## 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:

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:
In 1904, Helge von Koch describes a single complex continuous curve, generated with rudimentary geometry.
Around 1967, NASA physicists John Heighway, Bruce Banks, and William Harter discovered what is now commonly known as 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.
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??
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:

### 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.
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..)).
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.
Of course the process can also be reversed, allowing the curve to flow seamlessly from one space to another.
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).
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.

### 6.2 Hilbert’s Curtain

Here is the Hilbert Curve going through the same process, which I am aptly naming ‘Hilbert’s Curtain.’
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.

## The Petal Hypothesis

The petal has long been a surround for the reproductive parts of the flower, its varied forms and designs attract numerous species of animals and insects, enabling its existence to grow and spread. As a result, the petal will not only encounter pollen of its own species, but also that of many differing plants.

Taking people as the pollen. This caravanserai will attract people both day and night, providing a space for play and discussion, encouraging communication, observation and interaction.The Petal Hypothesis sits expressively within its setting. Exposing the raw structure of the plywood ‘petals by day and revealing the elaborate display of the EL wires by night.

Configured in a circular array, each ‘petal’ is construct from just two ‘pods’ which in themself only take 1 sheet of plywood to construct. Connected together to generate one ‘2d’ curve, the end points then bend around to complete the monocoque structure.

These pods are then mirrored to generate the ‘petal’ form and anchored to the ground. The act of fixing the extreme widths and mirroring the pods minimises the natural flex within the ‘pod’ and enables it to be a strong physical structure.

In place of the EL wires, a cloth stress skin has been incorporated to the top tier of petals. This not only provides shading during the day but also absorbs the light from the EL wires and distributes it across the whole surface.

Interactivity:

Observe – Sit around and within the ‘petals’ to observe the activities at its centre.

Inhabit – Climb the structure and occupy one of the many vantage points within the ‘petals’

Connect – Share memories and congregate either on mass at its centre or privately within the petals

## ORBIT

Narrative | ‘Orbit’, an aluminium tube pavilion stands as a playful take on the orbit of our solar system. A kinetic, inhabitable architectural structure that orbits around itself revealing a central, occupiable space that acts as a ‘center of the universe’  location within which the occupier will experience the rest of the world rotate around them.

Occupiers act as planets orbiting around one another, taking in the beautiful surroundings as each hammock level gently rotates as if it is floating, free from visible connections below, In order to reach these relaxing levels, the occupiers must scale its lightweight structure eventually reaching the central ‘ritualistic’ epicenter.

Physical Description | Orbit stands as a playfully abstract vision of the universes orbit around the sun. Visually the structure is very simple. A series of single recursively scaled down forms provide both the frame work in which to house multiple levels of hammock space to relax whilst also offering a highly structural climbing frame that is scaled in order to reach its epicentre.  It stands tall amongst its neighbours as a combination of both inhabitable architecture and a visually striking art piece.

The structure is composed of multiple interlocking aluminium tubes of varying diameter that hang from a single point supported by the main outer structural framework.  Within the opening at the bottom of each frame is space for hammock netting to be fitted to the aluminium tubing providing an inhabitable space to relax on.

The inset neon LED lighting on the inside of the aluminium tube frame enhances the proposals visual impact at night, illuminating to be seen from near and afar.

Interactivity | There are multiple levels for potential seating, each incorporating a hammock like mesh suspended between the aluminium structure. This provides a comfortable place to relax whilst the structure gently rotates about its axis. As with most exciting Burning Man installations, this structure is climbable with the final point to reach being the central frame large enough for one person to sit in whilst the rest of the structure rotates around them.

## The Cloud

So easily can fun and playfulness be neglected within Architecture. My proposal stands as an embodiment of these aspects, creating an area of inclusive participation, a space that can be explored and is only complete when occupied.

Fallen from the sky and tied down in the middle of Black Rock City ‘The Cloud’ stands as a mirage for weary-eyed travellers from far and wide, a beacon of sanctuary that creates spaces that provide respite from the harsh conditions of the desert using permeable fabric to create a cool atmosphere diffusing light within daylight and emitting a soft glow from within in the evening.

Walking through the dessert after a long journey along the silk road ‘The Cloud’ emerges as a whimsical mirage. Mimicking the form of a cloud the easily recognisable form is transformed into Architecture; a sinuous billowing form allowing us to fulfil a childhood dream, walking on clouds.

The principle structure of the cloud is composed of hollow rolled steel tubes ,sandwiched between thick perforated fabric, strategically placed to withstand the extreme wind conditions as well as human interaction. Elevated from the floor these tubes are secured to the ground using the kandy kane re-bar method.
Keeping the form soft and playful so that not only is the installation safe but also malleable, responding to people climbing and walking it, bungee rope is securely looped over the steel tubes and threaded through the ‘ground’ fabric to hold it up, as illustrated in the accompanying drawing.

Interactivity is an integral part of the installation. Bringing to life the stranded cloud people are encouraged to explore the piece climbing in, over and around it, finding intricate crevasses that provide discreet hidden entrances to the inner cloud where an intimate social environment softly illuminated by the diffused daylight, providing an area of solace.

DIMENSIONS // 5000mm(l) x 3100mm(w) x 4100mm(h)

## Thursday 12th December 2013

We just finished our last tutorials of the first term! Congratulations to all the students for the great three months and looking forward to the remaining two terms.

Students completed both briefs (brief01:systems and brief2A:festival) and are starting the case studies of events as part of our last brief (brief2B:realise).

Here are couple pictures of the projects we have seen during the last tutorials. Where do you suggest building the structures over the summer?

Merry Christmas & best wishes for the New Year!!

## Building Fractal Cult and Shipwreck at Burning Man 2013

We’re back from the desert! Very proud to have completed two beautiful projects at the Burning Man festival 2013 with our DS10 students and guests from the Architectural Association, Columbia University and UCL.

Credits to the team:

Team: Toby Burgess and Arthur Mamou-Mani a.k.a. Ratchet and Baby Cup (Project Directors), Thanasis Korras (Designer of Fractal Cult), Georgia Rose Collard-Watson (Designer of Shipwreck), Jessica Beagleman (Food & Meals), Natasha Coutts (Camp and Rentals), Sarah Shuttlesworth, Andy Rixson,  Luka Kreze, Tim Strnad, Philippos Philippidis, Nataly Matathias, Marina Karamali, Harikleia Karamali, Antony Joury, Emma Whitehead, , Jo Cook, Caitlin Hudson, Dan Dodds and Chris Ingram.

Engineers: Ramboll Computational Design (RCD) –  Stephen Melville, Harri Lewis, James Solly

Suppliers: Hess Precision (Plywood Laser Cutting), One-to-Metal, (Metal Punching and Folding), Safway (Scaffolding), West Coast Netting (Netting)

Special Thanks: BettieJune, Ben Stoelting, Kevin Meers, Caroline Holmes, Chloe Brubaker, Papa Bear,

Photos by Jo Cook, Arthur Mamou-Mani, Toby Burgess, Luka Kreze, Thanasis Korras, Antony Joury.

Here are couple more pictures of the finished projects:

Some images of the construction of Shipwreck, from the collection of the pieces all the way to the assembly

Images of the construction process of Fractal Cult until the burn:

Finally, how we made our camp look more like a home and less like a refugee camp:

A beautiful view of the festival itself at sunrise:

Here is a text that we wrote about the experience:

Diploma Studio 10:
Diploma Studio 10 at the University of Westminster is led by Toby Burgess and Arthur Mamou-Mani. They both believe that involvement is key to the process of learning and therefore always try to get their students to “get out and build” their designs in the real world. The studio starts the year with the study of systems, natural, mathematical and architectural systems of all sort, paired with intense software training in order to build up skills and a set of rules to design a small scale project which they will be able to build during a real event in the summer. Throughout the year, they build large scale prototypes and draw very accurate technical drawings, they also need to provide a budget and explain how it makes sense within the wider context of the festival, some of them will event start crowd-funding campaign to self-finance the projects. Our ultimate goal is to give them an awareness of entrepreneurship in Architecture and how to initiate projects as this is for us the best way to fight unemployment in our profession.
Burning Man and the 10 Principles:
The Burning Man festival takes place every summer in Black Rock desert, Nevada. It is a “participant-led” festival in which the activities are initiated by the people attending it. There are around 60,000 “burners” every year building a giant temporary city in which they create a social experiment which follows the 10 principles of Burning Man. They conclude the festival by burning a large sculpture of a Man.
What interested Toby and Arthur are the 10 principle which guide the “burners”: Radical Self-Reliance, Radical Inclusion, Gifting, Leaving No trace, to name a few. Designing with these rules in mind help students understand basic issues of sustainability. Designing for Burning Man also helps the students to design with “playfulness” in mind, as all the structures have to be climbable and interactive. We are not the only one inspired by these rules, Sergei Brin, co-founder of Google, asks all his staff to follow the principles when they come up with new ideas.
The Story:
On our first year at Westminster we found out that our student could submit their Burning Man proposals and receive a grant from the organizers. After receiving 20 submissions from the same school, the organizers were very intrigued and decided to contact us. The director of the Art Grant told us that she loved the project but that all of them were just not possible in the context. She decided to visit us in London to explain what we could do to submit better projects the following year which we did. On the second run, the festival chose two projects, Shipwreck by Georgia Rose Collard-Watson and Fractal Cult by Thanasis Korras.
These two projects are representative of the way we run our studio: Thanasis looked at Fractal on Brief01 and Georgia looked at ways to bend and assemble strips of wood together. They both explored these systems before submitting a project with a very strong narrative which fitted perfectly the burning man philosophy. Thanasis linked his Fractal to the symbol of “Merkaba” whereas Georgia told the story of a shipwreck which offered shelter from the dust storms.
Once the project got chosen, we partnered with an engineer, Ramboll and started researching for suppliers and fabrication facilities in the USA. We took the 3D files from concept all the way parametric models for fabrication. We started a Gantt chart with every step to take from rental of 24ft truck, collection of item all the way to demolition.
One of the main aspect that required a lot of planning was the camp. We had to plan every meal and food that would not perish under the extreme condition. We also found a way to rent a whole camp equipment from past burners.
On site:
The team grew little by little, many of our student could not afford the trip or could not take such a long time off so we asked around if anyone else would like to join us and thanks to our blog posts and active social networking online, students from the Architectural Association, Columbia or UCL started showing interest and joined the team.
Our first surprise on site was the power of the dust storm. One of our Yurt flew away and some of us got stuck in different places of the site seeking shelter. We were terrorised. Sleeping in tents was also extremely hard as you would be awaken by temperatures approaching 40degrees celcius, at the end of the construction, a lot of us would sleep in the foam hexayurts in which we were storing equipment at first.
We learned so much.

## A Week at Grymsdyke Farm

We just finished our week at Grymsdyke Farm, Buckinghamshire. Ten students spent about two nights each working on their individual projects, building a 1:1 to 1:5 prototype using the available technology: a CNC Milling Machine (with RhinoCam), a laser cutter, a Z-Corp and a RepRap 3Dprinter.

DS10 would like to thank Guan Lee, Ed Grainge and Kate for their precious help and patience on the CNC, Jessie Lee and Keith McDonald for their great advices!

Below are some pictures of the week.

Above: Dhiren Patel’s “Ear Parabola” being assembled

Above: Dan Dodds testing the fiber optic cables of his Sectionned Harmonograph

Above: Emma Whitehead cutting her convection cell models out of plywood

Above: Thanasis Korras’ CNC milled components for his giant fractal building.

## Scan-and-Solve for Rhino

Scan-and-Solve is a plug-in for Rhino which ‘completely automates basic structural simulation of Rhino solids. Unlike other analysis tools, no preprocessing (meshing, simplification, healing, translating, etc.) is needed.’ See http://www.scan-and-solve.com/ for additional information, tutorials and discussions or you can also find it through the www.food4rhino.com downloads list.

Attached are my initial explorations in the student license of the software to analyse a block for use in a reciprocal grid structure. As the images show, the software is very simple to use, simply choose a solid; a material from the drop down list; select the faces to act as restraints and then the loads to apply. View the results as a colour gradient showing displacement values or danger levels within the solid. The software also allows you to visualise the deformation. Unfortunately, you cannot perform analysis on multiple solids within a system currently and the student license is limited to a solid of 50 faces or less.

## A Year of Grasshopper Experiments with DS10

It has almost been already a year that Toby and I started tutoring DS10 at Westminster. One of our main ambitions was to link physical and digital experiments so that one feeds the other.

Physical reality is much more than surfaces on a screen therefore students created complex parametric models working as systems linked to many forces (gravity, environment, structure…etc…) and not just finished objects. These very precise digital models allow students to implement what they learn from their physical models, to simulate even more design options and further understand the rules behind them.

To do so, they used Grasshopper and its numerous plugins provided by generous developers. Grasshopper is a graphical algorithm editor integrated with Rhinoceros 3D modelling tool and a 18,000 strong community exchanging ideas and helping each other on the Grasshopper3d.com forum.

Below are most of the printscreens that I used to help the students with their journey into parametric modelling which is based on help that I also received previously. I hope that this will help others to design amazing things! If you have any questions on one of the images, please do not hesitate to ask.

Below is my favourite image: packing balloons on a surface using Kangaroo (with Emma Whitehead)