Asymptotic Grid Structure of a Triply Periodic Minimal Surface

Through extensive research into the construction of grid shells, as well as differential geometry, I present a design solution for a complex grid structure inspired by the highly symmetrical and optimised physical properties of a triply periodic minimal surface. The proposal implements the asymptotic design method of Eike Schling and his team at Technical University of Munich.

‘Minimal Matters’ utilises the several geometric benefits of an asymptotic curve network to optimise cost and fabrication. From differential geometry, it is determined asymptotic curves are not curved in the surface normal direction. As opposed to traditional gridshells, this means they can be formed from straight, planar strips perpendicular to the surface. In combination with 90° intersections that appear on all minimal surfaces (soap films) this method offers a simple and affordable construction method. Asymptotic curves have a vanishing normal curvature, and thus only exist on anticlastic surface-regions. 

Asymptotic curves can be plotted on any anticlastic surface using differential geometry.

On minimal surfaces, the deviation angle α is always 45 (due to the bisecting property of asymptotic curves and principle curvature lines). Both principle curvature networks and asymptotic curve networks consist of two families of curves that follow a direction field. The designer can only pick a starting point, but cannot alter their path.

(a) Planes of principle curvature are where the curvature takes its maximum and minimum values. They are always perpendicular, and intersect the tangent plane.

(b) Surface geometry at a generic point on a minimal surface. At any point there are two orthogonal principal directions (Blue), along which the curves on the surface are most convex and concave.
Their curvature is quantified by the inverse of the radii (R1 and R2) of circles fitted to the sectional curves along these directions. Exactly between these principal directions are the asymptotic directions (orange), along which the surface curves least.

(c) The direction and magnitude for these directions vary between points on a surface.

(d) Starting from point, lines can be drawn to connect points along the paths of principal and asymptotic directions on the respective surface.

Gyroid TPMS

The next step is to create the asymptotic curve network for the Gyroid minimal surface; chosen from my research into Triply Periodic Minimal Surfaces.

As the designer, I can merely pick a starting point on an anticlastic surface from which two asymptotic paths will originate. It is crucial to understand the behaviour of asymptotic curves and its dependency on the Gaussian curvature of the surface.

Through rotational symmetry, it is resolved to only require six unique strips for the complete grid structure (Seven including the repeated perimeter piece).

The node to node distance, measured along the asymptotic curves, is the only variable information needed to draw the flat and straight strips. They are then cut flat and bent and twisted into an asymptotic support structure.

Plywood Prototype: 600mm cubed

Eight fundamental units complete the cubit unit cell of a Gyroid surface. Due to the scale of the proposal, I have introduced two layers of lamellas. This is to ensure each layer is sufficiently slender to be easily bent and twisted into its target geometry, whilst providing enough stiffness to resist buckling under compression loads.

Minimal Matters’ aims to create an explorative, meditative and interactive experience for visitors. It is a strained grid shell utilising the geometrical benefits of an asymptotic curve network; digitally designed via algorithmic rules to minimise material, cost, and construction time.

Minimal Matters: Burning Man Proposal

Inspired by the highly symmetrical and optimised physical properties of a triply periodic minimal surface, ‘Minimal Matters’ aims to create an explorative, meditative and interactive experience for visitors. It is a strained grid shell utilising the geometrical benefits of an asymptotic curve network; digitally designed via algorithmic rules to minimise material, cost, and construction time.

The proposal takes the form of a crystalline structure found in nature, interpreted through parametric design into a timber grid art piece. In the sense of repeating themselves in three dimensions, a gyroid is an infinitely connected triply periodic minimal surface. A minimal surface is a single surface articulation which minimises that amount of surface needed to occupy space. The proposal represents restoring a balance in energy, taking only that of the earth’s resources required to fulfil the form. Our inability to distinguish our needs from our greeds leads to excessive desires for life’s commodities. The efficiency of the design complements the beauty of rotational symmetry of a single node.

The lattice structure will create foot and hand holds to help climbers onto the series of sloping platforms; allowing users to survey the desert camp from different perspectives.

More than just a climbable structure, Minimal Matters is to be a resting place for festival-goers and a shelter from the strong sun of the site. The layers of grids cast shadows of varied patterns throughout the day. At night, LED lighting along the lamellas will celebrate it’s form and illuminate the playa.

1.5mm Plywood Prototype – 600x600x600mm


Inspired by nature, the proposal brings a parametrically designed structure into the realm of physical interaction. The piece is a culmination of thorough research and physical exploration of timber’s potential. The combination of conceptual bravery matched with architectural reality seeks an architecture of playfulness and beauty which will respond to the inclusive environment of Burning Man. It will celebrate a new design method for timber grid construction, and symbolise the harmony between nature and computational design.

Omnis Stellae

Omnis Stellae – Redrawing your own constellation

“Only in the darkness can you see the stars”
Martin Luther King

 

This project involves the conception and design of a new way of mapping constellations, based on subdivision processes like Stellation. It explores how subdivision can define and embellish architectural design with an elaborate system of fractals based on mathematics and complex algorithms.

Example of Stellation diagram on a platonic polygon

An abstracted form of galaxy is used as an input form to the subdivision process called Stellation. In geometry, meaning the process of extending a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure.

Omnis Stellae – Daytime interior render view

The material used for this installation will be timber sheets of 1/3 of an inch thickness that will be laser-cut.The panels will be connected to each other with standard connection elements which have already been tested structurally based on an origami structure.

The lighting of the installation will consist on LED strips that will light with burners interactions.

Omnis Stellae – Daytime exterior render view

Although stars in constellations appear near each other in the sky, they usually lie at a variety of distances away from the observer. Since stars also travel along their own orbits through the Milky Way, the constellation outlines change slowly over time and through perspective.

There are 88 constellations set at the moment, but I would like to prove that there are infinite amount of stars that have infinite amount of connections with each other.The installation will show you all the possible connections between this stars, but will never rule which connection is the one you need to make.

Omnis Stellae – Daytime interior render view from the ground

I would like burners to choose their own stars and draw their own constellations. Any constellation that they can possibly imagine from their one and only perspective, using coloured lights that react to their touch.

The end result will have thousands of different geometries/constellations that will have a meaning for each one of the burners and together will create a new meaningful lighted galaxy full of stars.

 

Omnis Stellae – Nightime exterior render view

On a clear night, away from artificial light, it’s possible to see over 5000 stars with the naked eye. These appear to orbit the Earth in a fixed pattern, as if they are attached to a giant sphere that makes one revolution a day.This stars though are organised in Constellations.

The word “constellation” seems to come from the Late Latin term cōnstellātiō, which can be translated as “set of stars”. The relationship between this sets of stars has been drawn by the perspective of the human eye.

Omnis Stellae – Daytime interior render view from above

“Omnis Stellae” is a manifestation of the existence of different perspectives. For me, there is great value in recognising different perspectives in life, because nothing is really Black and White, everything relates to the point of view and whose point of view and background that is.

As a fractal geometry this installation embodies an endless number of stars that each person can connect and imagine endless geometries, that will only make sense from their own perspective. The stellated geometry will show you all the possible connections but will never impose any.

Omnis Stellae – Daytime and Nightime

“Omnis Stellae” is about creating your own constellations and sharing them with the rest of the burners, is about sharing your own perspective of the galaxy and create some meaningful geometries that might not mean anything to other people but would mean the world to you.

Omnis Stellae – Daytime interior render view

The grand finale is if it could become the physical illustration of all the perspectives of the participants at Burning Man 2018 shown as one.

With Love,

Maya

 

 

 

The Fractal Hourglass

The Fractal Hourglass counts down to the singularity, the moment that artificial super-intelligence triggers an unprecedented shift in human civilisation. The concept of recursively self-improved AI is portrayed by a tower of iterated fractal trusses, in which time is measured by a cascade of light.

Triangular steel trusses array to form a 15-foot tall hourglass silhouette, where scaled repetitions within each truss form a lattice of increasing complexity and infinite bounds. The visual density of each truss intensifies at each fractal iteration, culminating in the filling of the lower hourglass bulb, representing the finite time remaining until the singularity. At night, a dynamic cascade of LEDs will flow on and off from the upper to the lower bulb, a spectacle alluding to sand pouring through an hourglass.

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The steel tubes forming the piece range from a diameter of 1.5″ in lengths from 1 to 3-feet, which are hammered flat and bolted to form the main structure, and 0.5″ diameter tubes welded inside to form the decorative fractal repetitions.

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On approach, the tense drama of time running out is visible through the concentration of material in the bottom of the hourglass, provoking an instinct to stall the process. Burners have a choice of how to experience the hourglass- whether that is to ascend the structure to experience the inversion of the hourglass as the bulb empties, where ascension serves as a sanctuary from the saturation of technology and AI in the lower bulb. Or they can recline on the ground and let their eyes weave through the layers of trusses and bathe in the saturation and complexity of technological advancement. Or simply to turn away and let what effectively has become a natural process to take its course. At night, the cascading light display forms an even more immersive encounter with the hourglass, as waves of light repeat the process of time as it funnels through and fills the lower bulb, swarming anyone who is inside.

 

The finite nature of fractals in the hourglass represents the capacity for infinite artificial intelligence- each increment provides an equally stable steel structure, whilst having the capacity to use less and less material, but only to a point. It is not possible for this fractal to reach infinity and be constructed at a human scale. This poses the question of, at which point on the way to infinity do humans get before their intelligence can be overtaken by AI- the moment of the singularity. Is it too late to invert the hourglass and, given the choice, would you want to?

The Fractal Hourglass allows for Burners to take a moment to relish on their existence as humans, with the capacity to orchestrate their own experience, something which AI’s currently don’t possess. Artificial intelligence is currently an opportunity to shape a future experience where humans can outsource themselves, freeing up valuable time and energy. The hourglass serves as a visual symbol that human existence is fleeting so long as AI is permeating our lives, and provides a timer for the impending singularity, a moment that will transform the world as we know it, a reminder that we still have the alluring capacity to define and create.

 

‘The first ultra-intelligent machine is the last invention that man need ever make, provided that the machine is docile enough to tell us how to keep it under control.’

I J Good

 

 

The Wishing Well

something caught in between dimensions – on its way to becoming more.

Summary

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 notion of geometry in between dimensions is explored in a previous post: Shapes, Fractals, Time & the Dimensions they Belong to

 

Description

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.

Different Recursive Steps of a Dragon Curve

Ambition

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.

171108 - Burning Man Timber Brick Laying Proposal View 2.jpg

 

Philosophy

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
  • ► ORANGE – change, adaptability, spontaneity, concentration
  • ► 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.

171108 - Burning Man Timber Brick Laying Proposal View 1.jpg

Life’s First Flicker

LifeFirstFlicker_External_Night.png

Project Summary

Translucent fractal ball animated by dancing color changing lights symbolizing the very first spark at the beginning of the universe as well as the spark of life that our species is on both an eternal mission to keep alight and is in the final stages of creating anew with artificial intelligence.

Physical Description

A 20 foot diameter translucent form, made of twelve identical five petal polypropylene origami flowers arranged as a dodecahedron each with two smaller flower at their centers and intricately lazercut with a 2d filigree depicting the overall form. All the flowers are tied to each other and to a timber dodecahedral internal skeleton with concealed zip ties. The creased polypropylene held in tension by the origami folds themselves provide the rest of the stability. LED DMX lights sit in the five points which touch the ground, facing upwards, illuminating the entire form.

LifesFirstFlicker_External_Day

Philosophy

At the end of Isaac Asmiov’s book ‘The last question’ a disembodied AI is the last mark of the human race left in an empty entropic universe. It remains calculating the answer to the last question it was asked, and, unable to find a recipient for it’s solution it says ‘Let there be light’ and creation begins again.

This piece is shaped as a tangible interpretation of this spark. The spark at the first second of the universe and the spark of life which our civilization is racing to create in a sentient self learning AI. This moment, when our creations become self aware, is the theme of burning man 2018 and also likely to be the most important moment in our species’ history. Self teaching AI will rapidly become so powerful it will effectively be a deity. This pavilion’s purpose is to draw the visitors attention to this rapidly approaching moment and consider how we should design this mind before it is too late.

Burners can pass by or play with the spark and think it is just a cool shape, but those who climb up inside what is figuratively a snapshot right at the beginning of new life and the universe, can take a moment to pause and ponder from within, whether their life’s endeavor is relevant in the face of coming AI, what our species’ current knowledge may lack and how it could be codified and explained to a machine and how lucky we are as a generation to have both been born late enough to see AI’s birth, and early enough to have known life before it.


<p><a href=”https://vimeo.com/244593633″>Life’s First Flicker: Burning Man 2018 Art proposal</a> from <a href=”https://vimeo.com/user71835996″>Benjamin Street</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p>

Fractalized Gates – A journey of Self Discovery

renders01The project is inspired by the interaction between people. It celebrates the female and male union and brings together people that already know each other or are completely strangers. Its interactive, it challenges its visitors, its playful.

It invites people to take part and be interactive with each other. It is called like that as it challenges people to reach the final gate and come together. It acts indeed like a mistletoe. Mysterious and magical. The moment you stand beneath the final gate you stand where all the magic happens. You feel the power and need to kiss the other person. Its design celebrates this union and offers a magical mutation.

It offers a labyrinth of complex, intriguing that seek to nourish individual experiences.

Black rock city seems to be the promised land, the promise of freedom, of self-expression, of immediacy and creativity and community. In a community like Burning Man you can assume the right to approach any random person and have an interesting interaction. You can overly express interest and curiosity. The Desert Mistletoe aims to give people a chance to be more expressed, more playful. Its for those people who want deeper connections, more meaningful interactions, less seriousness and more play.

Hugs and affection are a particularly significant aspect in which to expect more from strangers. We all need love, hugs and kisses are one of the best ways to deliver it. Take the risk to go in for a kiss. Of course some people will be hesitant. The project challenges people to express affection and admiration and relax. Hugs and kisses bring us together. At Burning Man, the endless parade of people flaunting their unusualness brings joy and excitement.

The unusual is both delightful and challenging. People love the unusual, the extraordinary and anything out of our everyday lives and routines. At Burning Man you do not need to sacrifice your wonderful weirdness. On the contrary you are challenged to explore your playful impulses and discover or express your freaky freedom.

The project is inspired specifically by the people of Burning Man.

‘Burners respect the gift of being touched’

 

Inspiration – The Symbol of Sri Yantra

axonometric explanationSymbolism

The Sri Yantra is conceived as a place of spiritual pilgrimage. It is a representation of the cosmos at the macrocosmic level and of the human body at the microcosmic level. It is a Journey. A spiritual journey from the stage of material existence to the union of the individual soul with the divine. The spiritual journey is taken as a pilgrimage in which every step is an ascent to the center. It represents a movement beyond one’s limited existence and every level is nearer to the goal. All the stages are within a gated frame, which is called the ‘earth citadel’ and forms an enclosed space.

The geometry

The symbol consists of nine interlocking triangles, centered around a bindu ( the central point). The five downward pointing triangles represent Shakti; the female principle. The four upright triangles represent Shiva; the male principle. The nine interlocking triangles form 43 small triangles each representing a particular aspect of existence.

 

The Concept – Fractals & Koch Snowflake

The proposed project is inspired by the concept of fractals – a never ending pattern, applied to a specific geometry. They are infinitely complex patterns created by repeating a simple process over and over, creating patterns that are self-similar across different scales. Fractals are images of dynamic systems that are driven by recursion.

The idea is based on the Koch Snowflake, a mathematical curve which has a finite area bounded by an infinitely long line.

For the project a script was created in Grasshopper and different patterns were created based on the triangles extracted from the Sri Yantra symbol.

fractals-experimentsBelow: Acrylic model based on the fractal pattern that was created above

fractals-experiments-model-01The model below is based on the fractal pattern created by

  1. the triangles representing the Female principle
  2. the triangles representing the Male principle

Together they represent, like the Sri Yantra symbol, the union of female and male. A celebration of our Creation.

fractals-experiments-model-02

Developing the design and the creation of a journey through a structure

The initial idea was to extract some patterns and create something that will look like gates. Gates that could provide a journey to their visitors.

first-attemptFurther development of the gates was undertaken. The final concept consists of 4 gates which intersect with each other in order to provide a journey. There are 4 different designs for each gate. Each gate has a dense pattern in order to create something elegant and ornamental. The pattern is based on the Koch snowflake concept and has been applied to multiple directions. The gates have different heights and dimensions with the last one reaching up to 10m height. This was designed in such a way to provide a climax of the visitor journey at the final gate/stage.

final-gatesThe design is all about a journey. A personal journey with different experiences. A welcoming structure, inviting the visitors to interact with each other. It says a story, it represents a story and it invites you to create one.

acrylic-model

model-photos

Experiment model in order to understand at what points the gates could intersect. At this point, the gates are connecting only on the top.

This has as a result to have an ’empty’ structure on the sides.

initial-ideasFinal Design Concept

final-concept

All the gates are connected with each other not only at the top but also on the sides. To achieve best results and better connections, the 3 gates are duplicated and connected with each other. The second gate will be connected with the third one only at the bottom. The 4rth gate has three sides creating a triangular central space. The gates have been rotated along the 2 other sides allowing for its visitors to enter from different directions. This also creates an enclosed space, like the Sri Yantra symbol.

The proposed installation aims to be part of this beautiful journey of self-discovery. It takes part of expecting more from strangers, while noticing the weirdness of others and encourages everybody to express themselves. With expecting more of strangers it increases the likelihood that the people you meet will become a part of that sometimes elusive network of connections we call community.

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renders03night-render