were the third non-trivial examples of minimal surfaces (the first two were the catenoid and helicoid). The two surfaces are conjugates of each other.

Scherk’s first surface

Scherk’s first surface is asymptotic to two infinite families of parallel planes, orthogonal to each other, that meet near z = 0 in a checkerboard pattern of bridging arches. It contains an infinite number of straight vertical lines.

Scherk’s second surface

Scherk’s second surface looks globally like two orthogonal planes whose intersection consists of a sequence of tunnels in alternating directions. Its intersections with horizontal planes consists of alternating hyperbolas.

Other types are:

1. The doubly periodic Scherk surface
2. The Karcher-Scherk surface
3. The sheared (Karcher-)Scherk surface
4. The doubly periodic Scherk surface with handles
5. The Meeks-Rosenberg surfaces

In physics, elasticity (from Greek ἐλαστός “ductible”) is the ability of a body to resist a distorting influence or stress and to return to its original size and shape when the stress is removed.

This can be explained looking closer at the components which form the cytoskeleton – the cellular structure – formed of elastic and semi-elastic arrangements of proteins, which are adaptable to the cell’s requirements. Not only do they hold the structural integrity of the cell, but they also also perform functions of communication, transport combined with other “plug-in” proteins, whilst elastically responding to external forces or stimuli.

Elastic bending is used in both natural as well as man-made environments, expanding surfaces and volumes of various deployable structures.

The force responsible for elastic bending can be described in terms of the amount of deformation (strain) resulting from a given stress, a ratio known as Young’s Modulus. Hooke’s Law adds that the force responsible with restoring the initial shape of a bent material is proportional to the amount of stretch.

Using the formulas given by Young’s Modulus and Hooke’s Law, we can determine how much a certain material will bend when a force is applied to it.

Deploying structures using bending is a space and construction time saving method of producing structural elements which mimic the behaviour of natural systems.

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After the structure is deployed, equilibrium  of forces is required in order to keep the structure open and usable.

This can either be achieved by combining bending elements with meshes,

or by using the acting forces of bending elements against the reacting forces of other bending elements in both 2D and 3D.

Using these principles of acting forces vs. reacting forces, an elastic module in equilibrium is created, which can be stacked using its geometry to achieve various configurations.

Furthermore, using the same principles, a larger module is created containing groups of elements producing forces acting and reacting against each other, in order to achieve equilibrium.

However, due to the elastic properties of the newly formed module, the structure can bend and morph shape without losing integrity or equilibrium, in order to form various shapes, or respond to or external factors or users requirements, similarly to the cytoskeleton initially studied.

• Mihai Chiriac

An exploration of the simplest Hyperbolic Paraboloidic ‘saddle’ form has lead to the development of a modular system that combines the principles of the hypar (Hyperbolic Paraboloid) and elastic potential energy.

A hyperbolic paraboloid is an infinite doubly ruled surface in three dimensions with hyperbolic and parabolic cross-sections. It can be parametrized using the following equations:

Mathematical:   z = x² – y²  or  x = y z

Parametric:   x(u,v)=u   y(u,v)=v   z(u,v)=uv

The physical manifestation of the above equations can be achieved by constructing a square and forcing the surface area to minimalise by introducing cross bracing that has shorter lengths than the  square edges.

A particular square hypar defined by b = n * √2 (b=boundary, n=initial geometry or ‘cross bracing’) thus constricting the four points to the corners of a cube leads to interesting tessellations in three dimensions.

Using a simple elastic lashing system to construct a hypar module binds all intersections together whilst allowing rotational movement. The rotational movement at any given intersection is proportionally distributed to all others. This combined with the elasticity of the joints means that the module has elastic potential energy (spring-like properties) therefore an array of many modules can adopt the same elastic properties.

The system can be scaled, shaped, locked and adapted to suit programmatic requirements.

The inspiration for this research came from the Asian artist Ren Ri, who uses bees in order to generate his sculptural  work. He predefines the space for the bees to work with, and allows for a time period for the honeycombs to take shape.

There are three types of surface division that manage to fill up all the area with prime geometric space – triangular (S3), square (S4) and hexagonal (S6). Other types of surface division, either leave gaps between the prime elements, which need to be filled by secondary shapes, or are confined to irregular shapes.
Research shows that the most efficient way of dividing a surface is through a minimum number of achievable line intersections, or a maximum number of membranes. In either case, the hexagonal division fits the case. This type of organization is a second degree iteration from the triangular division. It is formed by identifying and connecting the triangular cell centroids.
Such as in the case of soap-bubble theory, these cells expand, tending to fill up all the surface area around them, and finally joining through communicating membranes.
From a structural point of view, the best integration is the triangular one, because of the way each element (beam) reacts to the variation of the adjacent elements.
By converting the elemental intersection in the hexagonal division from a single triple intersection to a triple double intersection, the structure would gain sufficient structural resistance. This can be done through two methods – translation or rotation. Translation implies moving the elements away from the initial state in order to open up a triangular gap at the existing intersection. This method results in uneven shapes. In the case of rotation, the elements are adjusted around each middle point until a sufficient structural component is created. It is through rotation that the shape is maintained to a relative hexagonal aspect, due to the unique transformation method.

Pursuing the opportunity to test the system through a 1:1 scale project, I was offered the chance to design a bar installation for a private event at the Saatchi Gallery. The project has been a success and represents a stage test for the system.

Moving further, the attempt was to implement dynamic force analysis to the design, through variation of the elemental thickness. The first test was a bridge design. The structure was anchored on 2 sides, and had a span of 5m.  

The next testing phase includes domed structures, replicating modular structures and double curved instances.

Geometry can be found on the smallest of scales, as is proven by the beautiful work of the butterfly in creating her eggs. The butterflies’ metamorphosis is a recognised story, but few know about the start of the journey. The egg from which the caterpillar emerges is in itself a magnificently beautiful object. The tiny eggs, barely visible to the naked eye, serve as home for the developing larva as well as their first meal.

White Royal [Pratapa deva relata] HuDie’s Microphotography

Clockwise: Hesperidae, Nymphalidae, Satyridae, Pieridae

Each kind of butterfly has its unique egg design, creating a myriad of beautiful variations.

These are some of the typical shapes that each family produce.

But it is the Lycaenidae family that have the most geometrical and intricate eggs.

Lycaenidae

Lycaenidae eggs from left to right: Acacia Blue [Surendra vivarna amisena], Aberrant Oakblue [Arhopala abseus], Miletus [Miletus biggsii], Malayan [Megisba malaya sikkima]. HuDie’s Microphotography

There are thousands of species of butterfly, each with their unique egg design. A truncated icosahedron for a frame, the opposite of a football. Instead of panels pushed out, they are pulled in.

Biomimetics, or biomimicry is an exciting concept that suggests that every field and industry has something to learn from the natural world. The story of evolution is full of problems that have been innovatively solved.

This research will underpin the design of a sculptural installation in which people can interact with live butterflies. With the ever-declining numbers of butterflies worldwide and in the UK, conservation and education are paramount.

The link between butterflies and humans in our ecosystem is one that is vital and should be conserved and celebrated.

I can imagine an ethereal space filled with dappled light where people can come for contemplation and perhaps their own personal metamorphosis.

—Tia

Imagine the solar system as our universe’s gigantic candyfloss maker. Candyfloss from Venus is recreating a mystic phenomenon of the solar system in a human scale and in the most fun approach.

The sacred geometry that exists between the cycles of the planets has been a source of fascination for humans for centuries. The Candyfloss from Venus is an art installation designed for the Burning Man Festival, inspired by the fivefold rosette pattern formed by the orbit of Venus around the Earth, when viewed from the geocentric position, also known as ‘the rose of Venus’. The translation of the dance of Venus around the Earth in more than one geometric plane creates a grid-shell rose pattern.

The fivefold rosette and the grid-shell rose pattern are informing the geometry of the Candyfloss from Venus. This pavilion is created as a result of digital and physical testings of the orbits of circles or spheres around a fixed centre, reminiscent of the function of a spirograph toy, which creates hypotrochoid and epitrochoid curves and by expanding the possibilities of the spirograph in three-dimentional space.

The main supporting structure of this pavilion is created by a wooden structure, of three concentric rose like, grid-shell frames. The laser cut plywood pieces of the frames are connected, interlocked and bolted together with metal brackets, forming a secure and climbable construction. The beautiful ‘rose of Venus’ is recreated within the grid-shell structure, by the weaving of white EL wire inside translucent, coloured and flexible PVC tubes, supported by the wooden skeleton. In the centre of this light hurricane, a candyfloss maker is continuing the orbital loops of matter, gifting to burners a galaxy to taste.

This art installation carefully considers the principles of Burning Man, by providing an interactive, playful, climbable pavilion that reveals the beauty of Venus’ orbit spiralling around the Earth, directed by the divine proportion of the golden ratio. The internal spaces created by the wooden framework and the loops of the ‘rose of Venus’ can be explored and utilised. The dense spiralling geometry of the coloured weaved PVC tubes generate beautiful shadows during the day. At night-time, the EL wires light up the rosettes, producing a visually mesmerizing effect, creating an illusion of depth and density. The whole structure is tactically tilted in order to permit burners to explore every dimension of these orbital flowers of light. Candyfloss from Venus is providing an interactive space for cherishing the beauty of the Playa’s starry night sky, while enjoying the colourful sweetness of some candy floss. The geometric secrets of the sky become a playground for every burner to discover. It is an art piece designed to reminisce everyone’s childhood and aspires to leave a sweet memory of the night sky to every burner.

Spiral Pleat Origami

The first part of the research focused on the pattern, folds and strips of a spiral pleat origami. A strip of rectangles or trapeziums is divided by equally spaced mountain folds,then sloping parallel valley folds are placed diagonally between the mountains to connect them. The complete pattern of folds can be drawn in one continuos zigzag line. Collapsed flat, it will resemble a circular rosette. It can also be made from shapes other than a rectangular strip, such as a very long triangle or a very long rhombus.The Helicoid

Discovered by Meusnieur in 1776. The helicoid is the only known ruled minimal surface,nother than a plane. It also has the interesting property that, while its topology is finite, its Gaussian curvature is infinite (when v = 0).

Definition

A helicoid has the following parametric equation:

x (u, v) a v Cos (u)
y (u, v) = a v Sin (u)
z (u, v) bu

Geometrically, the helicoid is defined by simultaneously rotating and translating a line at constant speed about an axis to which it is perpendicular. A parametrized helicoid. The function (u,v)=(ucosv,usinv,v) parametrizes a helicoid when (u,v) E D, where D is the rectangle [0,1]Å~[0,2]. The region D, shown as the rectangle in the first panel. The region D is divided into small rectangles which are mapped to “curvy rectangles” on the surface. You can click any of the borders of the small rectangles to highlight a curve where one of the variables is constant. Vertical lines in the rectangle D represent curves where u is constant; these correspond to curves that spiral up the helicoid. Horizontal lines in the rectangle D represent curves where v is constant; these correspond to lines that cut across the helicoid.The research into helicoid surfaces led to experiments with Curved Crease Origami. Experiments were done by cutting different shaped components out of the folded helicoid and connecting them together.Physical model

The structure is made out of six identical modules creating a volume. Each module consist of 14 arch-shaped strips that are laser cut and fixed together by attaching each strip on to a piece of fabric. This technique allows the surface to be folded in mountain and valley folds, where by introducing concentric folds to a flat surface allows the surface to bend freely and flexibly. Each module is placed in a circular pattern and laced together with string at each ends. Once attached, every second connection point is tied together towards the centre.

My initial studies stemmed from researching into Stellation. This, in simple terms, is the process of extending  polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in n dimensions, to form a new figure. Through researching the application of this process, I came across the sculptures created by George Hart, as he has experimented with stellated geometries to which are subdivided to create mathematical interweaving structures.

My Research into the method and calculations of George Hart’s Mathematical Sculpture’s focused on the sculpture ‘Frabjous’. Through rigorous testing and model making I have understood the rules behind the complex form. This is based on the form of a stellated icosahedron, whose shape is contained within a dodecahedron.

Lines are drawn from one point, to a point mirrored at one edge of the face of the dodecahedron form – as shown in the diagram. This creates intersecting lines at each face as you can see from the diagrams below. Each dividing line has two intersection points, with symmetry at the center of the line. The sculpture aims to avoid the intersections of these lines by introducing a sine curve with the domain 0 to 2*pi. As you can see, each component is exactly the same – for this model, 30 components are used.

`To simplify the construction of the sculpture, I extracted a build-able section which uses ten components in total. Two of these sections are then weaved together and joined up by a further ten single components to form the entire sculpture.

Following this research, I extracted the concept of avoiding the intersection and subdivided a cube with lines from each corner of the cube. These lines were then weaved around eachother using a sine curve with a domain of 0 to pi. I then mirrored the curves and rotated them to create an intertwining form.

Another test was created with the same process, however subdividing a cube using the midpoint of each face. – This creates an octahedral geometry.

Using this interweaving geometry, I have created different three dimensional arrays to create a spatial form. The concept of avoiding intersections naturally cause a structure to fail. To form a structurally efficient version of this geometry, I introduced the idea of a reciprocal structure, and allowed the beams to self support by resting on eachother. This did not create a structure strong enough to stand on, however through adding a cube whose dimensions are equal to the width of the beams, the structure became very strong.

Testing the component at a small scale required the design of a joint which allowed me to assemble these components together through interlocking elements. Each beam element slots into the joint; When two joints and two beams are connected together the curves naturally stay in place due to the angle cut into the joint. Three of these connected elements together form the component.

As mentioned previously, avoiding intersections create inefficient structures – For this small scale experimentation, the concept of Tensegrity was implemented. Tensegrity is a structural principle based on using isolated compression components within a net of continuous tension, allowing the compression members to not need to touch each other. This model was constructed using 1.5mm plywood which has been laser cut; the modularity of the system ensures minimal material wastage.

The three dimensional array of this geometry creates many interesting shapes and patterns when viewed from different angles – this is visible in the following video:

Thousand Line Construction :

Hamish Macpherson

A spatial exploration into the interplay of materials, construction techniques, and delicate and precise design.

Inspired by Hanakago; the craft of Japanese Bamboo basketry, to celebrate the western discovery of tea and its associated culture during the renaissance.

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. 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
Weight.

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.

-Tia

INSTALLATION SUBMISSION TO BURNING MAN 2016 – ‘Entwine’

Entwine is a timber frame structure which has been developed through rigorous physical and digital testing to ensure a safe climbing frame for all to enjoy. When exploring Entwine, the vast expanse of the playa is framed through beautiful intertwining curved plywood beams. Burners can view the event from glorious vantage points nestled amidst multiple communal spaces that encourage interaction and play.

The structure predominantly consists of strips of curved plywood which have been connected together using pioneering construction techniques, specifically the utilisation of conflicting forces, similar to those apparent in ‘Tensegrital’ design. Drawing inspiration from Leonardo Da Vinci and his various experimentations with physical form, ‘Entwine’ is a marvel of geometry. The piece is formed from an arrangement of 19 octahedral components, each consisting of six beams, which are paired and positioned upon one of three axis. These three elements represent the unity of man, nature and the universe that surrounds us.

Each modular component is tessellated to form an octahedral space frame structure. The rigidity resulting from this tessellation is in direct contrast to the curving structural beams which exude an organic aesthetic. As Burners view Entwine from different aspects, a remarkable array of different patterns and forms are revealed, many bearing resemblance to sacred geometry, specifically the Flower of Life, which was a significant study within Leonardo Da Vinci’s work.

Entwine is unorthodox in its composition, and this is a contributing factor to what makes it so unique: Each module is constructed through tensioning layers of ¼ inch thick plywood, which are then mechanically fixed together when a desired radius has been reached. By laminating the plywood in this manner, each component retains its curvature but remains in compression. These conflicting forces are integral to the design of Entwine: Each octahedral module is constructed from these compressed plywood elements, and are held together with tensioning ropes creating a structure of isolated components in compression within a net of continuous tension.The form of the structure is based on the octahedron, which is a Platonic solid composed of eight equilateral triangles; four of which meet at each vertex. One of the eight triangles acts as a base for the structure. This results in one edge creating a small cantilever, whilst the counter edge can be anchored to the ground. As previously studied by Buckminster Fuller, the geometry of an octahedron is particularly good at forming space frames with a strong cantilevers.

The participatory aspect of the installation voids the role of the ‘spectator’ and creates more active engagement. In many of Leonardo Da Vinci’s paintings, his subjects are framed by surreal, dreamlike landscapes. This is reflected within Entwine: As Burners become part of the installation, they are framed by the awe inspiring backdrop of Black Rock Desert: In many ways Entwine becomes the artist, the playa the canvas, and Burners the subjects.

“the artist is not a special sort of person, but every person is a special sort of artist.”

This is not only true in the sense of physical involvement but during the construction the ‘spectator’ becomes involved in making strategic decisions in the realisation of the work of art. The development, design and construction of the project embodies the principles of self-reliance and self-expression, whilst a proposal that is safe, interactive and beautiful will be gifted to the community at Burning Man.

Entwine’s curving form will be illuminated using LED spot lights to enhance the organic patterning existent within the structure. This allows the full form of the structure to be fully visible.

Project Summary

S(l)OSH (standing for ‘ slosh= to move through mud’) is a new Pop-Up Spa situated in Hackney Road, in East London. It is designed as an interactive relaxation area to be experienced through exploring and reflecting within a cavernous space, surrounded by mysterious voids, while soaking in a healing mud tub. S(l)OSH represents a new concept of fun mud house, that tells a different side of the wellness story.

The Spa aims to promote the cleaning and health rituals around the world and invite the users to become aware of the areas in need of healthy kickstarts. The new concept started from the idea that spas and relaxation areas are generally luxurious places to relax and heal and sometimes they are too expensive for the general citizen. S(l)OSH wants to bring healthy hedonism to the city while boosting urban areas that need a little support, while making the cleaning and health rituals accessible and fun to everyone.

Philosophy

Bathhouses, spas and saunas have long been part of cleaning and health rituals around the world. Mud baths have existed for thousands of years, and can be found now in high-end spas in many countries of the world. Mud wraps are spa treatments where the skin is covered in mud for a shorter or longer period. The mud causes sweating, and proponents claim that mud baths can slim and tone the body, hydrate or firm the skin, or relax and soothe the muscles. It is alleged that some mud baths are able to relieve tired and aching joints, ease inflammation, or help to “flush out toxins” through sweating.Opportunity

The design is composed of layers of horizontal wooden planks that follow the mathematical formula of a Scherk’s Minimal Surface geometry of a continuous surface, placed in and around a shipping container. The Spa has been designed after several form manipulation and shape iterations of the initial system, followed by massing of standard bath tubs in a tight space. The proposal stands somewhere between the realms of both sculpture and architecture – a spatial construct where movement through will encourage intimate social interaction, and a full emerge into the relaxation experience.

Physical Description

Visually, the main part of the Spa is composed of three main areas: the reception, the mud baths and the outdoor pools. The spas includes hot mud tubes, cold water plunges, a changing area, shower and relaxation platforms. The structure will be built from layers of horizontal CNC cut wooden planks stacked on top of each other and fixed together. Internally, the bathtubes will have a smooth concrete walls to hold the liquid and make the stay more pleasant for the sitting. Despite being designed to fit in one or two containers, the spa can expand even outdoors and other spaces.

As the Maker World develops, we want to have a greater impact on our environment, the spaces we live in. Auxetic Assemble gives you a chance to build your own furniture and have a input in your product. Auxetic Assemble gives you the chance to buy either the cut parts for the product or the instructions and CNC cutting files to then source your own material, cut your own product and assemble your work. The future of this adaptable design system allows the product to be fitted and designed to your require space.

The Pavilion

I have set up a Kickstarter Campaign in order to Fund a Pavilion in Hackney. The Pavilion is the next step in the journey to developing this system to its future potential. The construction process used for the plywood shelves will be developed at the larger scale to develop a pavilion to display the process of constructing the shelves and to exhibit the product. The Pavilion walls will become the seating, shelving and tables for the space. This is a chance to explore the system at a larger scale in order for further development for future of housing, an adaptable system that can be applied to a unique space and engages everyone as its workforce to build it.

Kickstarter Page

The Products

The design system has been developed into shelving product to sell as rewards, both in cardboard and plywood.

Follow the links to the Kickstarter Page to help this project be realized, Click on any image for the Kickstarter page link. Thank you.

As part of my research to inform my final thesis project on the London Housing Crisis, I have created a short multiple choice survey that would benefit greatly from the input of members of the WeWantToLearn community who have lived in London at any point over the past six years. The survey only takes a few minutes to complete and will directly influence the design progression of my project in the coming weeks. Please spare a few moments to participate, and/or share with friends and relatives who may be able to contribute also.

You can find the survey at the following link: Here

All survey responses are anonymous.

Thank you in advance.

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

(1+h(v)^2)*h(uu)-2*h(u)*h(v)*h(uv)+(1+h(u)^2)*h(vv)=0

(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

Neovius surface

Gyroid surface

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.

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

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.

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.

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

KICKSTARTER VIDEO & CAMPAIGN LINK: http://kck.st/1qGLHSw

PURSUIT is an interactive art installation that celebrates humanity’s ongoing quest for Peace, Freedom and Joy – in Life, Love and Art. The design aims to create an interactive and unique sculptural playground for visitors of the 2016 Burning Man Festival, which takes place from August 28th to September 5th in Black Rock Desert, Nevada.

THE PROJECT

PURSUIT emerges from the playa in tiers of intertwined timber elements that ascend seamlessly in unison to form a series of congregation and celebratory spaces. The final design is the result of a year-long study into the sensuous geometry generated from a mathematical theory known as Pursuit Curvature. This theory was explored as I wanted to utilise something that could fully embody the notion of people coming together from different places and striving towards a common goal. With Pursuit Curvature, each point starts at a unique position of a polygon, and moves incrementally towards the nearest adjacent point until they all converge in the centre. The path travelled is directly influenced by the points around it, so the final curves represent the effects all of the points have on one another as a group.

Central Space

Burners can rest inside the ornate central space of Pursuit, which frames the ongoings of the playa and provides burners with a place of respite from the open sun. The six inhabitable pillars connect the playa directly to the platforms that lie atop Pursuit. Here, a glorious vantage point in which to congregate and take in the festival is gifted to Burners. During the day the interiors of the pillars are concealed from the elements, and their curved form helps to guides burners ascent to the open air. Here they can bathe in the wondrous light of either sunrise or sunset, a truly magical playa experience indeed.

At night time each pillar’s interior is powerfully lit to envelop the burners in light, so they can experience a sense of weightlessness and freedom. The soft glow emanating from each of the pillars’ cores invites burners to commune atop Pursuit to celebrate the radiant beauty of the night sky.

OUR PURSUIT

Interior of each Pillar

“As I look back on my life, I realise that every time I thought I was being rejected from something good, I was actually being re-directed to something better.” – Steve Maroboli. 

Pursuit is a gift to the Burning Man community. Every year, we apply for funding from the Black Rock City LLC (Burning Man) to help fund our projects. Unfortunately this year, nobody received funding towards their project. Despite initial disappointment, I realised that this helped elevate the project’s intent and concept to a new level than originally planned. By crowdfunding the entirety of the project, we can manifest the collective Pursuit of people from all over the world to see this project built. This is not only tremendously exciting, but also a very humbling prospect, in that we have a passion to give this gift to the playa, but we need your help to give that gift. It is through this collective pursuit that we can embody the spirit of the festival and the project in a built architectural form.

REWARDS

To show our gratitude for any of your generous pledges, I have created some truly beautiful and unique rewards for all levels of contribution – Each inspired by the projects form and concept, that are all exclusive to this campaign. Please do go and have a look for yourself at them and support the campaign. If you can’t spare a donation at this time, then please share the campaign to as many people as you can – so that together, we can make the project a reality.

Thank you

Joshua

KICKSTARTER LINK: http://kck.st/1qGLHSw

The aim is to generate an architectural response through a playful loop between the digital and the physical. Digital tools such as Rhino and Grasshopper are used  in order to carry out analysis and generate buildable three-dimensional forms. Interplay between physical fabrication and digital experiments enable to become an inventor of a system. Here is mine.

TriNect is a flexible system of triangular elements with slots at their vertices. Elements interlock with one another creating different space filling polyhedra. The system can be applied in various scales and adapted for different needs.

TriNect

This project is a physical exploration of anamorphosis in three dimensions centred around the theme of duality. It aims to combine two widely recognisable figures into a pavilion that will attract burners, provoke debate, and catalyse interaction.

The theme of this project arose from the realisation that even the most widely recognisable symbols contain multiple layers of meaning and mystery.  Social, historical and sometimes even spiritual contexts give a symbol its perceived meaning. For example, while the Christian cross is a symbol of hope it is literally a scaled representation of an ancient torture device – an icon synonymous with good carries with it a darker elucidation. This interpretation led to the emergence of duality as a topic and a title. There are many symbols which have multiple meanings and nuances to those who interpret them.

I began by looking at the Ankh, the Egyptian symbol for life/fertility. The Loop of the Ankh represents the feminine discipline or the womb, while the elongated section represent the masculine discipline or the penis. These two sacred units then come together and form life. This is a perfect representation of man and woman in perfect union. I then was led to study the symbol for mercury, which is used in botany to indicate a flower with both male and female reproductive organs.

This duality of meaning in symbols led me to the desire to study how I could physically combine other symbols and forms to create one form. Anamorphosis, from the Greek anamorphōsis meaning ‘transformation,’ from ana- ‘back, again’ + morphosis ‘a shaping’, became an interesting opportunity to do just this.

I want to explore this theme using the iconic faces of Donald Trump and Kim Kardashian as instigators. From a random vantage point or even from up close, the subject matter of the piece is evidently unclear, the image changes until the viewer arrives at a specific pre-set location, only then does the likeness reveal itself. This echoes our warped perception of figures in limelight; anything the media choose to present to the world is an engineered production and if taken out of its context it becomes incomprehensible. My aim is to stir ambivalence among the burners, for them to engage in discussion with one another about these two incredibly famous personalities and what they seemingly represent.

As a physical entity, the sculpture is purposefully made durable enough to be able to endure the brunt of any elicited reactions. Its exposed surfaces are smooth, an open invitation to graffiti, carve or deface in any manner possible. It is large enough to climb and to gather within as a group – it only takes a spontaneous suggestion from a creative festival goer to give the sculpture another unforeseen use.

The aim of my proposed sculpture is to provoke an exchange of opinions and interactions between burners. It depicts two iconic and highly controversial public figures who personify two tremendously important issues that we as a society face today; political and social change.

As festival goers approach the installation, and the two widely recognisable faces reveal themselves, comments about the likenesses will spiral inevitably highlighting or at least touching upon the shift that these two personalities represent.

The sculpture’s physical form comprises of several spatial elements that lend themselves to fostering the kind of debates that I wished to promote. The hollow centre creates an enclosure, to enable hosting or housing for a meeting, it gives its participants a sense of protection; this is an open forum, please take part. The raised base on the peripheries can act as stages or podia. The expansive smooth external surfaces can act as billboards or banners, the skin of the sculpture will bear the physical outcome of the issues discussed here.

Whether people get photographed with it, or whether they deface, damage or even burn it to the ground, I will have succeeded if among any of the interactions the agenda was heard and a heartfelt reaction was made.

The sculpture will be made of 8mm CNC routed plywood sheets fixed to a heavy plywood formwork. Standing at 6m tall, one side will represent a 25:1 scale stencilled portrait of president-elect Donald Trump, the other side; the likeness of reality television personality and socialite Kim Kardashian. Much like the oblique anamorphosis incorporated in Holbien’s The Ambassadors, the sculpture’s subject matters will reveal themselves only from some 60m away, but from close up, the installation will seem like a mass of abstract wooden extrusions, something suggestive of an adult-sized climbing frame. Fluorescent LEDs recessed into junctions of the outer plywood skin layer will illuminate the piece at night.

The pavilion achieves the incredible feat of allowing the viewer to have a personal and intimate connection with it whilst also allowing for reflection. The two images are intended to bring moments of delight to viewers to allow for interaction even from a distance.

Combined with its symbolic and evocative power, it should indeed conjure a deeper sense of place and self, and bring a subtlety and complexity to what might have been just another pavilion.

Truth is a personal conquest which one attains through a mystery.

The Burning Man festival has evolved from a simple gathering of people on a beach in San Francisco into a spectacular spectacle. Burners travel from all over the world to meet in the Black Rock desert of Nevada where they form a city of temporary structures and burn a huge towering figure of a man. The theme of the 2017 burning man event will be Radical Ritual, an attempt to reinvent ritual in our post-post-modern world disregarding assertions of belief and concentrating instead on the immediate experience of play. Fractured cosmos seeks to provide this year’s burners with an edible artwork experience in the playa, a crystalline structure made from hard boiled sugar candy.

Fractured Cosmos draws the inspiration for its form from the Shri Yantra Mandala, a mystical diagram used in the Shri Vidya school of Hindu Tantra. The diagram, nine Isosceles triangles interlaced to form 43 smaller triangles, is said to be symbolic of the entire cosmos. The geometry of this symbol, usually depicted as a flat, two-dimensional construct, has been inclined, distorted and fractalized out of its two-dimensional plane to create a series of inclined planes for Burners to inhabit and play within.

Those cultures which meditate using the Shri Yantra symbol believe life exists between two planes, that of Samsara, this earth plane, and Nirvana, a perfect heavenly plane. One transcends from Samsara to Nirvana when they have gained enough genuine insight into impermanence and non-self reality. This notion has been reinterpreted as a series of transparent colourful planes on the playa, casting colourful light onto the ground beneath our feet. Each of the planes will be made from coloured rock candy, offering burners the opportunity to perceive an altered sweeter perspective of the world, as well as a delicious treat if they’re brave enough to lick it.

The use of hard boiled candy as the structural material for the pavilion means the structure will be made by mixing sugar, liquid glucose and Creme de Tartar with water. This concoction will be boiled to a temperature of 145 degrees celsius before pouring into formwork to cure. Waterproof LED lights will be added once the liquid has cooled to a sufficient temperature, and these will be used to light the structure at night.

The plinth of Fractured Cosmos must be strong enough to support the weight of the candy structure above plus the added weight of any burners, tie the structure down, capture any waste which falls off the structure as it is consumed and house the power source for the night time lighting. This has been designed using the same method of recursion as the candy structure to create a plinth using the mandala’s cosmic gate element as its inspiration. the final form generates a climbable podium for burners to ascend before inspecting and consuming.