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

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.

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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.

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PNEU + PACK

All living organisms are composed of cells, and cells are fluid-filled spaces surrounded by an envelope of little material- cell membrane. Frei Otto described this kind of structure as pneus.

From first order,  peripheral conditions or the packing configuration spatially give rise to specific shapes we see on the second  and third order.

This applies to most biological instances.  On a larger scale, the formation of beehives is a translated example of the different orders of ‘pneu’.

Interested to see the impact of lattice configuration on the forms, I moved on to digital physics simulation with Kangaroo 2 (based on a script by David Stasiuk). The key parameters involved for each lattice configuration are:

Inflation pressure in spheres
Collision force between the spheres
Collision force of spheres and bounding box
Surface tension of spheres
Weight.

 

Physical exploration is also done to understand pneumatic behaviors and their parameters.

This followed by 3D pneumatic space packing. Spheres in different lattice configuration is inflated, and then taken apart to examine the deformation within. This process can be thought of as the growing process of seeds or pips in fruits such as pomegranates and citrus under hydrostatic pressure within its skin; and dissections of these fruits.

As the spheres take the peripheral conditions, the middles ones which are surrounded by spheres transformed into Rhombic dodecahedron, Trapezoid Rhombic dodecahedron and diamond respectively in Hex Grid, FCC Grid,  and Square Grid. The spheres at the boundary take the shape of the bounding box hence they are more fully inflated(there are more spaces in between spheres and bounding box for expansion).

   

Physical experimentation has been done on inflatables structures. The following shows some of the outcome on my own and during an Air workshop in conjunction with Playweek led by Will Mclean and Laylac Shahed.

To summarize, pneumatic structures are forms wholly or mainly stabalised by either
– Pressurised difference in gas. Eg. Air structure or aerated foam structures
– liquid/hydrostatic pressure. Eg. Plant cells
– Forces between materials in bulk. Eg. Beehive, Fruits seeds/pips

There is a distinct quality of unpredictability and playfulness that pneumatic structures could offer. The jiggly nature of inflatables, the unpredictability resulted from deformation by compression and its lightweightness are intriguing. I will call them as pneumatic behaviour. I will continually explore what pneumatic materials and assembly of them could offer spatially in Brief 02. Digital simulations proved to be helpful in expressing the dynamic behaviours of pneumatic structures too, which I intend to continue.

Moiré Patterns

Dutch Invertuals – ‘Cohesion’

Moiré patterns are superimposed secondary patterns created when two static surface patterns are overlaid one on top of the other. By displacing or rotating one or both patterns a new visual pattern becomes visible separate to the geometry of the first two. This moiré effect is created in the eye of the viewer, disparate from the shapes formed by the individual patterns themselves.

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A moiré pattern generated by overlapping two identical patterns of concentric circles

Associated mathematical formulas can be used to determine the size and spacing of inferred moiré patterns from a series of regularly spaced overlaid patterns. The beauty of the moiré effect is the illusion of movement created through completely static overlays. This forms a naturally interactive experience for the participant, giving over control to the superimposed pattern through visual movement and rotation.

Physical Moiré experiments

The video above illustrates the moiré effect in two dimensions by overlaying static linear and concentric patterns, printed on acetate, and manipulating their motion and rotation in order to create a new visual pattern.

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Concentric and Linear patterns, printed on acetate overlays

This effect is not restricted to two dimensional patterns but can also be applied in three dimensions. These spatial patterns then utilise the motion of the viewer in order to manipulate the moiré effect. The video below illustrates how three dimensional sculpted elements, set on separate spatial planes can form a visual pattern and take advantage of simple motions by the viewer.

Scale model of the facade for Brisbane Girls School Creative Learning Centre – M3 Architects

The two primary resultant effects from the physical experiment above illustrate the potential of moiré to create alternate visual patterns and to generate the illusion of movement. These were then applied digitally to create an animation that controls these aspects to create a recognisable representation of motion to the viewer, as opposed to an abstract pattern.

Digital testing of the moiré effect in animation

The above digital animation illustrates the rotational movement of a circle through the movement of a linear overlay, created with the two static images below:

Moire Overlay   The linear moiré overlay       Resultant Shape

The moiré underlay  creating the circular motion 

This moiré underlay is created through a series of rules defined by the size of the overlay and the direction, factor and type of movement (linear or rotational). The diagram below explores the rules associated with this specific type of moiré animation.

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Rules for defining a moiré ‘underlay’ for linear animation

Whilst primarily a visual effect it is the ability to translate spatially which gives the moiré effect the potential to be applied in a design context, particularly given it’s interactive nature and the reliance on the involvement of participants in order to reveal it’s true beauty.

The video below takes this concept to the extreme, exploring the effects of imagining matter as nothing more than multi-dimensional moiré patterns……

Moiré – Julias Horsthuis

Updated Research:

Video illustrating various physical moiré experiments

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Rules for defining moiré patterns in linear gratings

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Mathematical rules for defining moiré patterns of rotation

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Physical model for experimenting with moiré rotation patterns

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Results from the physical model using sin curves & square gratings

Moiré patterns can be ‘programmed’ using a certain mathematical formula. If two variables are known; the base layer and the desired moiré pattern (in this instance a sin curve) the resultant reveal layer can be determined, allowing moiré patterns to be programmed to any shape.

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Digital tests and physical proofs of programming moiré

Moiré patterns work in both ‘positive’ and ‘negative’ constructions. Positive moiré can be classed as additive, constructing patterns consisting of lines to create the effect. Negative moiré conversely removes elements of material (in this instance circles from card) to create patterns when held at a distance. The bottom row of images shows the most successful variables for discerning negative moiré patterns.

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Negative moiré, set-up & physical experiment

The above experiment was digitally reproduced, modelling its negative space in order to understand how the variables of distance affect the reception of pattern.

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Digital experiments with distance variables

In order to move from the plane into a spatial exploration of the moiré effect, sin curve gratings were mapped onto the faces of a cube, at varied rotations. The effect is a spatial understanding of moiré patterns when the various faces of the cube overlap. The moiré effect can be created by two distinct methods; a movement by the user, distorting the areas of overlap and the movement of the cube itself, visually shifting patterns.

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Physical model exploring moiré patterns in three dimensions