# Category: sculpture

## Developing Space-Filling Fractals

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

### Foreword

*…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**.

**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

**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.

**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:

**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..)

### 4.2 Later Examples

**how much space they fill when iterated infinitely**.

**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.

**Nicolino’s Quatrefoil**.’ The following was created in

*Rhino*and

*Grasshopper,*in conjunction

*Anemone*.

#### On A Strange Note:

**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??**

*…*“

**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

**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

**passing only through the ones we choose**. This now becomes a well documented computational problem that has immediate ‘real world’ applications.

**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.

**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

**more organic and less controlled**.

*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**.

*Rhino*&

*Grasshopper – Kangaroo 2 –*Planar

**any two-dimensional surface**(or in three-dimensions (even higher spacial dimensions I guess..)).

*Rhino*&

*Grasshopper – Kangaroo 2 –*NonPlanar

*Rhino*&

*Grasshopper – Kangaroo 2 –*Single-Curved Stanford Rabbit

*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.

*Rhino*&

*Grasshopper – Kangaroo 2 & Anemone –*Octopus

*Rhino*&

*Grasshopper – Kangaroo 2 & Anemone –*BatmanDuck

# 6.0 Developing Fractal 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).

**a two-dimensional surface seamlessly transforming into a three-dimensional volume**.

### 6.1 Dragon’s Feet

**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*.

*Dragon’s Feet*.’

### 6.2 Hilbert’s Curtain

*Hilbert’s Curtain*.’

### 6.3 Developing Whale Curve

**this method fills a specific space in a less controlled manner**.

*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.

## Duality

This project is a physical exploration of anamorphosis in three dimensions centred around the theme of duality. It aims to combine two widely recognisable figures into a pavilion that will attract burners, provoke debate, and catalyse interaction.

The theme of this project arose from the realisation that even the most widely recognisable symbols contain multiple layers of meaning and mystery. Social, historical and sometimes even spiritual contexts give a symbol its perceived meaning. For example, while the Christian cross is a symbol of hope it is literally a scaled representation of an ancient torture device – an icon synonymous with good carries with it a darker elucidation. This interpretation led to the emergence of *duality* as a topic and a title. There are many symbols which have multiple meanings and nuances to those who interpret them.

I began by looking at the Ankh, the Egyptian symbol for life/fertility. The Loop of the Ankh represents the feminine discipline or the womb, while the elongated section represent the masculine discipline or the penis. These two sacred units then come together and form life. This is a perfect representation of man and woman in perfect union. I then was led to study the symbol for mercury, which is used in botany to indicate a flower with both male and female reproductive organs.

This duality of meaning in symbols led me to the desire to study how I could physically combine other symbols and forms to create one form. Anamorphosis, from the Greek *anamorphōsis* meaning ‘transformation,’ from *ana-* ‘back, again’ + *morphosis* ‘a shaping’, became an interesting opportunity to do just this.

I want to explore this theme using the iconic faces of Donald Trump and Kim Kardashian as instigators. From a random vantage point or even from up close, the subject matter of the piece is evidently unclear, the image changes until the viewer arrives at a specific pre-set location, only then does the likeness reveal itself. This echoes our warped perception of figures in limelight; anything the media choose to present to the world is an engineered production and if taken out of its context it becomes incomprehensible. My aim is to stir ambivalence among the burners, for them to engage in discussion with one another about these two incredibly famous personalities and what they seemingly represent.

As a physical entity, the sculpture is purposefully made durable enough to be able to endure the brunt of any elicited reactions. Its exposed surfaces are smooth, an open invitation to graffiti, carve or deface in any manner possible. It is large enough to climb and to gather within as a group – it only takes a spontaneous suggestion from a creative festival goer to give the sculpture another unforeseen use.

The aim of my proposed sculpture is to provoke an exchange of opinions and interactions between *burners*. It depicts two iconic and highly controversial public figures who personify two tremendously important issues that we as a society face today; political and social change.

As festival goers approach the installation, and the two widely recognisable faces reveal themselves, comments about the likenesses will spiral inevitably highlighting or at least touching upon the shift that these two personalities represent.

The sculpture’s physical form comprises of several spatial elements that lend themselves to fostering the kind of debates that I wished to promote. The hollow centre creates an enclosure, to enable hosting or housing for a meeting, it gives its participants a sense of protection; *this is an open forum, please take part*. The raised base on the peripheries can act as stages or podia. The expansive smooth external surfaces can act as billboards or banners, the skin of the sculpture will bear the physical outcome of the issues discussed here.

Whether people get photographed with it, or whether they deface, damage or even burn it to the ground, I will have succeeded if among any of the interactions the agenda was heard and a heartfelt reaction was made.

The sculpture will be made of 8mm CNC routed plywood sheets fixed to a heavy plywood formwork. Standing at 6m tall, one side will represent a 25:1 scale stencilled portrait of president-elect Donald Trump, the other side; the likeness of reality television personality and socialite Kim Kardashian. Much like the oblique anamorphosis incorporated in Holbien’s *The Ambassadors*, the sculpture’s subject matters will reveal themselves only from some 60m away, but from close up, the installation will seem like a mass of abstract wooden extrusions, something suggestive of an adult-sized climbing frame. Fluorescent LEDs recessed into junctions of the outer plywood skin layer will illuminate the piece at night.

The pavilion achieves the incredible feat of allowing the viewer to have a personal and intimate connection with it whilst also allowing for reflection. The two images are intended to bring moments of delight to viewers to allow for interaction even from a distance.

Combined with its symbolic and evocative power, it should indeed conjure a deeper sense of place and self, and bring a subtlety and complexity to what might have been just another pavilion.

## Sine Curve Orderly Tangle

In the 1970s and 1980s Alan Holden described symmetric arrangements of linked polygons which he called regular polylinks or orderly tangle. The fundamental geometric idea of symmetrically rotating and translating the faces of a platonic solid is applicable to both sculpture and puzzles.

The process started with making a frame out of the geometry; in this case a cube. All 6 faces are moved inwards with the central point of the original cube is used as the origin axis.

Using the same origin, the faces of the geometry are then rotated along their axis at a certain degree to create the orderly tangle.

The faces are then thicken to ensure all of them fixed together.

Using the method as stated before, an icosahedron is used and different length of movement and degree of rotation is used to suit the shape.

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*Some images can be scanned using augmented reality apps called Augment. *

*Links to Augment apps:*

*iOS: https://itunes.apple.com/us/app/augment/id506463171*

*Android: https://play.google.com/store/apps/details?id=com.ar.augment*

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In this experiment, the edges of the icosahedron is replaced with sine curve.

In this experiment, the edges of the icosahedron is replaced with triangular curve.

In this experiment, the edges of the icosahedron is replaced with steps curve.

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The icosahedron sine curve edges is used to continue with further design. The original sine curve is manipulated using grasshopper to enable the shape to intertwine through itself and interlock without major intersection. This provide more ways to control the curve and makes it easier to assemble.

A small model is built to see how it holds together.

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After making the first small scaled model, i started to study on a more efficient jointings needed for the sine curve component as well as the interlocking component needed to connect the face together.

A medium sized model is built with the new jointing design.

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The sine curve polylinks created an icosahedron space on the inside. Each triangle face of the icosahedron corresponds to the sine curve geometry due to the initial process of replacing all the edges with sine curves.

In icosahedron, there is always surface that pairs in a parallel to each other, in this case 10 pairs of the 20 triangle faces. Based on this, i tried to use the surface as a floor plate for the structure. The whole geomtery is rotated so that one of the surface lays flat on the ground. The excess part is then removed.

The section shows the space inside with one of the triangle face acts as a floor plate.

—————————- EXPERIMENTS —————————-

Different experiments were carried out using the system as the basis for design. The experiments focus more on a different form other than the spherical nature of the system.

In this experiment, the polylinks is divided into two halves (each half contains 10 modular shape) and the bottom half is move to the side on the x- axis while still intertwine with the top half portion. Due to the adjustment, the bottom half is also slightly moved up on the z-axis to ensure no major intersection. This creates a more elongated structure with the system still intact. The process is repeated with each time the geometry still intertwine between the top and bottom half.The process is then repeated along the y-axis to create a planar design based on the shape.

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In this experiment, the polylinks is divided into two halves (each half contains 10 modular shape) and the bottom half is removed. The top half contains two component face that is in the same plane but different angle. These two will be used as a sharing planes to array the whole structure. The top half structure is then copied to the adjacent with the parallel face is lined up. The structures will intertwine at the sharing planes helping it to stay in place. The process is repeated with each time the geometry still intertwine at the sharing planes on each iteration.

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In this experiment, using one of the component face as a floor plate, the structure is rotated to lay the component face on the floor and all the excess (bottom) are removed. The opposite component face, which is in parallel to the one used as floor plate, will be used as the second floor plate. All the excess (top) are removed as well. The structure is then mirrored along the x and y plane to get a tower shape structure. The trimmed part where the excess are removed will connect with the new mirrored structure making them all connected.

—————————- BURNING MAN PROPOSAL —————————-

Desert Petal