Developing Space-Filling Fractals

Delving deeper into the world of mathematics, fractals, geometry, and space-filling curves.

 

Foreword

Following my last post on the “…first, second, and third dimensions, and why fractals don’t belong to any of them…“, this post is about documenting my journey as I delve deeper into the subject of fractals, mathematics, and geometry.
The study of fractals is an intensely vast topic. So much so that I’m convinced you could easily spend several lifetimes studying them. That being said, I chose to focus specifically on single-curve geometry. But, keep in mind that I’m only really scratching the surface of what there is to explore.

4.0 Classic Space-Filling

Inspired by Georg Cantor’s research on infinity near the end of the 19th century, mathematicians were interested in finding a mapping of a one-dimensional line into two-dimensional space – a curve that will pass through through every single point in a given space.
Jeffrey Ventrella writes that “a space-filling curve can be described as a continuous mapping from a lower-dimensional space into a higher-dimensional space.” In other words, an initial one-dimensional curve is developed to increase its length and curvature – the amount of space in occupies in two dimensions. And in the mathematical world, where a curve technically has no thickness and space is infinitely vast, this can be done indefinitely.

4.1 Early Examples

In 1890, Giuseppe Peano discovered the first of what would be called space-filing curves:

Peano-space-filling-Curve_-four-approximations_-version-A_1 4i.gif
4 Iterations of the Peano Curve
An initial ‘curve’ is drawn, then each element of the curve is replace by the whole thing. Here it is done four times, and it’s easy to imagine how you can keep doing this over and over again. One would think that if you kept doing this indefinitely, this one-dimensional curve would eventually fill all of two-dimensional space and become a surface. However it can’t, since it technically has no thickness. So it will be as close as you can get to a surface, without actually being a surface (I think.. I’m not that sure..)
A year later, David Hilbert followed with his slightly simpler space-filing curve:
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8 Iterations of the Hilbert Curve
In 1904, Helge von Koch describes a single complex continuous curve, generated with rudimentary geometry.
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7 Iterations of the Koch Curve
Around 1967, NASA physicists John Heighway, Bruce Banks, and William Harter discovered what is now commonly known as the Dragon Curve.
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13 Iterations of the Dragon Curve

4.2 Later Examples

You may have noticed that some of these curves are better at filling space than others, and this is related to their dimensional measure. They fall under the category of fractals because they’re neither one-dimensional, nor two-dimensional, but sit somewhere in between. For these examples, their dimension is often defined by exactly how much space they fill when iterated infinitely.
While these are some of the earliest space-filling curves to be discovered, they are just a handful of the likely endless different variations that are possible. Jeffrey Ventrella spent over twenty-five years exploring fractal curves, and has illustrated over 200 hundred of them in his book ‘Brain-Filling Curves, A Fractal Bestiary.’ They are organised according to a taxonomy of fractal curve families, and are shown with a unique genetic code.
Incidentally, in an attempt to recreate one of the fractals I found in Jeffery Ventrella’s book, I accidentally created a slightly different fractal. As far as I’m concerned, I’ve created a new fractal and am unofficially naming it ‘Nicolino’s Quatrefoil.’ The following was created in Rhino and Grasshopper, in conjunction Anemone.
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5 Iterations of Nicolino’s Quatrefoil
You can find beautifully animated space-filling curves here:
(along with some other great videos by ‘3Blue1Brown’ discussing the nature of space-filling curves, fractals, infinite math, and more)

On A Strange Note:

It’s possible to iterate a version of the Hilbert Curve that (once repeated infinity) can fill three-dimensional space.
As an object, it seems perplexingly difficult to categorize. It is a single, one-dimensional, curve that is ‘bent’ in space following simple, repeating rules. Following the same logic as the original Hilbert Curve, we know that this can be done indefinitely, but this time it is transforming into a volume instead of a surface. (Ignoring the fact that it is represented with a thickness) It is a one-dimensional curve transforming into a three-dimensional volume, but is never a two-dimensional surface? As you keep iterating it, its dimension gradually increases from 1 to eventually 3, but will never, ever, ever be 2??
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Nevertheless this does actually support a statement I made in my last post suggesting “there is no ‘first’ or ‘second’ dimension. It’s a bit like pouring three cups of water into a vase and asking someone which cup is the first one. The question doesn’t even make sense…

5.0 Avant-Garde Space-Filling

In the case of the original space-filling curve, the goal was to fill all of infinite space. However the fundamental behaviour of these curves change quite drastically when we start to play with the rules used to generate them. For starters, they do not have to be so mathematically tidy, or geometrically pure. The following curves can be subdivided infinitely, making them true space-filling curves. But, what makes them special is the ability to control the space-filling process, whereas the original space-filling curves offer little to no artistic license.

5.1 The Traveling Salesman Problem

Let’s say that we change the criteria, from passing through every single point in space, to passing only through the ones we choose. This now becomes a well documented computational problem that has immediate ‘real world’ applications.
Our figurative traveling salesman wishes to travel the country selling his goods in as many cities as he can. In order to maximize his net profit, he must make his journey as short as possible, while of course still visiting every city on his list. His best possible route becomes exponentially more challenging to work out, as even just a handful of cities can generate thousands of permutations.
There are a variety of different strategies to tackle this problem, a few of which are described here:
The result is ultimately a single curve, filling a space in a uniquely controlled fashion. This method can be used to create single-lined drawings based on points extracted from Voronoi diagrams, a topic explored by Arjan Westerdiep:
Traveling Salesman Portrait.png
This illustration, commissioned by Bill Cook at University of Waterloo, is a solution to the Traveling Salesman Problem.

5.2 Differential Growth

If we let physics (rather than math) dictate the growth of the curve, the result becomes more organic and less controlled.
In this example Rhino is used with Grasshopper and Kangaroo 2. A curve is drawn on a plain, broken into segments, then gradually increased in length. As long as the curve is not allowed to cross itself (which is achieved here with ‘Collision Spheres’), the result is a curve that is pretty good at uniformly filling space.
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Differential Growth with Rhino & Grasshopper – Kangaroo 2 – Planar
The geometry doesn’t even have to be bound by a planar surface; It can be done on any two-dimensional surface (or in three-dimensions (even higher spacial dimensions I guess..)).
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Differential Growth with Rhino & Grasshopper – Kangaroo 2 – NonPlanar
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Differential Growth with Rhino & Grasshopper – Kangaroo 2 – Single-Curved Stanford Rabbit
Additionally, Anemone can be used in conjunction with Kangaroo 2 to continuously subdivide the curve as it grows. The result is much smoother, as well as far more organic.
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Differential Growth with Rhino & Grasshopper – Kangaroo 2 & Anemone – Octopus
Of course the process can also be reversed, allowing the curve to flow seamlessly from one space to another.
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Differential Growth with Rhino & Grasshopper – Kangaroo 2 & Anemone – BatmanDuck
Here are far more complex examples of growth simulations exploring various rules and parameters:

6.0 Developing Fractal Curves

In the interest of creating something a little more tangible, it is possible to increase the dimension of these curves. Recording the progressive iterations of a space filling curve allow us to generate what is essentially a space-filling surface. This new surface has the unique quality of being able to fill a three-dimensional space of any shape and size, while being a single surface. It of course also shares the same qualities as its source curves, where it keep increasing in surface area (and can do so indefinitely).
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Surface Unrolling Study
If you were to keep gradually (but indefinitely) increasing the area of a surface this way in a finite space, the result will be a two-dimensional surface seamlessly transforming into a three-dimensional volume.

6.1 Dragon’s Feet

Here is an example of turning the dragon curve into a space-filling surface. Each iteration is recorded and offset in depth, all of which inform the generation of a surface that loosely flows through each of them. This was again achieved with Rhino and Grasshopper.
I don’t believe this geometry has a name beyond ‘the developing dragon curve’, so I’ve called it ‘Dragon’s Feet.’
Adding a little thickness to the model allow us to 3D print it.
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Developing Dragon Curve: Dragon’s Feet – 3D Print

6.2 Hilbert’s Curtain

Here is the Hilbert Curve going through the same process, which I am aptly naming ‘Hilbert’s Curtain.’
3D Printed Developing Hilbert Curve
Developing Hilbert Curve: Hilbert’s Curtain – 3D Print
3D Printing Space-Filling Curves with Henry Segerman at Numberphile:
‘Developing Fractal Curves’ by Geoffrey Irving & Henry Segerman:

6.3 Developing Whale Curve

Unsurprisingly this can also be done with differentially grown curve. The respective difference being that this method fills a specific space in a less controlled manner.
In this case with Kangaroo 2 is used to grow a curve into the shape of a whale. Like before, each iteration is used to inform a single-surface geometry.
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Iterative Steps of the Differentially Grown Whale Curve

3D print of the different recursive steps of a space-filling curve
Developing Whale Curve – 3D Print

Desire for Immortality

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.

DesireforImmortalityNight

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.

DesireforImmortalityCombined

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.

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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

Mixpinski’s Myriad

In mathematics a fractal is an abstract object used to describe and simulate naturally occurring objects. Artificially created fractals commonly exhibit similar patterns at increasingly small scales. It is also known as expanding symmetry or evolving symmetry.  Mandelbulb 3D allows us to explore fractals in 3D, creating a seamless amalgamation of maths, art and science.

Understanding how this geometry can become infinite and how it can be built within the constraints of the physical reality was part of the philosophy of my piece.

Mandelbulb 3d fractals:

 

161117 FRACTAL sheet 1

From these specific chosen 3d Fractals I noticed a clear correlation with the natural formation of crystalline structures, in particular Hopper crystals.

Hopper crystals form when there is more rapid growth at the outer edges of a face than at the centre. This results in what appears to be a hollowed out step lattice formation, as if someone had removed interior sections of the individual crystals. This missing part was never actually developed as the crystals grow so rapidly that there is never time for this to be developed. Hopper crystals are very similar to the cubic halite skeletal crystals formed from extreme supersaturation in salt lakes existing in nature. Hopper crystals can be found in rose quartz, gold, calcite, bismuth, salt and ice. I looked at the growth of these crystals to better understand the structural qualities.

Hopper Crystal Formation:

161117 FRACTAL sheet 3.1

From looking at the crystalline structure it became apparent that the connection between the tapered levels was very important to the structure and adaptability of the proposal. The versatility of this connection allows for flexibility and movement within the module. The connector can be placed on any material simply by adapting the end nodes width to factor for the material depth. By creating this modular junction I can join all the stepped timber elements of the proposal in such a way that they are all supporting each other.

Connection options:

161117 FRACTAL sheet 2

Hopper crystal growth is never as predicted due to outside influences such as movement and temperature change. These influences creates the beautiful images we see of their crystalline forms and without these the fractal crystal growth would be predictable and simplistic. It is with these outside interactions that the crystals have their own idiosyncrasies. By combining the hopper crystal growth with the organic forms created with the 3d fractal generator, I created a pavilion proposal. Using a stepped form and the junction designed above I could use the unpredictable growth lines to create an interesting pavilion which can be experienced in the same way that crystals would grow, naturally and not within their algorithmic form. Nature does not always conform to predictability. The pavilion expresses this individuality and in turn expresses the way in which we grow as individuals, adapting to our environments and moulded by our experiences.

This project is a physical exploration of crystal formation centred around the theme of fractals. It aims to combine one joint in order to create a crystalline structure. Inspired by the geometry from the crystalline growth the lattice structure provides sanctuary and calm in a sea of dust and at night mesmerising myriads of stepped lights will illuminate the playa.271117 render night.jpgThe proposed installation will be formed of a mixture of 2 x 4 timber with CNC curved plywood pieces incorporated into the structure. Each 2 x 4 will have a joint or a pocket in order for it to slot into and support the weight of the neighbouring beam or column. The project will appear out of the sand as an elegant stepped fractal structure which gives the proposal an ecclesiastical ambiance.

231117 close up night

The intertwined stepped lattice timber elements form congregation and celebratory spaces, whilst capturing special views of the playa. The stepped elements promote Burners to climb and crawl between the spaces created by the overlapped timbers. At night when you ascend through the individual spaces the lights will constantly change and oscillate. With the lights constantly changing and staggering further through the elements the stepped structure will be enhanced.  The project aims to play with the burners’ perception of depth where the lattice stepped geometry is staggered and rotated. At night this perception is further confused by  the LED coloured strips oscillating along the staggered stepped beams and columns. The burners can seek sanctuary in a space in which dimensionality and form is confused and adapted.

271117 renderday

 

 

 

 

The beauty of error

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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

Summary 

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.

Physical description

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.

Philosophy 

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.

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Shapes, Fractals, Time & the Dimensions they Belong to

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.

Foreword

In this post, I’m going to try my best to explain the 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. But before getting into the nitty-gritty of the subject, I think it’s worth prefacing this post with a short note on the nature of mathematics itself:

Alain Badiou said that mathematics is a rigorous aesthetic; it tells us nothing of real being, but forges a fiction of intelligible consistency. That being said, I think it’s interesting to think about whether or not mathematics were invented or discovered – whether or not numbers exist outside of the human mind.

While I don’t have an answer to this question (and there are at least three different schools of thought on the subject), I do think it’s important to keep in mind that we only use math as a tool to measure and represent ‘real world’ things. In other words, our knowledge of mathematics has its limitations as far as understanding the space-time continuum goes.

 

1.0 Traditional 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.

It’s important to note that there is no ‘first’ or ‘second’ dimension. It’s a bit like pouring three cups of water into a vase and asking someone which cup is the first one. The question doesn’t even make sense.

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Excerpt from ‘minutephysics’
We usually arbitrarily pick a dimension and calling it the ‘first’ one.

 

1.1 – Zero Dimensions

 O2FFp

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.

1.2 – One Dimension

R1

A line or a curve gives us a one-dimensional object, and is typically bound by two zero-dimensional things.
Only one coordinate is required to define a point on the curve.
Similarly to the point, a curve can inhabit higher dimension (i.e. you can plot a curve in three dimensions), but as an object, it only possesses one dimension.
Another way to think about it is: if you were to walk along this curve, you could only go forwards or backward – you’d only have access to one dimension, even though you’d be technically moving through three dimensions.

 

1.3 – Two Dimensions

 R2

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&yy&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 surface can live in three+ dimensions, but still only possesses two dimension. Two coordinate are required to define a point on a surface. For example a sphere is a three-dimensional object, but the surface of a sphere is two-dimensional – a point can be defined on the surface of a sphere with latitude and longitude.

 

1.4 – Three Dimensions

rotated-cube

A volume gives us a three-dimensional shape, and can be bound by two-dimensional shapes (surfaces).

Shapes in three dimensions are most commonly represented in relation to an x, y and z axis. If a person were to swim in a body of water, their position could be defined by no less than three coordinates – their latitude, longitude and depth. Traveling through this body of water grants access to three dimensions.

 

2.0 Fractal Dimensions

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.

 

2.1 – Between the First & Second Dimensions

A line or a curve gives us a one-dimensional object that allows us to move forwards and backwards, where only one coordinate is required to define a point on them.

Surfaces give us two-dimensional shapes, where two coordinate are required to define a point on them.

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Here is a shape that cannot be classified as a one-dimensional shape, or a two-dimensional shape. It can be plotted in two dimensions, or even three dimensions, but the object itself does not have access the two whole dimensions.

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.

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Each fractal has a unique dimensional measure based on how much space they fill.

 

2.2 – Between the Second & Third Dimensions

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Developing Koch Snowflake
The same logic applies when exploring fractals plotted in three dimensions:

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.

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Fourth Iteration Menger Sponge
The Menger Sponge for example has (mathematically) a volume of zero, but an infinite surface area.

 

2.3 – Calculating Fractal Dimensions

The following are three methods of calculating Hausdorff-Besicovitch dimension:
• The exactly self-similar method for calculating dimensions of mathematically generated repeating patterns.
• The Richardson method for calculating a dimensional slope.
• The box-counting method for determining the ratios of a fractal’s area or volume.

 

On another note:

In theory, higher (non-integer) dimensional fractals are possible.
As far as I’m concerned however, they’re not particularly good for anything in a three-dimensional world. You are more than welcome to prove me wrong though.

 

3.0 Higher Dimensions

Sadly, living in a three-dimensional world makes it especially difficult to think about, and nearly impossible to visualise, higher dimensions. This is in the same way that a two-dimensional being would find it impossibly hard to think about our three-dimensional world, a subject explored in the novel ‘Flatland’ by Edwin A. Abbott.

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.

 

3.1 – The Fourth Dimension (Temporal)

If we consider time an additional variable, then despite the fact that we live in a three dimensional world, we are always subject to (even if we cannot visualize) a fourth dimension.

Neil deGrasse Tyson puts it quite plainly by saying:
“[…] you have never met someone at a place, unless it was at a time; you have never met someone at a time, unless it was at a place […]”

Suppose we call our first three dimensions x, y & z, and our fourth t: latitude, longitude, altitude and time, respectively. In this instance, time is linear, and time & space are one. As if the universe is a kind of film, where going forwards and backwards in time will always yield the exact same outcome; no matter how many times you return to a point in point time, you will always find yourself (and everything else) in the exact same place.
However time is only linear for us as three-dimensional beings. For a four-dimensional being, time is something that can be moved through as freely as swimming or walking.
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Excerpt from ‘Seeker’

 

3.2 – The Fourth Dimension (Spacial)

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.

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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.

Cube-Rotation

 

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.

  • A line is bound by two zero-dimensional things
  • A square is bound by four one-dimensional shapes
  • A cube is bound by six two-dimensional surfaces
  • A hypercube, bound by eight three-dimensional volumes

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.

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3D representation of eight cubes folding in 4D space to form a Hypercube

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.

 

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3.3 – The Fifth and Sixth Dimensions

On the temporal side of things, adding the ability to move ‘left & right’ and ‘up & down’ in time gives us the fifth and sixth dimensions.

(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.

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In spacial dimensions, it is theoretically possible to fold four-dimensional objects with a fifth dimension. However, it becomes increasingly difficult for us to visualise what is happening to the shapes that we’re folding.
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In theory, objects can keep being folded together into higher and higher spacial dimensions indefinitely. (R1, R2, R3,R4,R5, R6, Retc.)

There’s a terrific explanation of what happens to platonic solids and regular polytopes in higher dimensions on Numberphile: https://youtu.be/2s4TqVAbfz4

 

3.4 – Even Higher Dimensions

If we can take a point and move it through space and time, including all the futures and pasts possible, for that point, we can then move along a number line where the laws of gravity are different, the speed of light has changed.

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.

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Cross section of the quintic Calabi–Yau manifold
There may very well be a link between temporal dimensions and spacial dimensions. For all I know, they are actually the same thing, but thinking about it for too long makes my head hurt. If the topic interests you, there is a philosophical approach to the nature of time called ‘eternalism’, where one may find answers to these questions. Other dimensional models include M-Theory, which suggests there are eleven dimensions.

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.