something caught in between dimensions – on its way to becoming more.
The Wishing Well is the physical manifestation, a snap-shot, of a creature caught in between dimensions – frozen in time. It is a digital entity that has been extracted from its home in the fractured planes of the mathematical realm; a differentially grown curve in bloom, organically filling space in the material world.
The piece will be built from the bottom-up. Starting with the profile of a differentially grown curve (a squiggly line), an initial layer will be set in pieces of 2 x 4 inch wooden studs (38 x 89 millimeter profile) laid flat, and anchored to the ground. Each subsequent layer will be built upon and fixed to the last, where each new layer is a slightly smoother version than the last. 210 layers will be used to reach a height of 26 feet (8 meters). The horizontal spaces in between each of the pieces will automatically generate hand and foot holes, making the structure easily climbable. The footprint of the build will be bound to a space 32 x 32 feet.
The design may utilize two layers, inner and out, that meet at the top to increase the structural integrity for the whole build. It will be lit from within, either from the ground with spotlights or with LED strip lights following patterns along the walls.
At the Wishing Well, visitors embark on a small journey, exploring the uniquely complex geometry of the structure before them. As they approach the foot of the well, it will stand towering above them, undulating organically across the landscape. The nature of the structure’s curves beckons visitors to explore the piece’s every nook and cranny. Moreover, its stature grants a certain degree of shelter to any traveller seeking refuge from the Playa’s extreme weather conditions. The well’s shape and scale allows natural, and artificial, light to interact in curious ways with the structure throughout the day and night. The horizontal gaps between every ‘brick’ in the wall allows light to filter through each layer, which in turn casts intriguing shadows across the desert. This perforation also allows Burners to easily, and relatively safely, scale the face of the build. Visitors will have the opportunity to grant a wish by writing it down on a tag and fixing it to the well’s interior.
If you had one magical (paradox free) wish, to do anything you like, what would it be?
Anything can be wished for at the Wishing Well, but a wish will not come true if it is deemed too greedy. Visitors must write their wish down on a tag and fix it to the inside of the well. They must choose wisely, as they are only allowed one. Additionally, they may choose to leave a single, precious, offering. However, if the offering does not burn, it will not be accepted. Visitors will also find that they must tread lightly on other people’s wishes and offerings.
The color of the tag and offering are important as they are associated with different meanings:
► PINK – love
► RED – happiness, joy, success, good luck, passion, vitality, celebration
► YELLOW – nourishment, warmth, clarity, empathy, being free from worldly cares
► GREEN – growth, balance, healing, self-assurance, benevolence, patience
► BLUE – conservation, healing, relaxation, exploration, trust, calmness
► PURPLE – spiritual awareness, physical and mental healing
► BLACK – profoundness, stability, knowledge, trust, adaptability, spontaneity,
► WHITE – mourning, righteousness, purity, confidence, intuition, spirits, courage
The Wishing Well is a physical manifestation of the wishes it holds. They are something caught in between – on their way to becoming more. I wish for guests to reflect on where they’ve been, where they are, where they are going, and where they wish to go.
As our studio dipped into the complexity of fractals, it became easy to get lost. Suddenly, these geometries were everywhere. Trees, clouds, coastlines, our own bodies – all examples of fractals. Systems, that are made up of smaller self-simular parts until they reach infinity. Systems, that travel between dimension (more about it here https://wewanttolearn.wordpress.com/2017/10/18/shapes-fractals-time-the-dimensions-they-belong-to/). Wanting to understand these geometries better, I found a Fractal plugin for Grasshopper by albertovalis on Food4Rhino. Playing around with various parameters and GH components gave me interesting shapes, but which seemed far away from an architectural object. I then decided to give it a try and allow the program to randomly select elements by assigning different true/false patterns. Finally, an error happened and it was beautiful.
Error 101 is a visual representation of relationships between machines and humans. It illustrates what we can learn from each other (what does this mean?). The geometry was generated through a combination of fractal mathematics properties, parametric design tools and finally a computer error, which were all guided by human decisions.
The artwork will be made out of ‘chaotically’ arranged ribbons that, together, form a tetrahedron. From far, the geometry will look well defined – a triangle or pyramid. As you get closer you notice the complexity. When you experience is physically, you find logic in the chaos. Inside the tetrahedron is a void.
Error 101 will be constructed using bent cross-laminated timber modules that are interlocked together with flitch plates. Their arrangement will allow the object to be self-supporting. The whole piece is 18’x18’x18’ (5.5 metres). Timber strips create the outer shell and are 25 inches wide (635 mm). Their surface will be treated to achieve a smooth finish to protect both the piece and visitors. Light strips will be fixed to edges of timber curves and turned on at night. Assembly will be completed on site.
Interactivity and Mission
Error 101 is left open to interpretation – everyone can have their personal take on the piece. Visual and emotional perception of Error 101 may change depending on how close you get to it. It may encourage visitors to think of it, as something that travels between dimensions, which is a liberating allegory of how one thing can become another and how the whole is just a collection of its parts. Just like water can be liquid, ice or vapor, Error 101 can be a triangle, pyramid or chaotic curves.
The structure is climbable and each of many unique curves can be treated as a nest. Occupying empty spaces on different levels may make burners feel like a part of the ‘chaos’, that has a space for everyone. Different curvature can suggest different positioning of a body that may influence visual as well as physical experience. Entering the structure’s core shifts the visitor’s focus away from the idea of a pyramid and allows them to focus on what’s within. Such study erases preconceptions and allows new ideas to be born. This notion is also enhanced with the use of lights at night.
Error 101 is a product of human ability to perceive beauty, and computer’s power to process complex mathematics. Its development started with an attempt to try to understand fractal geometries that only became possible to study in the recent years due to the development of computer processing power. A continuous human-computer-human processes that involved both logic and error allowed for the piece to be born.
Error 101 is a common error in Internet browsing. A simple solution to it is clearing browsing history and cache. It may also appear in other spheres of digital world when software or a device is out of date. Burning Man participants are invited to clear their mind, update the ‘software’ and reset their system to become a new advanced version of themselves. The final steps of error 101 creation involved chance and error. The chaos led to something beautiful. We, as humans, can learn from this – learn to let go, to acknowledge and even appreciate mistakes, complexity of the world and our own selves. The geometry of an artwork is essentially a continuous strip that can be unrolled into one flat curve on the ground. This idea of continuity and interdependence is an allegory of a world’s structure.
The closer you get to Error 101, the more you can learn from it. A 2D triangle turns into 3D pyramid and then into a collection of overlaying shapes that are not truly from our dimension. With the speed of the modern world we tend to simplify things, which leads to inability to see details. Visitors are invited to come take time to study and appreciate the complexity of the Error, and to realize the beauty of a whole. From this, they may find that, in fact, all processes in our lives have a similar structure. Chaos generates order and order generates chaos.
Language has a strong symbolic meaning to the mankind. It is not just a sound but with meanings which then allows to self-express, communicate and inspire. The mechanism of the sound system of languages is translated into visually represented geometries using Chladni’s Law.
3-Dimensional computer generated Chladni Patterns
When the frequencies increase, the pattern gets more complicated.
Tower of Babel – The origin of different languages
“Come, let’s make bricks and bake them thoroughly. […] Come let us build ourselves a city, with a tower that reaches to the heavens, so that we may make a name for ourselves and not be scattered over the face of the whole earth.” (Genesis 11:3~4)
(The Tower of Babel by Pieter Bruegel)
It is the story from the Bible but also architectural structure found in Mesopotamia Civilisation – called Ziggurat. It was made of asphalt and baked bricks with total dimensions of 90m x 90m, 90m high. This is equivalent 30th floor building.
The united humanity spoke a single language and agreed to build a city and a tower that is ‘tall enough to reach heaven’. God found such behaviour as rude and disrespectful. He confounded man’s speech so that they could no longer understand each other.
Concept Development through systematic studies of Ziggurat
The frequency and nodes of the word is analysed and recreated as two geometrical forms. They are proportioned according to the Ziggurat Algorithm ratio and timber pieces are stacked up vertically reaching the highest deck at 8m above. The structure encourages to climb complex geometry.
While reaching the top, less intense the space becomes. The LEDs are placed underneath the timber pieces which are concentrated on the top of the tower and scattered following the central void of the structure. Lights illuminate with the voice reactive sensor placed at the top of the tower.
Human always wanted to reach higher points either physical or spiritual. The height of architecture symbolised one’s power and control. This can be observed from the tower of Babel and continues in architectural history. Such expression of the desire of heights lead to competition of building higher structure.
High rise buildings were often found in religious architecture where they had few typical characteristics. First, it was the only tower to observe your land and the only tower which can be seen from everywhere in town. It has a visual meaning that the land within the perspective is the land within control. Second, religious architecture often had music instruments embedded within. This represented the control of the land where music reaches. And finally, high-rise tower was a representation of the centre of universe and sacred space in religious term. The tower, architecture of height is a spatial symbol of man’s deep desires.
The ritual is all about finding the true desire of your own. This begins with constructing the tower where the ritual follows the biblical story of Babel. Climbing up 8m high construct is a challenge then the climbers are rewarded with the beautiful panoramic view of Black Rock city. The climbers will also interact with the installation by continuously stacking up the Babels with anything they can find. Eventually it will deform from the original shape. Then the Babels will be the collective symbol of the Burners’ pure desire.
Whatever your creed your reliance on the sun is unquestionable.
We have worshipped it as a God.
Spent lifetimes studying it through science.
Yet human hands will never touch its surface.
Celestial Field brings our sun to the Playa for us to dance in its glory.
Triggering our own solar flare.
From the dawn of time the sun has been a constant in human life. It has been central to the beliefs of nearly every civilisation throughout history. What was once an astrological wonder sustaining life; dictating when to plant and harvest our crops; evolved into a god and deity, woven into the stories and teachings of nearly every culture, from the Egyptians to the Ancient Greeks and even Christianity.
The oldest man-made structures in the world have resounding astrological connections to both the sun and constellations, covered in carvings they unquestionably align to major astrological events.
Newgrange in Boyne Valley, Ireland, thought to be built in 3500BC, has a tomb in which sits a stone basin lit by a single beam of light at sunrise on the winter solstice.
The Egyptians, Greeks, and Christians have all referenced the sun within their religion and beliefs.
The Egyptians in 3000BC had Ra, the Greeks in 400BC believed Helios to be God of the sun, and Christians have often depicted Jesus in front of what is thought by many to be the solar cross.
In the past the sun has been depicted as a 2Dimensional disk of light travelling across the sky before dying only to be reborn at sunrise.
The Ancient Greeks believed Helios to be the personification of the sun. A man with a many rayed crown of light, pulling the sun across the open sky with a horse drawn chariot. The Helios named after the Greek god has been used and adapted through the ages, with one of the most recognisable iterations being the logo of global corporation BP which symbolises “a number of things – not least the greatest source of energy … the sun itself..” – bp.com
This once celestial being has now become a tangible thing. Through advances in our technological and scientific capability we have gained an understanding of the suns chemical make-up, uncovering many of its secrets from sun spots to solar flares. Although we have developed an increased understanding of the forces driving the sun, it is still no more accessible to us mere humans than on the first day on earth remaining an impenetrable sphere in the sky only to viewed from a far.
Digital animation of lighting tests
The suns surface has taught us much. Galileo’s sun spot diagrams unknowingly demonstrated the unique fluidity of the suns chromosphere. Further study of these sun spots and magnificent solar flares proved that the surface of the sun is covered in billions of interlaced magnetic fields all interacting together to form the whole. When these fields cross swirling plasma burst in an instance out into the corona bringing with it immense light displays that can be seen on earth as the aurora.
In an age where endless streams of newfound knowledge are accessible with the touch of a finger – it is easy to lose our sense of innocent amazement and unquestioning awe. We have a constant need for explanation of why and how phenomena exist, no longer blindly excepting their beauty and revelling in it.
The indescribable beauty of these gigantic magnetic fields can often be lost and forgotten in the mundane when scaled down to earthly objects. Viewing them at a micro scale allows us to connect with their other-worldly nature.
Science has taught us how a magnet attracts and repels enabling use in industry, medicine and everyday life. And as our knowledge expands, we move from child to adulthood and our desire to play diminishes – burdened by explanations and reasoning; we are no longer in awe of our ability to make metal move without laying hands on it. It has become the norm and the expected, it is no longer ‘magic’.
Life should be fun and full of mesmerising moments. Our increased knowledge should enable and enhance our experience of ‘magic’ not hinder it.
Celestial Fields captures the unexplainable wonderment the sun once held and makes it accessible through modern mediums, combining two worlds; science and enchantment, imbedding them on the Playa at Black Rock City, Nevada, for people to explore and lose themselves in.
Thousands of swaying rods made of tubes of one-way mirror form an undulating field, rising high above your head, and falling like the plasma pulled in all directions by the phenomenal magnetic forces found on our sun.
By day a field of mirrors reflect and intensify the suns natural beauty and power. Creating a maze of ever changing light to explore, push through and play within. At sunset everything transforms. The field morphs, bursting into a sea of glowing beams reacting to movement and mimicking the fluid, almost pulsing nature of the suns corona.
Like the chromosphere, magnetic fields have informed density and pattern, creating patches of pure brightness and areas as dark as sunspots. With each rod built on a spring loaded base it can be pushed a manipulated, enabling you to forge your own path through the densest areas of Celestial Field, parting rods like magnets repelling polarised iron.
Movement through the sprung rods creates interest not only for the participants but also onlookers. During daylight hours people weaving in and out can be seen across the playa through the constant glinting of the sun on the reflective rods. An ever changing shimmer, like sunlight dancing across water in the distance, drawing people in from all directions out of wonder and intrigue.
Once the sun has set the lights come on, and the show only gets better. The rods now glow and pulse, changing colour, transforming the world around them – each equipped with a sensor so as to react to movement as people push past; creating tracks of swirling light shifting like the turbulent surface of the sun. Areas of intense and overwhelming light can occur when people team together to trigger a cluster of rods forming a concentration of light evocative of a solar flare.
The sun is not solely about light, with it comes inevitable darkness. Shadows too have been used throughout time as a symbol in opposition to that of the sun; and in this instance the areas of shadow formed in the magnetic layout create areas of calm within the thrill of the lights where one can sit and ponder everything from the dessert to the sky and the sun that brings life to earth.
What was once worshipped as a distant god and celestial being can now exist on the surface of the earth as a Celestial Field in Nevada. The sun has risen and set, bringing with it heat and light; powering life on earth since the dawn of time, a focus of incomprehensible wonder and fascination for each and every culture across the globe.
Celestial Field intends to reignite our faith in the intangible, while showing us there are powers and beauty still to be found in the modern world.
The installation centers around the natural forms of feathers and pine cones, both being significant symbols in Native American culture, specifically relating to the Agai-Dicutta Numu. Feathers were often used as an expression of a certain virtue or to reflect a certain individual or significant event. As such feathers were viewed in a similar manner to a gift from a high official, representing a link between the individual and divinity.
The sculpture starts in the form of a feather to reflect on the link between man and divine. The bright colours painted across the platforms represent those painted across the barbs of the modern feathers used in Native American ceremonial dress and ornamentation.
As burners ascend they will make their way up the vane of a feather they will complete their own initiation, drawing similarity between the great events that feathers were rewarded for in the past. As the platforms open they will begin to transform in shape to bear visual resemblance to pine cones.
The pine nuts are important in particular for the Northern Paiutes to survive the winter; and have their own festival in the third week of September. The change in form will reflect the process of change from receiving a symbol from the divine, to receiving bountiful harvest and personal fulfillment. The lighting will help to accentuate the fluctuations in shape throughout the day and night, representing the omnipresent divinity and how it can be seen in many different forms of nature.
Form finding and colour explorationInteractivity
The piece will be both physically and visually interactive, with a changing form that will alter over the course of the festival. Burners can twist the colorful platforms around their central core allowing them to climb to the higher levels. As they do so they will be able to read messages of wisdom from other burners and leave their own along the way. The bridging links are fixed and thus can be used by the burners to sit and relax, observing the surrounding playa. The structure can also be used as a shelter, by rotating the platforms to create shade from the sun’s direct heat as well as forming a barrier from wind.
Special holes carved through some of the platforms will facilitate themselves as climbing holds to aid burners on their ascent. The bridging links are fixed and thus can be used by the burners to sit and relax, observing the surrounding playa. As the platforms are rotated, light will interact with the sculpture in different manners, pouring through the holes and creating intricate shadows. At night, the project will illuminate and create a light show, which will make the climbing possible even during night time.
Model – scale 1:1
Model- scale 1:50
Materials and components
The varying thickness platforms will be comprised of 12 and 25 mm layers of plywood, depending on it’s span and form. In each platform will be a ring of 10 mm acrylic between the top layers allows for LEDs to cast out a unique visual form at night. Metal ball bearings will separate each platform facilitating the change of form as it is used by burners.
All this will be held up by steel tubes measuring between 5 and 6 meters in height with a diameter of 160mm. The steel tubes will be welded to a metal plate buried in the sand and secured with ground anchors. The feathers are also structurally supporting each other, through the 4 bridge link, made from the same plywood.
Through the inviting but challenging spirals I want to bring you to the inside of the dust storms, to the dead zone where you don’t feel anything, not even a single hint of air or dust but you will still enjoy the beauty and the magic.
The symbol of the spiral
Through the inviting but challenging spirals I want to bring you to the inside of the dust storms, to the dead zone where you don’t feel anything, not even a single hint of air or dust but you will still enjoy the beauty and the magic.
What is an spiral?
A curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.
A three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a helix. By definition describes a planar curve, that extends in both of the perpendicular directions within its plane; the groove on one side of a record closely approximates a plane spiral (and it is by the finite width and depth of the groove, but not by the wider spacing between than within tracks, that it falls short of being a perfect example); note that successive loops differ in diameter. In another example, the “center lines” of the arms of a spiral galaxy trace logarithmic spirals.
What does the spiral mean to us?
Spirals are some of the oldest geometric shapes in ancient artwork dating back at least to the Neolithic period, the product of people thousands of years away from having access to writing. As such, we know very little about their religious beliefs and can, at best, guess about general meanings of symbols based on context.
Spiral to me means CHANGE, means CYCLE OF LIFE, means EVOLUTION all these is what The eye of the Storm is to me. A structure that will symbolize change as the same way Burning Man changes people; the cycle of life, the merging of completely different kinds of lives into the same place or same art structure or same festival; and evolution because it will attract you to the inside and will make you push yourself and challenge you towards the center where you will feel the peace of the eye of a storm as the same way life does to us.
To me The eye of the Storm symbolizes THE LIFE CHANGING, every frame of each spiral is an action, a decision you make in life that leads you to the place you want to be or the person you want to become. Although like in a windy storm that pushes you around, these events are not necessary your choice, many factors in life can change your path like a new elect president in the country or falling in love with a person, but all these merged together will consolidate our lives.Hence that is our task, putting all these frames together and connect them to become whatever we want to be and finally find the peace inside of each one of us.
Breaking down a six sided box The eye of the storm has six spirals, each one of them in a different orientation, symbolizing different kinds of paths in life. But each one of these spirals merge in the center of the cube creating a space of peace. Like the feeling of going through the Burning Man festival until the day the Temple burns when everything goes silent and calm.
The spirals, made out of standard size timber, will attract you to the center but these will be challenging like going towards a storm.The shape of the structure could become an obstacle but if you success you will get your reward of peacefulness in the center, protected by all these wooden spirals and with the chance to share these moment with whoever makes it through the storm with you.
Making the storm real
Starting from a box and breaking its frame into spirals is the basic starting geometry of The eye of the Storm. Going through many different testings and alterations on the parameters and after an structural analysis leading to the final shape.
Looking at the materializing of the structure and mimicking the aesthetics of a storm, the overlaying wood planks give the sense of it. Structured but at the same time crazy, just like Black Rock City.
Testing of different materials
Building the eye of the storm
The structure is composed of 6 identical modules called twists, arranged in cuboid manner. Each twist is formed of frame elements and ribs connecting those frames, which act as ruled surfaces.
Vertical loads are spread between the outer frames and the central bottom twist that acts in vending and axial force to carry the vertical load to the ground. Lateral forces are resisted using the perimeter frames acting as bracing or the in-plane rigidity of the modules themselves.
The frames with a cross section of 50 by 50 mm will be the main structure. Eleven frames on each side of the 6 sides of the cube, making a total of 66 frames.
The overall length of this out of the shelve timber will be 45.5 meters.
By the interpolation of the division points on the main frames we create the secondary structure which will support and interlock the main frame, with a cross section of 10 by 50 mm.
The overall length of this out of the shelve timber is 672 meters
The structure is arranged in a cubic form of 14 x 14 x 14 feet and will invite the people to climb,interact and inhabit it. This can represent a health and safety risk. In order to ensure the safety of all actors interacting with the sculpture at each stage of the project, we will be following the right safety requirements.
Feeling the dust at Black Rock City
Sand storms that make you lose sense of time, space, people or anything else that happens around you create these magical moments in Playa when every burner feels alone in a place full of people.
With The Eye of the Storm I want to bring you to the inside of these dust storms, to the dead zone where you don’t feel anything not even a single hint of air or dust but you still enjoy the beautiful magic of the storm. Through the challenging spiral structure inviting burners to come in, I want them to come inside the eye to the safety of being inside the storm and at the same time experiencing all the dust storms happening in Playa which are a natural ritual that happens every year.
We can compare the plague from the Exodus with any of the dust storms that we go through at burning man, The eye of the storm is a structure calling at the sun on these occasions.
The experience from the outside is compared with a tornado in a sand storm. Symbolizing a constant sand storm, kept in a broken box, which often happens in Playa. Within the overlaying of the wood frames experiencing the messiness of a storm but at the same time the spiral of the rotating structured nature force.While in the inner geometry you will sense none of this in the same way as in the dead zone or in the eye of the storm. All the layers of the structure will protect everyone inside from any sand storms in the outside. It wont be easy to reach that inner point as this all wooden layer will be complicated to cross like when you go through a storm and you are not able to find anyone around you and you feel in a place of nowhere.
The overall dimensions of the structure are 14 x 14 x 14 feet. Allowing people to get inside and experience the space in between the frames.
The shadows from the structure will project a similar form as an spider web.Simulating as the sun goes around through the day, a tornado moving through The Playa.
The reward to cross all these layers of wood through is to be in this protected space inside the Eye and experience the storm from the inside until you are ready to leave the peace and call Ra the god of the sun as in Egyptian rituals when the Pharaoh called the God to clear the skies from the storm and get back the sun to Black Rock City.
Once the ritual is finished the burners will enjoy a view of the dust storm in a protected space while they wait the answer of the call to Ra the god of sun, who will finish with the dust storm and bring the sun back again.
The LED lights will be located on each one of the frames on every side of the cube. They will be located on the inner face of these frames flowing from the outside to the inside of the cube, giving the sensation of the spiral coming inwards the structure.
TOTAL LENGTH : 45.5 meters
LENGTH ON 1/6 OF THE SIDE OF THE CUBE: 7.6 meters
At night the structure will become and spiral of colorful thunder lights that you will recognize from everywhere in The Playa.
By Maialen Calleja , Master of Architecture ( University of Westminster)
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
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.
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.
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.