4.0 Classic Space-Filling
4.1 Early Examples
In 1890, Giuseppe Peano discovered the first of what would be called space-filing curves:
Diploma Studio 10 at Westminster University School of Architecture
Delving deeper into the world of mathematics, fractals, geometry, and space-filling curves.
In 1890, Giuseppe Peano discovered the first of what would be called space-filing curves:
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.
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.
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
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 notion of geometry in between dimensions is explored in a previous post: Shapes, Fractals, Time & the Dimensions they Belong to
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:
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.
Since the last post on the 23rd October our students have been exploring how to materialise their research into fractals (which they generated with Mandelbulb3D). The conflict between endless geometry and finite material world creates a creative tension that pushes innovation in digital design and fabrication. From parametric equations to parametric design, students have explored fractals as self-generating computer images and attempted to control them, first through changing their variables and then by extracting the most appealing fragments and recreating them using Grasshopper3D . From pure voxel-based images to NURBS or meshes and to 3D printing, laser-cutting, thermo-forming, casting..etc… students are confronted to the limitation of the computer’s memory and processing power as well as materials and numerical control (NC) programming language such as Gcode.
Navigating through fractals, exploring their recursive unpredictability to create more finite prototypes is like walking through the forest and noticing a beautiful flower to design your next building – it helps to let go of a fully top-down approach to architecture, it encourages a collaborations with your computer and a deep understanding of machines and materials. It anticipates a world in which the computers will have an intelligence of their own, where the architect will guide it onto a learning path instead of giving him instructions. Using infinite fractals to inspire designs helps instill infinity within the finite world – bringing a spiritual dimension to our everyday life.
Below is a selection of our students Brief01 journey so far:
Diploma Studio 10 is back with 21 talented architecture students from 4th and 5th year working on the Brief01:Fractals. Here is an overview of their experiments so far after 4 weeks of workshops.
We are back after a year exploring Symbols & Systems, and an inspiring unit trip to South India, visiting the Hempi Valley and Auroville. This year our focus is on Fractals, not just as forms but as tools to understand how geometry can become infinite and how it can be built within the constraints of the physical reality. Fractals gives the opportunity to expand confined spaces, to let the mind fill the gap that reality had to stop. Therefore it also provides a great tool for the second brief, which is the Tiny Home movement, society’s need to create more compact, efficient homes to face the environmental and economical crisis. As per our previous briefs, we would like our students to build their projects, whether it is a giant fractal at a festival or an actual home within a space that would otherwise be left empty, we want students to raise funds and make, using digital fabrication tools combined with off-the-shelf material. Our goal is to continue training the entrepreneur-makers of tomorrow. Below is a breakdown of our briefs as they are being drafted:
The 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
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 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.
Below: Acrylic model based on the fractal pattern that was created above
The model below is based on the fractal pattern created by
Together they represent, like the Sri Yantra symbol, the union of female and male. A celebration of our Creation.
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.
Further 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.
The 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.
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.
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.
Arthur C. Clarke’s documentary on Fractals:
A basic set with fractal behaviour is the set of Complex numbers (C):
No matter how much one zooms in or out, the set is self-similar with infinite detail.
The typical fractal sets (Mandelbrot, Julia, Fatou) follow a pattern of 3 infinities: an infinite number of points is run an infinite number of times through a recursive polynomial and it will/will not reach infinity:
To make the step to 3d, the major issue is that the 2d rules cannot be generalized because there is no corresponding set of numbers for 3d space. 1d space has Real numbers, 2d space has Complex numbers, but there is no 3d equivalent. However, Quaternions (hypercomplex numbers) are a theoretical set of points corresponding to a 4d space. Therefore there are two possible approaches: a) Define a three dimensional set of points in polar coordinates and switch them back to a cartesian coordinates in order to build it in computer space. b) Build a theoretical 4d fractal using hypercomplex numbers and cast its 3d shadow in 3d.
Either way, results are similar:
Dealing with infinite numbers, infinite iterations and infinite sets of points, computation times become an issue. One way around this is to build ray-traced images estimating distances to a virtual fractal (not physically storing the points of the fractal in memory):
The image above is magnified ~3.10e13 times. In other words, presuming the size of the sectional model is 1m, it scaled up to roughly the size of the Solar System.
Building 3d models in computer space is slightly trickier because of the huge number of points involved to define even a limited section of a fractal. The issue is to define an algorithm for the correct order of the points in order to build a mesh. A rather neat solution is to ray-trace consecutive sections through a fractal (ray-tracing involves a Z-buffer anyway) and work from there. Here is an example (a 38,000,000 face-mesh obtained from 1,000 sections):
An important tool in exploring 3d fractals is building Julia sets (the only difference is that they use a constant increment at each iteration rather than the initial step):
Software used for my project: Processing, ImageJ FIJI, MeshLab, Netfabb Studio, Jesse’s MandelBulb 3D, Autodesk 3dS Max, Chaos Pro, Adobe Premiere.
Here is the book that I kept mentionning in the tutorial: The Nature of Code by Daniel Schiffman
The book explains many algorithm that attempt to reproduce natural systems (including swarms and fractals) using Processing, the java-based scripting interface.
You can download the book and make a donation or buy the hard copy. Try some examples, register to the Processing forum and to StackOverflow.com. Ask for help on the Processing IRC Channel.
For some example, you will need to need to download the Toxiclibs library and you might want to use the Eclipse IDE to speed up your workflow. You can also follow the great Plethora-Project.com tutorials by Jose Sanchez.
“Fractal Cult” is an installation consisting of two types of structures that aim to create an intriguing, mesmerising, explorative, playful and interactive experience for visitors of the 2013 Burning Man festival, an annual art event and temporary community based on radical self-expression and self-reliance in the Black Rock Desert of Nevada.
The geometry of the installation is based on the work of Swedish mathematician Niels Fabian Helge von Koch and in particular his invention of the Koch Snowflake, one of the earliest fractal curves to be described. Specifically, the structures are an adaptation of Koch Snowflake’s principles into a three-dimensional environment that essentially starts with a regular tetrahedron and recursively generates new tetrahedrons on each of its faces resulting in a complex, yet simply and efficiently defined, end result.
The installation is consisted by 4 timber-made, fractal pods that symmetrically surround a space frame-like structure of a similar fractal nature and with climbing nets dressing the faces of the geometrical shape that is created.
The timber pods, during daylight, are the first structure that a visitor encounters and both initiate and welcome the exploration of its symmetric but complex structure. Visitors are also able to enter the pods and experience an even more intriguing spectacle of the formation of faces and joints that create a kaleidoscope-like effect. They can also be used as temporary shelters from wind and sun, or even a meditation space. During the night, these timber fractal pods are illuminated from their interior, creating magnificent patterns of lighting that will attract visitors and welcome them to explore the site.
The imposing, central structure, during daylight, attracts visitors with its fractal nature, yet simple construction, and invites visitors to climb and engage with it with in all sorts of ways. Climbing the exterior and attempting to reach the top or even getting inside the interior and enjoy the complexity that the multiple layers of nets create. Moreover, the structure can definitely be seen as becoming a much more live thing during the festival, with people using the nets to create temporary shelters from the sun by weaving cloth materials or similar, forming a patchwork effect on the structure’s faces. It is difficult to predict exactly the kind of behaviour that visitors will have towards such a structure but more likely than not its lightweight nature, great size and the multifunctional nature of nets will allow for several different scenarios which would be great to observe. During the night, the structure maintains the same use but it is symmetrically lit with stage lights pointing from the ground up that will give the structure an illuminating effect and hopefully attract visitors from far away.
Last but not least, the geometry of the structures is strongly spiritually connected to Mekabah, a divine light vehicle allegedly used by ascended masters to connect with and reach those in tune with the higher realms. “Mer” means Light. “Ka” means Spirit. “Ba” means Body. Mer-Ka-Ba means the spirit/body surrounded by counter-rotating fields of light, (wheels within wheels), spirals of energy as in DNA, which transports spirit/body from one dimension to another.
Overall, “Fractal Cult” aims to offer a great variety of fun and explorative options, as well as serving as a place able to transform to temporary shelter or meditation space for visitors, while at the same time impose beauty through its fractal and symmetric nature.