Foreword
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:Â 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:Â 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 aim is to generate an architectural response through a playful loop between the digital and the physical. Digital tools such as Rhino and Grasshopper are used  in order to carry out analysis and generate buildable three-dimensional forms. Interplay between physical fabrication and digital experiments enable to become an inventor of a system. Here is mine.
TriNect is a flexible system of triangular elements with slots at their vertices. Elements interlock with one another creating different space filling polyhedra. The system can be applied in various scales and adapted for different needs.