4.0 Classic Space-Filling
4.1 Early Examples
In 1890, Giuseppe Peano discovered the first of what would be called space-filing curves:
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 proposal reflects on immortality and how our lives would look like if we could reach it. Evolution has sentenced us to the process of aging, and ultimately to death, but as we understand it more and more, we may be able to outwit it. Sounds like paradise? Wouldn’t you want to be immortal?
The art installation is composed of cone shaped cells that divide itself creating new cells, which in turn develop into new ones and the process repeats. The components are made of laser cut, rolled thin sheets of plywood and are connected with metal screws. The structure, measuring approximately 20 feet long and 26 feet high, becomes stronger with every iteration, is structurally stable and self-supporting but on the other side almost invisible and very fragile in appearance. By joining the cone-like shaped cells, a set of domes at different scale is formed which are composed into pavilion serving as shelter to partially protect from sun and wind and casting beautiful shadows at the same time.
The pavilion is providing an opportunity to lay down, calm and contemplate. Look around and reflect on the surroundings – is it the blurred, crowded playa that attracts your attention? Or the cells of the structure that interest you? You have a chance to hide away for a moment and meditate. At night, the structure becomes illuminated from the inside, which highlights the pattern, casting even more beautiful shapes than during the day. You can move the bulb around and play with the light to explore different parts of the structure and look closer into the cells and how they divide themselves.
The concept was born during my research on fractals and their exploration through the Mandelbulb 3D software where by composing different formulas and changing their parameters, I could create beautiful, endless shapes. Infinity is one of the main feature of fractals, therefore, trying to materialize the experiments into physical models was the biggest challenge. To represent endlessness, I started looking at cell division and unicellular organisms, such as bacteria and paramecia, which multiply by dividing themselves. The duration of the cell ends with the division, but the line can be considered immortal.
The life span of a cell usually has specific limits due to telomerase and a separate genetic program of aging and death of complex organisms that evolved only about a billion years ago. Single-celled organisms that lived on Earth before that did not experience either aging or death and at a certain stage of maturity, they divided into two new cells. The first death occurred, when the sexual reproduction appeared – evolution has sentenced us to the process of aging, and ultimately, to destruction. However, recent developments in the field of physiology and medicine show that the elixir of life does not sound like a myth anymore and may become a reality in the future. And what if it becomes a reality? Does it scare you or does it make you happy? The aim of the proposal is to reflect on immortality and how our live would look like if we could achieve it.
The Amazing Surf is a complex fractal geometry which ascends toward the light, symbolizing our obsession with reaching for the stars. We use our increasingly digital world to help us extend our reach, but at what point do the shadows we cast reach out above us?
The shape is inspired by the Amazing surf fractal which is generated by a mathematical formula and visualized in Mandelbulb3D. A visually imposing 25ft tall Ply wood hyperbolic structure, with intricate evolving folded panels. Each folded panel is digitally unrolled into a 2D net and CNC milled, the resultant ply components will be glued to a layer of fabric and folded back to their original 3D shape. This construction technique removes the need for a supporting frame, keeping the complex geometry unobstructed from view. A few panels have been removed at the base to make way for an entry point. Neon strips attached to each panel will produce dramatic light patterns on the surface at night. The installation will orient toward the sunset, where the sun appears at it’s closest.
The piece is intended to be used as an impromptu climbing frame, a ladder to ascend burners above the desert and into the stars. Sunlight will bounce off the multi-faceted shapes, creating intricate patterns of light and shadow. Burners are invited to dance in the light shafts and seek shelter in the shadows. As the shape begins to flatten toward the top, the folded panels can be used as armchairs, where vision will be limited to that of the sky and light above; burners can sit and watch as the sky transforms from day to night.
“Keep your face always toward the sunshine – and shadows will fall behind you”
As a race we strive to advance, developing new tools and machines to help us in this process. There will come a point in the not too distant future where the machines we have developed to help us will supersede us; we will become so reliant on technology, it will begin to control us. I see the Amazing Surf installation as a juxtaposition to this potential future; on the one hand we are using technology to create built environments that are intricate, beautiful and unique, on the other hand these environments are only attainable through the use of technology. If only we took a moment to look back into the shadows, we could avoid the fate that we are gradually bringing upon ourselves.
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.
dis/integration[loops], inspired by the composer William Basinski’s seminal works of the same name, explores the limitations of digital processes in our world – and the chaos that can unfold from overreliance on them.
A towering array is assembled from recursive fragments of an inherently destructive process. It explores the tension that exists between the digital and physical realms; challenging an immortal, digital world, the glorious ruin of the analogue realm confronts the perceived perfection of the artificial.
Existing in a state of intended incompleteness, dis/integration[loops] eschews vanity in favour of exhibiting procedural rawness; the power of ruinous accident reveals itself through the tarnishing of idyllic digitalism.
Pressure-laminated plywood modules, form-found through iterative casting experiments, connect to form a pervious, fragmented structure; it’s transcience and impermanence exaggerated as night follows day.
In the same way that Basinski’s fragile recordings were destroyed upon being processed by the human ear, dis/integration[loops] exists in a contented, lush and shimmering state prior to being activated by human presence.
Proximity-controlled LED lighting impregnates the structure. When combined with sounds inspired by those Basinski’s (de)generative process created, this affords a level of animated deconstruction upon activation; visually and sonically, the imperfect presence of humanity causes dis/integration[loops] to be engulfed in chaotic ripples of distortion.
It’s most perfect (yet still decidedly imperfect) state is one in which it lies dormant and peaceful, undiscovered by the presence of people. It experientially disintegrates upon activation.
The fragmented structure exaggerates ever-changing natural light conditions and provides shelter, as well as an intimate, tactile space withi it’s permeable walls.
‘And then as the last crackle faded and the music was no more, I took in my surroundings and looked around at the faces and I was right there with everybody and we were alive.’
dis/integration[loops] is a reminder than everything we encounter eventually falls apart and returns to dust. It challenges the perfect, edited, occularcentrism that blights our social lives, explores the sound of decay, and the beauty that can exist in destruction. It is a meditation on death and loss, and exploration on a theme that some things are better left untouched.
The experience of life – a gradual disintegration – is simultaneously enriched and eroded by the imperfect nature of our encounters; pristine digitalism deserves a tarnished, ruinous quality symbolic of our experiences.
‘and I was right there with everybody and we were alive.’
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.
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.
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.
First, second and third dimensions, and why fractals don’t belong to any of them, as well as what happens when you get into higher dimensions.
In physics and mathematics, dimensions are used to define the Cartesian plains. The measure of a mathematical space is based on the number of variables require to define it. The dimension of an object is defined by how many coordinates are required to specify a point on it.
Something of zero dimensions give us a point. While a point can inhabit (and be defined in) higher dimensions, the point itself has a dimension of zero; you cannot move anywhere on a point.
Surfaces or plains gives us two-dimensional shapes, and are typically bound by one-dimensional shapes (lines/curves).
A plain can be defined by x&y, y&z or x&z; more complex surfaces are commonly defined by u&v values. These variable are arbitrary, what is important is that there are two of them.
A volume gives us a three-dimensional shape, and can be bound by two-dimensional shapes (surfaces).
Fractals can be generally classified as shapes with a non-integer dimension (a dimension that is not a whole number). They may or may not be self-similar, but are typically measured by their properties at different scales.
Felix Hausdorff and Abram Besicovitch demonstrated that, though a line has a dimension of one and a square a dimension of two, many curves fit in-between dimensions due to the varying amounts of information they contain. These dimensions between whole numbers are known as Hausdorff-Besicovitch dimensions.
Surfaces give us two-dimensional shapes, where two coordinate are required to define a point on them.
If you were to walk along the shape starting from the base, you could go forwards and backwards, but suddenly you have an option that’s more than forwards and backwards, but less than left and right.
You cannot define a point on this shape with a single coordinate, and a two coordinate system would define a point off of the shape more often than not.
Surfaces give us two-dimensional shapes, where two coordinate are required to define a point on them.
A volume gives us a three-dimensional shape where a point could be defined by no less than three coordinates.
While these models live in three dimensions, they do not quite have access to all of them. You cannot define a point on them with two coordinates: they are more than a surface and less than a volume.
That being said, it’s plausible that we experience much higher dimensions that are just too hard to perceive. For example, an ant walking along the surface of a sphere will only ever perceive two dimensions, but is moving through three dimensions, and is subject to the fourth (temporal) dimension.
If we explore spacial dimensions, a four-dimensional object may be achieved by ‘folding’ three-dimensional objects together. They cannot exist in our three-dimensional world, but there are tricks to visualise them.
We know that we can construct a cube by folding a series of two-dimensional surfaces together, but this is only possible with the third dimension, which we have access to.
If we visualise, in two dimensions, a cube rotating (as seen above), it looks like each surface is distorting, growing and shrinking, and is passing through the other. However we are familiar enough with the cube as a shape to know that this is simply a trick of perspective – that objects only look smaller when they are farther away.
In the same way that a cube is made of six squares, a four-dimensional cube (hypercube or tesseract), is made of eight cubes.
It looks like each cube is distorting, growing and shrinking, and passing through the other. This is because we can only represent eight cubes folding together in the fourth dimension with three-dimensional perspective animation.
Perspective makes it look like the cubes are growing and shrinking, when they are simply getting closer and further in four-dimensional space. If somehow we could access this higher dimension, we would see these cubes fold together unharmed the same way forming a cube leaves each square unharmed.
Below is a three-dimensional perspective view of hypercube rotating in four dimensions, where (in four-dimensional space) all eight cubes are always the same, but are being subjected to perspective.
(For example: x, y, z, t1, t2, t3)
This is a space where one can move through time based on probability and permutations of what could have been, is, was, or will be on alternate timelines. For any one point in this space, there are six coordinates that describe its position.
There’s a terrific explanation of what happens to platonic solids and regular polytopes in higher dimensions on Numberphile: https://youtu.be/2s4TqVAbfz4
Dimensions seven though ten are different universes with different possibilities, and impossibilities, and even different laws of physics. These grasp all the possibilities and permutations of how each universe operates, and the whole of reality with all the permutations they’re in, throughout all of time and space. The highest dimension is the encompassment of all of those universes, possibilities, choices, times, places all into a single ‘thing.’
These ten time-space dimensions belong to something called Super-string Theory, which is what physicists are using to help us understand how the universe works.
While we don’t have experimental or observational evidence to confirm whether or not any of these additional dimensions really exist, theoretical physicists continue to use these studies to help us learn more about how the universe works. Like how gravity affects time, or the higher dimensions affect quantum theory.