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Natural Systems

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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

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)

FRD

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

D_prime

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)

FRD_prime

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_double

Gyroid surface

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

gyroid

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.

 

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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.

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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.

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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.

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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
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.

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.

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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

White Royal [Pratapa deva relata] HuDie’s Microphotography

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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.

lyc

Lycaenidae

Other eggs

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

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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.

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There are thousands of species of butterfly, each with their unique egg design. 3A truncated icosahedron for a frame, the opposite of a football. Instead of panels pushed out, they are pulled in.

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Fractals are commonly occurring in nature, and can be described as a never-ending pattern on different scales. People are subconsciously familiar with fractals, so are inherently more relaxed when surrounded by them.

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3D Printing is a relatively new technology that is set to change our world. Innovations in the uses of 3D printers, combined with falling costs, means that they could be a ubiquitous tool in every home and industry. 3D printers and scanners are already used a great deal in everything from the biomedical field to art studios, and experiments are currently being done to construct entire homes. This technology is in its infancy, and it is exactly for this reason that every effort should be taken to research its potential. It is common to use 3D printers in architecture to show small working models, I would like to now use it to make a large and complex structure at full scale.

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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.

Interior

—Tia

A geometric wall of fire burning on the sands of the Black Rock Desert. This immobile blaze stands as an edifice to Burning Man’s original figurehead. A burning yet fireless wall of plywood and acetate that can be encountered, entered and sheltered in.

20150125 Day Render w rays20150127 Night Render

This sculpture stands as an abstract image of flames sent by Vulcan the Roman God of fire, an emblem of the festival’s name. Created from a series of plywood shapes and acrylic, Vulcan’s Flame is a blazing wall of light and colour. The structure is created to both imitate and juxtapose chemical fire, sharing real fires beauty but opposing its destructive tendencies. The sculpture is designed as a wall of shelter, behind which burners can be shielded from the desert’s unforgiving sun.

Born from Ancient Egyptian ‘Cairo tiling’, the sculpture is created from morphing polyhedra. The lowest section of the fire is created from cubes which gradually deform into rhombic dodecahedrons – a cubist interpretation of a flames movement. Internally every shape is painted to mimic fire’s bright hues and coloured acetate panels within the wall will project red and yellow tones onto the surrounding desert floor. At night internal spotlights will illuminate the entire structure, creating a glowing inferno of colour. These lights will flicker to create the illusion of movement.

Visually the main structure consists of three main forms;

  • The outer zone: the sparse cubic section of the sculpture, representing the hottest part of a flame, the region of complete combustion
  • The middle zone: this is the central area in which the cubic deformation begins to occur.
  • The inner zone: this is the coolest space, the most densely packed red area of the sculpture. Burners can crawl into this space – sheltered by four layers of dodecahedrons.

Rendered Plan

Physical Description:

Vulcan’s Flame is a long, low plywood structure, the installation is the geometric interpretation of a flame, a curving sculpture of deforming polyhedral that slowly transform from a cube to a rhombic dodecahedron. The sculpture is created from 55 plywood polyhedra constructed from hand cut plywood boards and secured with cable ties. Internally each shape is painted using natural, organic paints, as the shapes change their internal colour alters from yellow to red. Coloured acetate panels in the uppermost faces of each shape will mirror the shapes internal hue, these panels will allow sunlight through during the day casting beautiful coloured shadows on the desert floor. At night the sculpture will be lit internally with fluctuating spot lights, this will create the illusion of flickering movement. The acetate panels will be secured with nails.

Construction Sequence

The structure sits on a base of 23 plywood shapes, secured to the ground with rebar stakes. The sculpture is very stable as the base is the widest section, the rest of the sculpture tapers away towards the top. Each new shape rest on the 4 corners of the shapes below, bolted through the vertices and then secured with rope. The final and highest rhombic dodecahedron is stabilised with a steel column. The highest point on the entire structure is just over 11 feet above ground level and consists of 4 stacked shapes. A full sized version of one of the shapes has already been constructed and load tested confirming that it can support human weight, all of the cable ties securing the structure will be meticulously rubbed down to ensure they are not sharp.

The sculpture curves in a gentle arc – creating a central area of shelter from the wind and sun. At ground level Burners can crawl inside the structure and rest in it’s shady, tinted interior.


20150129 Single Component

Inspired by previous research of pyritohedrons, these structures are an addition to a series of other models based on polyhedral deformation. Previous models have experimented with density, altering colour and infill panels.

Previous Models

Glow_Burning Man Final_night        Glow is an interactive light installation designed for the Burning Man festival. It is a touchable, playful, glowing structure. Three long branches form an interweaving structure, which is branching out into different directions, eventually ending by forming bunches of freely floating and glowing strings. With its appearance it is reminiscent to the willow tree which is a symbol of inspired imagination. Glow would be an experiential piece of art and one can experience it according to ones imagination.Glow as an artwork doesn’t intend to make a reference to any script that one would need to know which is not evident in it itself. Glow invites to engage with perceptual issues through experiencing it through its visual and tangible qualities.The installation will have the strongest impact in the night, as this would be time when it will reveal its qualities the most. Glow – is a captured light in the object. When the light is sparked at the bottom of the structure it runs through the paths created by the matter of plastic tubes and strings until eventually it escapes and disperse into the atmosphere, from that moment it becomes no more visible to the human eyes.

Glow is an object which embodies the constant flow of light. It tells a story about the light’s unique quality to make shapes visible to us. Glow is not designed as an object to be lit externally lit,it is an objects which glows itself. Glow traps the light for a moment and then light tries to escape it is being constantly mirrored, bouncing through the plastic it makes its shape visible.

Glow_Burning Man model

Philosophical Statement:

Inti: The Incan Sun God, his face portrayed as a gold disk from which rays and flames extended. Inti is the Sun and controls all that implies: warmth, light and sunshine. During the festival of Inti Ramyi, held during the Summer Solstice, Inti is celebrated with much drinking, singing and dancing - special statues are made of wood are burned at the end of the festival. This sculpture is an extended physical manifestation of this; decadent ritualism and a spiritual experience.

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Inti incorporates 288 petals are self-assembled into 12 concentric rings, with each petal representing the hours of the day and each ring every month of the year. These are held together using mirror polished circular brackets, designed to catch the light and reflect circles of sunlight around the structure interior. Inti's focus is the sunrise; as the sun rises on the playa, Inti is designed to catch the light at this precise moment and funnel through the piece, enveloping and bathing the burners inside with it's warmth and spirit.

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