4.0 Classic Space-Filling
4.1 Early Examples
In 1890, Giuseppe Peano discovered the first of what would be called space-filing curves:
Delving deeper into the world of mathematics, fractals, geometry, and space-filling curves.
In 1890, Giuseppe Peano discovered the first of what would be called space-filing curves:
something caught in between dimensions – on its way to becoming more.
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
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.
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.
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:
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.
First, second and third dimensions, and why fractals don’t belong to any of them, as well as what happens when you get into higher dimensions.
In physics and mathematics, dimensions are used to define the Cartesian plains. The measure of a mathematical space is based on the number of variables require to define it. The dimension of an object is defined by how many coordinates are required to specify a point on it.
Something of zero dimensions give us a point. While a point can inhabit (and be defined in) higher dimensions, the point itself has a dimension of zero; you cannot move anywhere on a point.
Surfaces or plains gives us two-dimensional shapes, and are typically bound by one-dimensional shapes (lines/curves).
A plain can be defined by x&y, y&z or x&z; more complex surfaces are commonly defined by u&v values. These variable are arbitrary, what is important is that there are two of them.
A volume gives us a three-dimensional shape, and can be bound by two-dimensional shapes (surfaces).
Fractals can be generally classified as shapes with a non-integer dimension (a dimension that is not a whole number). They may or may not be self-similar, but are typically measured by their properties at different scales.
Felix Hausdorff and Abram Besicovitch demonstrated that, though a line has a dimension of one and a square a dimension of two, many curves fit in-between dimensions due to the varying amounts of information they contain. These dimensions between whole numbers are known as Hausdorff-Besicovitch dimensions.
Surfaces give us two-dimensional shapes, where two coordinate are required to define a point on them.
If you were to walk along the shape starting from the base, you could go forwards and backwards, but suddenly you have an option that’s more than forwards and backwards, but less than left and right.
You cannot define a point on this shape with a single coordinate, and a two coordinate system would define a point off of the shape more often than not.
Surfaces give us two-dimensional shapes, where two coordinate are required to define a point on them.
A volume gives us a three-dimensional shape where a point could be defined by no less than three coordinates.
While these models live in three dimensions, they do not quite have access to all of them. You cannot define a point on them with two coordinates: they are more than a surface and less than a volume.
That being said, it’s plausible that we experience much higher dimensions that are just too hard to perceive. For example, an ant walking along the surface of a sphere will only ever perceive two dimensions, but is moving through three dimensions, and is subject to the fourth (temporal) dimension.
If we explore spacial dimensions, a four-dimensional object may be achieved by ‘folding’ three-dimensional objects together. They cannot exist in our three-dimensional world, but there are tricks to visualise them.
We know that we can construct a cube by folding a series of two-dimensional surfaces together, but this is only possible with the third dimension, which we have access to.
If we visualise, in two dimensions, a cube rotating (as seen above), it looks like each surface is distorting, growing and shrinking, and is passing through the other. However we are familiar enough with the cube as a shape to know that this is simply a trick of perspective – that objects only look smaller when they are farther away.
In the same way that a cube is made of six squares, a four-dimensional cube (hypercube or tesseract), is made of eight cubes.
It looks like each cube is distorting, growing and shrinking, and passing through the other. This is because we can only represent eight cubes folding together in the fourth dimension with three-dimensional perspective animation.
Perspective makes it look like the cubes are growing and shrinking, when they are simply getting closer and further in four-dimensional space. If somehow we could access this higher dimension, we would see these cubes fold together unharmed the same way forming a cube leaves each square unharmed.
Below is a three-dimensional perspective view of hypercube rotating in four dimensions, where (in four-dimensional space) all eight cubes are always the same, but are being subjected to perspective.
(For example: x, y, z, t1, t2, t3)
This is a space where one can move through time based on probability and permutations of what could have been, is, was, or will be on alternate timelines. For any one point in this space, there are six coordinates that describe its position.
There’s a terrific explanation of what happens to platonic solids and regular polytopes in higher dimensions on Numberphile: https://youtu.be/2s4TqVAbfz4
Dimensions seven though ten are different universes with different possibilities, and impossibilities, and even different laws of physics. These grasp all the possibilities and permutations of how each universe operates, and the whole of reality with all the permutations they’re in, throughout all of time and space. The highest dimension is the encompassment of all of those universes, possibilities, choices, times, places all into a single ‘thing.’
These ten time-space dimensions belong to something called Super-string Theory, which is what physicists are using to help us understand how the universe works.
While we don’t have experimental or observational evidence to confirm whether or not any of these additional dimensions really exist, theoretical physicists continue to use these studies to help us learn more about how the universe works. Like how gravity affects time, or the higher dimensions affect quantum theory.
‘The legend of a dream catcher began long time ago, when the child of a Woodland chief fell ill. Unsettled by fever, the child was plagued with bad dreams and unable to sleep. In an attempt to heal him, the tribe’s Medicine Woman created a device that would ‘catch’ these bad dreams. Forming a circle with a slender willow branch, she filled the centre with sinew, using a pattern borrowed from our brother the Spider, who weaves a web. This dream catcher was then hung over the bed of the child. Soon the fever broke, and the child slept peacefully.
It is said that at night, when dreams visit, they are caught in the dream catcher’s web, and only the good dreams are able to find their way to the dreamer, filtering down through the feather. When the warmth of the morning sun arrives, it burns away the bad dreams that have been caught. The good dreams, now knowing the path,visit again on other nights.’ (Oberholtzer, 2012, p9).
Dreamcatchers originated with the Ojibwe, a tribe of Native Americans scattered throughout the areas of the lake country in northern Michigan, Wisconsin, and Minnesota, and along the southern border of Canada, along the shores of Lakes Huron, Superior and Michigan, whose survival relied on fishing, hunting and trapping.
Traditionally, the dream catchers were made by tying sinew strands onto a few inches in diameter round or tear-shaped frames of willow and were often wrapped in leather.
The spiritual life of the Ojibwa centred around the Midewiwin, the Grand Medicine Society and focused on the individual spiritual growth, gaining the insight through their dreams or visions.
My project is a re-interpretation of the beliefs that dreams have magical qualities with the ability to change or direct one’s path in life. The bawaajige nagwaagan intends to create a mystical experience, where people are caught inside, similar to the way that bad dreams are caught in the dreamcatcher’s web, and good dreams escape through the centre. The participants are encouraged to climb through the centre and escape their bad dreams and feelings, releasing their spirit through the enclosure. Now they can sleep in the peaceful environment, stimulated by the fantasy of glowing feathers and luminescent rope structures. The pavilion aims for people to sleep, relax and free themselves from stress while being protected by the magical webs of the dream catcher.
The proposal is a celebration of the romantic essence of the Native American Culture. The large scale, three dimensional net is inspired by the native methods and techniques of making dream catchers. It is a manifestation of the traditions and significance of the Native Americans, paying respect and pledging support to the indigenous people of America.
The structure situated in the Burning Man festival commemorates the ceremonies of Native Americans, dedicated to acquiring an insight through dreams and visions. Fasting, or giving up of certain necessities for a certain length of time was a common practice used to enhance one’s ability to access different dreams or visions. Another method was to pour water over hot rocks to produce steam, which enhanced the occurrence of dreams, used as source of introspection. These rituals relate to the festival’s assertion of disconnecting from the necessities of our contemporary world, supplemented by the extreme weather conditions, which are hoped to encourage reflection.
The pavilion responds and works together with the Black Rock desert’s environment, and adds to the wider cultural context of leaving behind the essentials and expectations of the contemporary world while creating a moment for contemplation and tranquility in the magical weaves of the dream catcher.
The structure will be composed out of three, seven meters in diameter, dream catchers, tilted to form a tetrahedron. Each dream catcher’s net will be made out of 275 meters, 18mm, synthetic hemp rope which will be entwined in 1320 meters of 3mm fluorescent cord. Attached to the frame uv lights will make the fluorescent rope glow at night. Three rings hold the net structure together, with the bottom ring anchored to the ground, made out of T-shape plywood frames. The web of the frame will be 4 layers of 15mm ‘banana’ shaped pieces which will create a circle, together with 4 layers of 230mm x 2400mm x 9mm flange pieces bent in shape of the banana edge. Smaller rings, supporting the centre of the dreamcatcher net will be of similar structure, with 2 layers of banana pieces and 2 layers of 150mm x 2400m x 9mm flange pieces, bent in shape. The frame will be wrapped in 13500 meters of 8mm synthetic hemp with attached fluorescent fabric feathers.
Assembly of the net is inspired by a macrame knotting technique rather than weaving which means that the net could be made out of smaller 15 meters long pieces, rather than one 275 m coil of rope, making it easier to assemble and repair. Rope is anchored to the frame with thimbles and shackles, attached to the bolted staple on the plate. The rope is connected with simple S-shape stainless steel hooks. After testing the net I found that although these are easy to assemble, they can create some movement in a connection, therefore I am planning on exploring the idea of ferrules, which could be crimped in place.
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.
Love at Burning Man
The Heart is an internationally known Symbol for Love.
The Love Nest is a Pavilion designed for Burning Man as a destination to express your Love to another. Wedding ceremonies will take place within the Heart structure, this is the biggest gesture of Love, joining together as one, declaring your Love for everyone to see
The wooden hearts floating toward the sky create a Tower of Love
The origin of the Heart Symbol
Symbols as a System
At Burning Man within the Temple (also known as the Temple of Love) people leave messages to remember there loved ones that they have lost, the temple is then burnt so the messages can get to those who are being remembered. The Idea of the below design was inspired by a book, being a place of words, people add to the pages of the book of love.
Every time someone is married within the Tower a coloured heart is added to the Structure.
Developing the design to be more fluid and natural, as love isn’t hard and spiky.
Looking at different ways of bending wooden Hearts to be able to work with the newly designed form and being able to attach to the ‘Ribs’ of the design. I looked at two ways of achieving this:
By laser cutting a line pattern into the heart I achieved a material that bends easily
Cutting a slit from the bottom of the heart to the centre and taking the two half and bending one on top of the other
Results – The Laser Heart bent easily but did not stay in place and became more fragile whereas the folded heart keeps its shape and is a more solid form
Looking at what effects the curvature of the heart.
Variable – Length of cut for bending
Results – The Longer the cut the smoother the curve
The Final Design looks at creating a form by connecting the folding hearts, removing the structural ‘Ribs’ from the previous design and creating a system to achieve a Form
A small selection of photos from DS10s unit trip to Southern Spain.
For more Images follow: http://issuu.com/josh-haywood/docs/bewk?e=10839685/7097420