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:

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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:
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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

BIY – Build It Yourself

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Tiny house movement, which is a theme of DS10 second brief, is closely linked to the idea of self-build. It is commonly thought that both emerged simply as a response to housing crisis. However, compelled limitations can also encourage other unexpected movements. Here, I want to talk about how the self-build practice allowed discovering new possibilities that go beyond economy.

In 1980s, Berlin-born modernist architect Walter Segal proposed a solution to shorten the housing waiting list by allowing people to build their own homes. With the bold step of London borough of Lewisham, an ‘awkward’ piece of sloping land, that was found unsuitable for council’s programme, was donated for the experiment. People that got randomly chosen from the waiting list were allocated a site and given a basic induction in how to saw a straight line and drill a hole. Segal believed that a house should adapt to its occupants. Each household was invited to participate in the design stage, while the construction principles included lightweight timber frames and stilt foundations, meaning the layout could always be adjusted. New residents of Walter’s Way all worked collaboratively from the project commencement and as a result formed a tight community. At the end of the project, new occupants were given a chance to buy their homes. Walter’s Way became UK’s first self-built council housing project.

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In spring 2016 I visited the site to interview residents for my undergraduate dissertation project. I was keen to understand how design principles of community influenced the level of happiness of it’s residents and therefore affected sustainable behaviour. Walter’s Way was a unique case study. Walking onto the street felt nothing like the rest of London and more like an eco-village in a countryside. You could sense a spirit of community that seemed to work great almost naturally. The road slopes down and main entrances to houses are oriented in such way, that you always see your neighbours as soon as they walk out their private spaces. The central core is used for weekly community activities and as a kids playground. All residents were happy to talk and during the conversation it was common to hear “I’m not as sustainable as my neighbour, but I learn from them”, which is almost like a good eco-competition. That is without saying that certain homes achieve carbon savings of 73% (according to SuperHomes).

Today, all houses are private with many owners occupying their homes for over 20 years. Everyone remembers the history of Walter’s Way and feels proud to be a part of it. Many continued the legacy of self-build by attaching extensions or upgrading buildings. The fact that all houses were designed by those in need and built with their own hands allowed for an activist community to be born. It not only challenges the traditional approach to solving housing issue, but creates new opportunities in the city by building on ‘abandoned’ sites, creating a new model for urban life and teaching others sustainable lifestyles.

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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.

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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.

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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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Finding Crystal.

Project Summary

Crystal is lost within a sea of flashing lights. She is surrounded by a 6.4×12.8ft cubic lattice structure. She reacts to motion and touch. Walk through the interactive cube that holds Crystal hostage. Can you find her before she fades, and becomes one with the cube?

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Physical Description

Finding Crystal is an interactive installation that takes the principles of Crystallography and Bravais Lattices, and uses these principles to create a structural lattice cube. This cube is made of 0.4×0.4ft cubic pieces, fixed on top of, and next to each other to create a 6.4×12.8×12.8ft cube. I have chosen to use a cubic body centered lattice, removing the given structural frame, to allow the internal arrangement to determine the overall structure. The structure is made of steel spheres, steel rods and motion sensor LED lights. Each 0.4×0.4ft structure holds 3 spheres, 4 rods and 3 LED light. These pieces are put together in a puzzle manner to create the entirety of the structure. The art piece is modular and can be easily assembled on site.

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Interactivity & Mission

Finding Crystal takes principles from science and combines it with design to create a truly interactive art piece. It uses the science behind crystal lattice structures to produces a structural modular that encompasses the principles of crystallography. Each modular is sized specifically to produce a lightweight structure, reflecting back on the idea of a crystal flake and its lightweight properties.

The entire structure consists of 16384 0.4×0.4ftcubic lattice pieces, with 49152 LED lights. This creates a platform for the experience. The cube is programmed to displays ghost like figures that walk “through” the cube, the figures use human motion to “follow” or “escape” the users. This evokes a form of playfulness between the figure, which is Crystal, and its users.

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Philosophy

Crystal is a human. The cube is a symbol of an AI being that is attempting to deceive Crystal. Crystal is trying to escape the hold of the cube, but she is finding it extremely difficult to, the cube keeps misleading her into thinking that she is dependent on it. This piece explores addiction, in any form, by using the cube to confine the users perception of Crystal. Crystal doesn’t believe she can escape the cube, but ultimately, she is the only one that can help herself escape it.

I am deploying this piece in hope that it creates a curiosity behind the connection between technology, science and design. As well as creating a better understanding of someone who is in Crystals position. I’d like to evoke a desire to take these principles and explore with them, I want this piece to inspire and motive people to be creative. I want it to awaken the users curiosity. I want the user to want to explore, understand and interact with the artwork, and take this curiosity further by exploring their own artistic expression.

The intention of this piece is for it to be used as an interactive play experience. In the end, we are just trying to save crystal from herself.

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 Amazing Surf

Project Summary

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?

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Physical Description

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.

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Interactivity & Mission

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.

Philosophy

“Keep your face always toward the sunshine – and shadows will fall behind you”

-Walt Whitman

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.

(IN)Finitely Bound

Project Summary

(IN)Finitely Bound is a recursive fractal geometrical form, similar to that found in nature. It symbolises the universe and its finite boundary, and is an expression to show us the limitations to which technology can take us. As nothing can be bigger and more powerful than the universe.Fractal Geometry

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(IN)Finitely Bound is a recursive fractal Dodecahedron form, consisting of lengths of 2 by 4 timber held into place using bolts and metal plate joints. The structure will be fully burnable and will be both approximately 7m high and wide. On approach to the piece the structures beauty will be be hard to work out, symbolising our confusion with the colossal scale of the universe but as you get closer you realise the receptive nature of the form and come rest on the structure and understanding as we zoom in on aspects which make up the universe and include ourself within it. The piece will be lit up to gently to allow for meditation, contemplation and open our bodies up to mindfulness.

 

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Interactivity and Mission

The initial singularity was a singularity of infinite density thought to have contained all of the mass and space-time of the Universe. The standard model of cosmology predicts that the universe is infinite and flat. However, cosmologists in France and the US are now suggesting that space could be finite and shaped like a dodecahedron instead. They claim that a universe with the same shape as the twelve-sided polygon can explain measurements of the cosmic microwave background – the radiation left over from the big bang – that spaces with more mundane shapes cannot. In a world where “computational power is increasing exponentially, much like the singularity which created the universe” realising our own finite boundaries is where we take power back from the robots and become masters of our own minds, bodies and universe. The piece through its self-replicating fractal structure creates a dodecahedron (Universal) Boundary defined by perspective. In defining our boundary we are then able to instead of focus our mind inwards, a symbol towards mindfulness.

 

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Burning Man night render