1 Traditional Dimensions

 

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 surface can live in three+ dimensions, but still only possesses two dimension. Two coordinate are required to define a point on a surface. For example a sphere is a three-dimensional object, but the surface of a sphere is two-dimensional – a point can be define on the surface of a sphere with latitude and longitude.

 

1.4 – Three Dimensions

2.1 – Between the First & Second Dimensions

A line or a curve gives us a one-dimensional object that allows us to move forwards and backwards, where only one coordinate is required to define a point on them.

Surfaces give us two-dimensional shapes, where two coordinate are required to define a point on them.

Here is a shape that cannot be classified as a one-dimensional shape, or a two-dimensional shape. It can be plotted in two dimensions, or even three dimensions, but the object itself does not have access the two whole dimensions.

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.

Each fractal has a unique dimensional measure based on how much space they fill.

 

2.2 – Between the Second & Third Dimensions

3 Higher Dimensions

Sadly, living in a three-dimensional world makes it especially difficult to think about, and nearly impossible to visualise, higher dimensions. This is in the same way that a two-dimensional being would find it impossibly hard to think about our three-dimensional world, a subject explored in the novel ‘Flatland’ by Edwin A. Abbott.

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.

 

3.1 – The Fourth Dimension (Temporal)

If we consider time an additional variable, then despite the fact that we live in a three dimensional world, we are always subject to (even if we cannot visualize) a fourth dimension.

Neil deGrasse Tyson puts it quite plainly by saying:

“[…] you have never met someone at a place, unless it was at a time; you have never met someone at a time, unless it was at a place […]”

Suppose we call our first three dimensions x, y & z, and our fourth t: latitude, longitude, altitude and time, respectively. In this instance, time is linear, and time & space are one. As if the universe is a kind of film, where going forwards and backwards in time will always yield the exact same outcome; no matter how many times you return to a point in point time, you will always find yourself (and everything else) in the exact same place.

However time is only linear for us as three-dimensional beings. For a four-dimensional being, time is something that can be moved through as freely as swimming or walking.

 

3.2 – The Fourth Dimension (Spacial)

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.

• A line is bound by two zero-dimensional things
• A square is bound by four one-dimensional shapes
• A cube is bound by six two-dimensional surfaces
• A hypercube, bound by eight three-dimensional volumes

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.

 

We are back after a year exploring Symbols & Systems, and an inspiring unit trip to South India, visiting the Hempi Valley and Auroville. This year our focus is on Fractals, not just as forms but as tools to understand how geometry can become infinite and how it can be built within the constraints of the physical reality. Fractals gives the opportunity to expand confined spaces, to let the mind fill the gap that reality had to stop. Therefore it also provides a great tool for the second brief, which is the Tiny Home movement, society’s need to create more compact, efficient homes to face the environmental and economical crisis. As per our previous briefs, we would like our students to build their projects, whether it is a giant fractal at a festival or an actual home within a space that would otherwise be left empty, we want students to raise funds and make, using digital fabrication tools combined with off-the-shelf material. Our goal is to continue training the entrepreneur-makers of tomorrow. Below is a breakdown of our briefs as they are being drafted:

• Studio A4 describing the briefs
• Slide Show Unit Presentation

Diploma Studio 10 is back with 21 talented architecture students from 4th and 5th year working on the Brief01:Fractals. Here is an overview of their experiments so far after 4 weeks of workshops.

The Fractal Hourglass counts down to the singularity, the moment that artificial super-intelligence triggers an unprecedented shift in human civilisation. The concept of recursively self-improved AI is portrayed by a tower of iterated fractal trusses, in which time is measured by a cascade of light.

Triangular steel trusses array to form a 15-foot tall hourglass silhouette, where scaled repetitions within each truss form a lattice of increasing complexity and infinite bounds. The visual density of each truss intensifies at each fractal iteration, culminating in the filling of the lower hourglass bulb, representing the finite time remaining until the singularity. At night, a dynamic cascade of LEDs will flow on and off from the upper to the lower bulb, a spectacle alluding to sand pouring through an hourglass.

The steel tubes forming the piece range from a diameter of 1.5″ in lengths from 1 to 3-feet, which are hammered flat and bolted to form the main structure, and 0.5″ diameter tubes welded inside to form the decorative fractal repetitions.

On approach, the tense drama of time running out is visible through the concentration of material in the bottom of the hourglass, provoking an instinct to stall the process. Burners have a choice of how to experience the hourglass- whether that is to ascend the structure to experience the inversion of the hourglass as the bulb empties, where ascension serves as a sanctuary from the saturation of technology and AI in the lower bulb. Or they can recline on the ground and let their eyes weave through the layers of trusses and bathe in the saturation and complexity of technological advancement. Or simply to turn away and let what effectively has become a natural process to take its course. At night, the cascading light display forms an even more immersive encounter with the hourglass, as waves of light repeat the process of time as it funnels through and fills the lower bulb, swarming anyone who is inside.

The finite nature of fractals in the hourglass represents the capacity for infinite artificial intelligence- each increment provides an equally stable steel structure, whilst having the capacity to use less and less material, but only to a point. It is not possible for this fractal to reach infinity and be constructed at a human scale. This poses the question of, at which point on the way to infinity do humans get before their intelligence can be overtaken by AI- the moment of the singularity. Is it too late to invert the hourglass and, given the choice, would you want to?

The Fractal Hourglass allows for Burners to take a moment to relish on their existence as humans, with the capacity to orchestrate their own experience, something which AI’s currently don’t possess. Artificial intelligence is currently an opportunity to shape a future experience where humans can outsource themselves, freeing up valuable time and energy. The hourglass serves as a visual symbol that human existence is fleeting so long as AI is permeating our lives, and provides a timer for the impending singularity, a moment that will transform the world as we know it, a reminder that we still have the alluring capacity to define and create.

‘The first ultra-intelligent machine is the last invention that man need ever make, provided that the machine is docile enough to tell us how to keep it under control.’

I J Good

The idea that nature can inspire us to create robotic aided designs that have a positive impact on humans is something that fascinates me. Ever since I learned of the Stenocara Beetle’s unique ability to harvest water from the air, I knew I had to produce a design inspired by this. I hope this aspect alone will create some wonderment, questioning and reflection around what I personally deem to be one of the biggest crisis facing the planet: dwindling water supply.

 

A 13.13.13 feet structure made from light, translucent, flexible polypropylene material. The structure follows a complex geometric form made of Borromean rings and Mobius strip which consist of a single surface winding along three double ellipses and features a unique textured surface throughout.

Assembly always takes place at the base of the project as the sheets are assembled piece by piece using a thermoforming process. These large bumps take on an important structural role in helping form the sheets together. The peaks of the building follow attractor points and also become a unique artistic installation that reflect light in a unique way at night.

Project Summary*

Moon rocks are about challenging the way we experience voids within confined geometries.Inspired by the nature of the sea sponge and its porosity, it is designed to stand as a contrasting installation that references natural geometries confined within a perfect cube, a shape achieved only by human intervention. Ideally the cube will be made out of the earth, naturally and locally sourced, binding naturally with the playa landscape. An internal geometry that looks like something that has been naturally formed by the passage of time, confined within the geometrical precision of the cube,  it is a manifesto of nature and manufacture, a very apparent, very human, extension of nature. A temple honoring the natural playa and celebrating our ability in erecting structures that cohere with nature. Also, they look like pieces of the moon!

Physical Description*

In order to achieve the complex geometry of the moon rocks, we will need to create a mould and fill it with a viscous structural material. I’m currently experimenting with plaster because plaster is just great at picking up detail!

Balloons of varying sizes are inflated to a preferable shape, bound, put in a box, anchored in position to prevent any displacement of the balloons and poured over with plaster. The viscous material that is used to fill the voids inside the balloon box may vary from plaster to concrete,to…anything that behaves like a liquid enough to crawl in all the crevices and eventually dries to solid form. When building on the playa, what better than to use the earth that we will find beneath our feet to create the giant porous sand cube. If it’s one thing that’s easy to find in the desert, its soil.

The project is envisioned basically as a porous sand cube achieved by pouring wet sand (with a bit of natural adhesive) in a 16x16x16 foot cubic mould, potentially built out of prefabricated timber, brought to the desert in a container and fixed together easily on site.Sand will be collected using buckets and shovels, then, a single layer prefabricated inflatable PVC skin, that is basically a bunch of inflatable spheres of various sizes, stitched together, will be placed in the mould, inflated and poured over with the wet sand mix. After the sand crawls in all the corners of the mould, the sand should be compressed to maximum density so it withstands standing under its own weight and the harsh desert conditions throughout the festival. After all is done, the spheres can be deflated or popped like balloons and removed along with the box mould, leaving behind the porous sand cube, or moon rock! When the festival is over, we can knock down the moon rock and spread the sand, returning it to where we found it. Another idea is to pour a mix of wood chips and adhesive to fill the mould instead of sand. Then it can be Burned at the end of the Festival!

Interactivity and Mission*

Moon rocks purpose is to question the way we create spaces, proposing learning from irregular natural geometries. It’s designed to create a clear juxtaposition between its fluid internal structure and its linear cubic confinement and acts as a call to inspire integrating organic natural geometries within our strict linear manmade lives. But let my intent aside for a bit. Seeing an unusual giant sand cube, that looks like a piece of the moon, in the desert, just has so many interpretations, I’m sure the expression of the piece will be received in multiple forms.

Burners are invited to let go of their previous experience of structures by observing inside the cube and exploring its intricate internal form, discovering more and more interconnected naturally fluid spaces of varying sizes, the deeper within the cube their vision ventures. The piece offers shelter from the harsh weather conditions and a nice place to sit and rethink what kind of space we feel comfortable being in. Cubic or fluent? Simple or complex? Natural or artificial?

It’s an object that leaves the observer’s conclusion about the piece with the freedom to wander. But whatever the conclusion may be, I hope it will have to do with the beauty found in nature.

Philosophy the piece*

Creating a piece showing appreciation for the intricacy and beauty found in the natural world.

The philosophy behind the piece is one that suggests that we learn from naturally formed geometries when we create things. I aspire to make visitors rethink how we build, by creating an object that looks like something between a man-made object and a naturally formed shape. The Moon rocks are the embodiment of the threshold of the artificial and the natural. A structure that mimics nature but also imprisons it within a cube, expressing that we are still far from building like nature does and calls for humanity to realise the importance of building in coherence to our surroundings,  and work harder towards recreating nature.

Project Summary

(IN)Finitely Bound is a recursive fractal geometrical form, similar to that found in nature. It symbolises the universe and its finite boundary, and is an expression to show us the limitations to which technology can take us. As nothing can be bigger and more powerful than the universe.

(IN)Finitely Bound is a recursive fractal Dodecahedron form, consisting of lengths of 2 by 4 timber held into place using bolts and metal plate joints. The structure will be fully burnable and will be both approximately 7m high and wide. On approach to the piece the structures beauty will be be hard to work out, symbolising our confusion with the colossal scale of the universe but as you get closer you realise the receptive nature of the form and come rest on the structure and understanding as we zoom in on aspects which make up the universe and include ourself within it. The piece will be lit up to gently to allow for meditation, contemplation and open our bodies up to mindfulness.

Interactivity and Mission

The initial singularity was a singularity of infinite density thought to have contained all of the mass and space-time of the Universe. The standard model of cosmology predicts that the universe is infinite and flat. However, cosmologists in France and the US are now suggesting that space could be finite and shaped like a dodecahedron instead. They claim that a universe with the same shape as the twelve-sided polygon can explain measurements of the cosmic microwave background – the radiation left over from the big bang – that spaces with more mundane shapes cannot. In a world where “computational power is increasing exponentially, much like the singularity which created the universe” realising our own finite boundaries is where we take power back from the robots and become masters of our own minds, bodies and universe. The piece through its self-replicating fractal structure creates a dodecahedron (Universal) Boundary defined by perspective. In defining our boundary we are then able to instead of focus our mind inwards, a symbol towards mindfulness.

Project Summary

The Amazing Surf is a complex fractal geometry which ascends toward the light, symbolizing our obsession with reaching for the stars. We use our increasingly digital world to help us extend our reach, but at what point do the shadows we cast reach out above us?

Physical Description

The shape is inspired by the Amazing surf fractal which is generated by a mathematical formula and visualized in Mandelbulb3D. A visually imposing 25ft tall Ply wood hyperbolic structure, with intricate evolving folded panels. Each folded panel is digitally unrolled into a 2D net and CNC milled, the resultant ply components will be glued to a layer of fabric and folded back to their original 3D shape. This construction technique removes the need for a supporting frame, keeping the complex geometry unobstructed from view. A few panels have been removed at the base to make way for an entry point. Neon strips attached to each panel will produce dramatic light patterns on the surface at night. The installation will orient toward the sunset, where the sun appears at it’s closest.

Interactivity & Mission

The piece is intended to be used as an impromptu climbing frame, a ladder to ascend burners above the desert and into the stars. Sunlight will bounce off the multi-faceted shapes, creating intricate patterns of light and shadow. Burners are invited to dance in the light shafts and seek shelter in the shadows. As the shape begins to flatten toward the top, the folded panels can be used as armchairs, where vision will be limited to that of the sky and light above; burners can sit and watch as the sky transforms from day to night.

Philosophy

“Keep your face always toward the sunshine – and shadows will fall behind you”

-Walt Whitman

As a race we strive to advance, developing new tools and machines to help us in this process. There will come a point in the not too distant future where the machines we have developed to help us will supersede us; we will become so reliant on technology, it will begin to control us. I see the Amazing Surf installation as a juxtaposition to this potential future; on the one hand we are using technology to create built environments that are intricate, beautiful and unique, on the other hand these environments are only attainable through the use of technology. If only we took a moment to look back into the shadows, we could avoid the fate that we are gradually bringing upon ourselves.

Omnis Stellae – Redrawing your own constellation

“Only in the darkness can you see the stars”

Martin Luther King

This project involves the conception and design of a new way of mapping constellations, based on subdivision processes like Stellation. It explores how subdivision can define and embellish architectural design with an elaborate system of fractals based on mathematics and complex algorithms.

An abstracted form of galaxy is used as an input form to the subdivision process called Stellation. In geometry, meaning the process of extending a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure.

The material used for this installation will be timber sheets of 1/3 of an inch thickness that will be laser-cut.The panels will be connected to each other with standard connection elements which have already been tested structurally based on an origami structure.

The lighting of the installation will consist on LED strips that will light with burners interactions.

Although stars in constellations appear near each other in the sky, they usually lie at a variety of distances away from the observer. Since stars also travel along their own orbits through the Milky Way, the constellation outlines change slowly over time and through perspective.

There are 88 constellations set at the moment, but I would like to prove that there are infinite amount of stars that have infinite amount of connections with each other.The installation will show you all the possible connections between this stars, but will never rule which connection is the one you need to make.

I would like burners to choose their own stars and draw their own constellations. Any constellation that they can possibly imagine from their one and only perspective, using coloured lights that react to their touch.

The end result will have thousands of different geometries/constellations that will have a meaning for each one of the burners and together will create a new meaningful lighted galaxy full of stars.

On a clear night, away from artificial light, it’s possible to see over 5000 stars with the naked eye. These appear to orbit the Earth in a fixed pattern, as if they are attached to a giant sphere that makes one revolution a day.This stars though are organised in Constellations.

The word “constellation” seems to come from the Late Latin term cōnstellātiō, which can be translated as “set of stars”. The relationship between this sets of stars has been drawn by the perspective of the human eye.

“Omnis Stellae” is a manifestation of the existence of different perspectives. For me, there is great value in recognising different perspectives in life, because nothing is really Black and White, everything relates to the point of view and whose point of view and background that is.

As a fractal geometry this installation embodies an endless number of stars that each person can connect and imagine endless geometries, that will only make sense from their own perspective. The stellated geometry will show you all the possible connections but will never impose any.

“Omnis Stellae” is about creating your own constellations and sharing them with the rest of the burners, is about sharing your own perspective of the galaxy and create some meaningful geometries that might not mean anything to other people but would mean the world to you.

The grand finale is if it could become the physical illustration of all the perspectives of the participants at Burning Man 2018 shown as one.

With Love,

Maya

Project Summary

Crystal is lost within a sea of flashing lights. She is surrounded by a 6.4×12.8ft cubic lattice structure. She reacts to motion and touch. Walk through the interactive cube that holds Crystal hostage. Can you find her before she fades, and becomes one with the cube?

Physical Description

Finding Crystal is an interactive installation that takes the principles of Crystallography and Bravais Lattices, and uses these principles to create a structural lattice cube. This cube is made of 0.4×0.4ft cubic pieces, fixed on top of, and next to each other to create a 6.4×12.8×12.8ft cube. I have chosen to use a cubic body centered lattice, removing the given structural frame, to allow the internal arrangement to determine the overall structure. The structure is made of steel spheres, steel rods and motion sensor LED lights. Each 0.4×0.4ft structure holds 3 spheres, 4 rods and 3 LED light. These pieces are put together in a puzzle manner to create the entirety of the structure. The art piece is modular and can be easily assembled on site.

Interactivity & Mission

Finding Crystal takes principles from science and combines it with design to create a truly interactive art piece. It uses the science behind crystal lattice structures to produces a structural modular that encompasses the principles of crystallography. Each modular is sized specifically to produce a lightweight structure, reflecting back on the idea of a crystal flake and its lightweight properties.

The entire structure consists of 16384 0.4×0.4ftcubic lattice pieces, with 49152 LED lights. This creates a platform for the experience. The cube is programmed to displays ghost like figures that walk “through” the cube, the figures use human motion to “follow” or “escape” the users. This evokes a form of playfulness between the figure, which is Crystal, and its users.

Philosophy

Crystal is a human. The cube is a symbol of an AI being that is attempting to deceive Crystal. Crystal is trying to escape the hold of the cube, but she is finding it extremely difficult to, the cube keeps misleading her into thinking that she is dependent on it. This piece explores addiction, in any form, by using the cube to confine the users perception of Crystal. Crystal doesn’t believe she can escape the cube, but ultimately, she is the only one that can help herself escape it.

I am deploying this piece in hope that it creates a curiosity behind the connection between technology, science and design. As well as creating a better understanding of someone who is in Crystals position. I’d like to evoke a desire to take these principles and explore with them, I want this piece to inspire and motive people to be creative. I want it to awaken the users curiosity. I want the user to want to explore, understand and interact with the artwork, and take this curiosity further by exploring their own artistic expression.

The intention of this piece is for it to be used as an interactive play experience. In the end, we are just trying to save crystal from herself.

The proposal reflects on immortality and how our lives would look like if we could reach it. Evolution has sentenced us to the process of aging, and ultimately to death, but as we understand it more and more, we may be able to outwit it. Sounds like paradise? Wouldn’t you want to be immortal?

The art installation is composed of cone shaped cells that divide itself creating new cells, which in turn develop into new ones and the process repeats. The components are made of laser cut, rolled thin sheets of plywood and are connected with metal screws. The structure, measuring approximately 20 feet long and 26 feet high, becomes stronger with every iteration, is structurally stable and self-supporting but on the other side almost invisible and very fragile in appearance. By joining the cone-like shaped cells, a set of domes at different scale is formed which are composed into pavilion serving as shelter to partially protect from sun and wind and casting beautiful shadows at the same time.

The pavilion is providing an opportunity to lay down, calm and contemplate. Look around and reflect on the surroundings – is it the blurred, crowded playa that attracts your attention? Or the cells of the structure that interest you? You have a chance to hide away for a moment and meditate. At night, the structure becomes illuminated from the inside, which highlights the pattern, casting even more beautiful shapes than during the day. You can move the bulb around and play with the light to explore different parts of the structure and look closer into the cells and how they divide themselves.

The concept was born during my research on fractals and their exploration through the Mandelbulb 3D software where by composing different formulas and changing their parameters, I could create beautiful, endless shapes. Infinity is one of the main feature of fractals, therefore, trying to materialize the experiments into physical models was the biggest challenge. To represent endlessness, I started looking at cell division and unicellular organisms, such as bacteria and paramecia, which multiply by dividing themselves. The duration of the cell ends with the division, but the line can be considered immortal.
The life span of a cell usually has specific limits due to telomerase and a separate genetic program of aging and death of complex organisms that evolved only about a billion years ago. Single-celled organisms that lived on Earth before that did not experience either aging or death and at a certain stage of maturity, they divided into two new cells. The first death occurred, when the sexual reproduction appeared – evolution has sentenced us to the process of aging, and ultimately, to destruction. However, recent developments in the field of physiology and medicine show that the elixir of life does not sound like a myth anymore and may become a reality in the future. And what if it becomes a reality? Does it scare you or does it make you happy? The aim of the proposal is to reflect on immortality and how our live would look like if we could achieve it.

Tiny house movement, which is a theme of DS10 second brief, is closely linked to the idea of self-build. It is commonly thought that both emerged simply as a response to housing crisis. However, compelled limitations can also encourage other unexpected movements. Here, I want to talk about how the self-build practice allowed discovering new possibilities that go beyond economy.

In 1980s, Berlin-born modernist architect Walter Segal proposed a solution to shorten the housing waiting list by allowing people to build their own homes. With the bold step of London borough of Lewisham, an ‘awkward’ piece of sloping land, that was found unsuitable for council’s programme, was donated for the experiment. People that got randomly chosen from the waiting list were allocated a site and given a basic induction in how to saw a straight line and drill a hole. Segal believed that a house should adapt to its occupants. Each household was invited to participate in the design stage, while the construction principles included lightweight timber frames and stilt foundations, meaning the layout could always be adjusted. New residents of Walter’s Way all worked collaboratively from the project commencement and as a result formed a tight community. At the end of the project, new occupants were given a chance to buy their homes. Walter’s Way became UK’s first self-built council housing project.

In spring 2016 I visited the site to interview residents for my undergraduate dissertation project. I was keen to understand how design principles of community influenced the level of happiness of it’s residents and therefore affected sustainable behaviour. Walter’s Way was a unique case study. Walking onto the street felt nothing like the rest of London and more like an eco-village in a countryside. You could sense a spirit of community that seemed to work great almost naturally. The road slopes down and main entrances to houses are oriented in such way, that you always see your neighbours as soon as they walk out their private spaces. The central core is used for weekly community activities and as a kids playground. All residents were happy to talk and during the conversation it was common to hear “I’m not as sustainable as my neighbour, but I learn from them”, which is almost like a good eco-competition. That is without saying that certain homes achieve carbon savings of 73% (according to SuperHomes).

Today, all houses are private with many owners occupying their homes for over 20 years. Everyone remembers the history of Walter’s Way and feels proud to be a part of it. Many continued the legacy of self-build by attaching extensions or upgrading buildings. The fact that all houses were designed by those in need and built with their own hands allowed for an activist community to be born. It not only challenges the traditional approach to solving housing issue, but creates new opportunities in the city by building on ‘abandoned’ sites, creating a new model for urban life and teaching others sustainable lifestyles.

Foreword

4.0 Classic Space-Filling

4.1 Early Examples

4.2 Later Examples

On A Strange Note:

5.0 Avant-Garde Space-Filling

5.1 The Traveling Salesman Problem

5.2 Differential Growth

6.0 Developing Fractal Curves

6.1 Dragon’s Feet

6.3 Developing Whale Curve

Foreword

7.1 – Single Curved Surfaces

Analytic geometry is created in relation to a Cartesian planes using mathematical equations in relations to coordinate systems. Synthetic geometry is essentially free-form geometry that isn’t defined by coordinates or equations. The following shapes were created via Synthetic geometry with the use of u and v curves.

These curves highlight each dimension of the two-dimensional surface. In this case only one of the two ‘curves’ is actually curved, making this shape developable. This means that if, for example, it was made of paper, you could flatten it completely.

In this case, one of them grows in length, but the other still remains straight. Since one of the dimensions remains straight, it’s still a single curved surface – capable of being flattened without changing the area. Singly curved surfaced may also be referred to as uniclastic or monoclastic.

7.2 – Double Curved Surfaces

These can be classified as synclastic or anticlastic, and are non-developable surface. If made of paper, you could not flatten them without tearing, folding or crumpling them.

In this case, both curves happen to be identical, but what’s important is that both dimensions are curving in the same direction. In this orientation, the dome is also under compression everywhere.

The surface of the earth is double curved, synclastic – non-developable. The surface of a sphere cannot be represented on a plane, a topic explored by Michael Stevens: https://www.youtube.com/watch?v=2lR7s1Y6Zig

This one was formed non-uniformly sweeping a convex parabola along a concave parabola. It’s internal structure will behave differently, depending on the curvature of the shell relative to the shape. Roof shells have compressive stresses along the convex curvature, and tensile stress along the concave curvature.

This shape was achieved by sweeping a straight line over a straight path at one end, and another straight path at the other. This will work as long as both starting lines are not parallel. Although I find this shape perplexing; it’s double curvature that you can create with straight lines, yet non-developable, and I can’t explain it..

 

8.0 Geodesic Curves

These are singly curved curves, although that doesn’t sound very confusing. A simple way to understand what geodesic curves are, is to give them a width. As previously explored, we know that curves can inhabit, and fill, two-dimensional space. However, you can’t really observe the twists and turns of a shape that has no thickness.

A ribbon essentially is a line with thickness, and when used to follow the curvature of a surface (as seen above), the result is a plank line. The term ‘plank line’ can be defined as a line with an given width (like a plank of wood) that passes over a surface and does not curve in the tangential plane and whose width is always tangential to the surface.

Since one-dimensional curves do have an orientation in digital modeling, geodesic curves can be described as the one-dimensional counterpart to plank lines, and can benefit from the same definition.

The University of Southern California published a paper exploring the topic further: http://papers.cumincad.org/data/works/att/f197.content.pdf

8.1 – Basic Grid Setup

For simplicity, here’s a basic grid set up on a flat plane:

We start by defining two points anywhere along the edge of the surface. Then we find the geodesic curve that joins the pair. Of course it’s trivial in this case, since we’re dealing with a flat surface, but bear with me.

We can keep adding pairs of points along the edge. In this case they’re kept evenly spaced and uncrossing for the sake of a cleaner grid.

After that, it’s simply a matter of playing with density, as well as adding an additional set of antagonistic curves. For practicality, each set share the same set of base points.

He’s an example of a grid where each set has their own set of anchors. While this does show the flexibility of a grid, I think it’s far more advantageous for them to share the same base points.

First comes the shell (a barrel vault vault in this case), then comes the grid. The symmetrical nature of this surface translates to a pretty regular (and also symmetrical) gridshell. The use of geodesic curves means that these gridshells can be fabricated using completely straight material, that only needs to bend in one dimension.

The same grid used on a conical surface starts to reveal gradual shifts in the geometry’s spacing. The curves always search for the path of least resistance in terms of bending.

This case illustrates the nature of geodesic curves quite well. The dome was free-formed with a relatively high degree of curvature. A small change in the location of each anchor point translates to a large change in curvature between them. Each curve looks for the shortest path between each pair (without leaving the surface), but only has access to single curvature.

Structurally speaking, things get much more interesting with anticlastic curvature. As previously stated, each member will behave differently based on it’s relative curvature and orientation in relation to the surface. Depending on their location on a gridshell, plank lines can act partly in compression and partly in tension.

While geodesic curves make it far more practical to fabricate shells, they are not a strict requirement. Using using non-geodesic curves just means more time, money, and effort must go into the fabrication of each component. Furthermore, there’s no reason why you can’t use alternate grid patterns. In fact, you could use any pattern under the sun – any motif your heart desires (even tessellated puppies.)

Here are just a few of the endless possible pattern. They all have their advantages and disadvantages in terms of fabrication, as well as structural potential.

Gridshells with large amounts of triangulation, such as Buckminster Fuller’s geodesic spheres, typically perform incredibly well structurally. These structure are also highly efficient to manufacture, as their geometry is extremely repetitive.  

Gridshells with highly irregular geometry are far more challenging to fabricate. In this case, each and every piece had to be custom bend to shape; I imagine it must have costed a lot of money, and been a logistical nightmare. Although it is an exceptionally stunning piece of architecture (and a magnificent feat of engineering.)

8.3 – Gridshell Construction

In our case, building these shells is simply a matter of converting the geodesic curves into planks lines.

The whole point of using them in the first place is so that we can make them out of straight material that don’t necessitate double curvature. This example is rotating so the shape is easier to understand. It’s grid is also rotating to demonstrate the ease at which you can play with the geometry.

This is what you get by taking those plank lines and laying them flat. In this case both sets are the same because the shell happens to the identicall when flipped. Being able to use straight material means far less labour and waste, which translates to faster, and or cheaper, fabrication.

An especially crucial aspect of gridshells is the bracing. Without support in the form of tension ties, cable ties, ring beams, anchors etc., many of these shells can lay flat. This in and of itself is pretty interesting and does lends itself to unique construction challenges and opportunities. This isn’t always the case though, since sometimes it’s the geometry of the joints holding the shape together (like the geodesic spheres.) Sometimes the member are pre-bent (like Pompidou-Metz.) Although pre-bending the timber kinda seems like cheating to me thought.. As if it’s not a genuine, bona fide gridshell.

9.0 Form Finding

Having studied the basics makes exploring increasingly elaborate geometry more intuitive. In principal, most of the shells we’ve looked are known to perform well structurally, but there are strategies we can use to focus specifically on performance optimization.

9.0 – Minimal Surfaces

These are surfaces that are locally area-minimizing – surfaces that have the smallest possible area for a defined boundary. They necessarily have zero mean curvature, i.e. the sum of the principal curvatures at each point is zero. Soap bubbles are a great example of minimal surfaces.

Soap film inherently forms shapes with the least amount of area needed to occupy space – that minimize the amount of material needed to create an enclosure. Surface tension has physical properties that naturally relax the surface’s curvature.

We can simulate surface tension by using a network of curves derived from a given shape. Applying varies material properties to the mesh results in a shape that can behaves like stretchy fabric or soap. Reducing the rest length of each of these curves (while keeping the edges anchored) makes them all pull on all of their neighbours, resulting in a locally minimal surface.

Here are a few more examples of minimal surfaces you can generate using different frames (although I’d like stress that the possibilities are extremely infinite.) The first and last iterations may or may not count, depending on which of the many definitions of minimal surfaces you use, since they deal with pressure. You can read about it in much greater detail here: https://tinyurl.com/ya4jfqb2

Here we have one of the most popular examples of minimal surface in architecture. The geometry of these domes was derived from a series of studies using clustered soap bubbles. The result is a series of enormous shells built with an impressively small amount of material.

Triply periodic minimal surfaces are also a pretty cool thing (surfaces that have a crystalline structure – that tessellate in three dimensions):

http://facstaff.susqu.edu/brakke/evolver/examples/periodic/periodic.html

9.2 – Catenary Structures

Another powerful method of form finding has been to let gravity dictate the shapes of structure. In physics and geometry, a catenary (derived from the Latin word for chain) curves are found by letting a chain, rope or cable, that has been anchored at both end, hang under its own weight. They are look similar to parabolic curves, but perform differently.

A net shown here in magenta has been anchored by the corners, then draped under simulated gravity. This creates a network of hanging curves that, when converted into a surface, and mirrored, ultimately forms a catenary shell. This geometry can be used to generate a gridshell that performs exceptionally well under compression, as long as the edges are reinforced and the corners are braced.

White I would be remiss to not mention Antoni Gaudí on the subject of catenary structure, his work doesn’t particularly falls under the category of gridshells. Instead I will proceed to gawk over some of the stunning work by Frei Otto.

Of course his work explored a great deal more than just catenary structures, but he is revered for his beautiful work on gridshells. He, along with the Institute for Lightweight Structures, have truly been pioneers on the front of theoretical structural engineering.

1.0 Phyllotaxis Spirals

1.1 Rational Numbers

1.2 Irrational Numbers

1.3 Quantifiable irrationality

1.4 Phi

Source: Infinite fractions and the most irrational number: www.youtube.com/watch?v=CaasbfdJdJg

1.5 The Metallic Means & Other Constants

Source: The Silver Ratio & Metallic Means: www.youtube.com/watch?v=7lRgeTmxnlg

      Timber is a rather attractive option when it comes to sustainable building. The fact that it is a renewable resource that takes away carbon dioxide in the process of it’s life-cycle makes it a very viable option for new communities to arise. But it does little good, if the to turn a tree into architecture, a bunch of pollutant and in some cases unattainable materials are required.  What i’m trying to achieve with this project is not complete independence from other materials but that the minimization of their use to the bear minimum when it comes to construction. The goal of the project so far was to find a way to join timber parts together that go beyond planar or perpendicular joints in order to construct architectures that are not predetermined in shape or form but only in construction technique.  A simple construction system that requires no factories, but only hand tools and utilizes trees as close to their natural form as possible.

The System utilizes dry timber joints and dowels for the main skeleton of whatever is to be built, allowing for double curved, intuitive growth to take place when constructing a building, and making the building a dynamic entity instead of a static object that can grow or shrink towards any direction, filling the gaps and meeting the requirements of its inhabitant, whatever they may be.  A method that does not predetermine functional or spacial conditions but acts as a tool to create architecture prone to customization.

The system utilizes linear timber elements in the form of sawn timber or potentially logs. Using  linear elements means  that the process from tree to construction material is minimized without necessarily compromising the form of the architecture.

My research starts with the art of origami folding with paper and continues to investigates the techniques of kinetic folding with thickened materials. I find it interesting how easily a paper is folded into an origami shape, which can be folded and unfolded easily. Throughout my design research I would like to find an easy way of folding a wooden structure.

Thickness plays a large role in folding paper. It can be observed with simple folding of a sheet of paper, the more it is folded the more difficult it gets to fold it. The traditional origami is done using paper. It can be observed that after folding paper too many times, it losses its folding ability. Most origami rely on the fibers locate within the paper, and their ability to bend and stretch slightly during the fold. That is why the thicker the paper or the material used the more difficult it is to bend it.

The folding technique using a thin membrane, like fabric, to imitate the behaviour of the hinge joint is used. The membrane is treated as a zero-thickens element, hence it behaves in a similar manner to paper origami. For a 180 degree fold the gap between panels equals to 2 times the thickness of the panel, however if the desired angle is smaller, that distance can decrease.

That investigation led to creating a triangular grid representing the possibilities of the membrane technique.

Reduction of the size of the triangle in the grid, creates higher number of bend axis. However by doing so, clash points are created, in places where the new grid wants to bend, and the old one has no chamfer to accommodate it. Because of that the first row of the reduced grid, behaves in a similar way to its bigger neighbour.

The fabrication process involves the use of a CNC machine. The fist step is to prepare a digital file and using a using a CNC machine cut out all the pieces. The edges are then sanded and arranged according to their size. The pieces are places on a calico canvas and attached to it using a PVA glue. The fabric is then cut to size and the edges are rolled to avoid ripping.

Fabric used for the final model is calico canvas. It is a plain-woven textile made from cotton. It is undyed and unfinished. The fabric worked much better than the synthetic mesh used for the initial prototypes. Also it is more resistant to breaking than the Garden Weed Control Fabric, which with time tends to tear apart.

The model shows that the smaller the size of the triangles the better more flexible the structure becomes. This technique can be used in creating shelters, as well as, sculptural pieces.

The initial aspiration of the project was to produce a method of simplifying the construction of the 5 regular platonic solids (Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron) using only bend active timber and simple bolted connections, eliminating the need for complex nodal connections as seen in geodesic dome construction and other compound angle connections.

Bend active timber being the main research topic, the structural capabilities and bending radii of plywood were physically tested incorporating the several thicknesses and crucially the direction of the bend either being parallel or perpendicular to the grain, resulting in an informative results matrix.

Digital explorations were undertaken for each of the platonic solids creating various sized volumetric frame structures. The resultant sizes were due to each component being constrained to fit on a standard 2440x1220mm plywood sheet and the resultant bending radius

Following physical investigations of each, the cube was taken forward as it was; more efficient in terms of material usage, easier method of assembly comparatively and unintentionally produced a deployable mechanism similar to the famous Hoberman’s sphere.

The system utilizes identical components meaning the fabrication process can be quickly and easily be performed using a CNC machine, with assembling being intuitive, not requiring different parts or specialised assembly instructions. The components were cut using my own CNC and were then simply assembled by hand using bolted connections to create the skeletal frame. Assembly was extremely quick, from flat components to finished volumetric module taking only 20 minutes.

The pre-fabricated modules can then be replicated and scaled to suit various habitable typologies a community would need or could be used as a deployable shelter for the homeless/emergency relief.

Touching on topics of personal interest including sustainability, synchronicity with the environment and parametric design, my view to create an eventual large-scale structure represented an exciting opportunity to engage with wood, a material I had not yet handled in great detail. It is renewable, can in some form be  sourced locally throughout the world and is incredibly versatile with infinite characteristics that create both solutions and challenges when implemented to the built space.

In this project I begin by reviewing the varying available species of wood types that would carry potential for me to work with towards a final piece, taking account of their sustainable qualities I wanted to delve into wood and explore the material’s extremes. Following this I look into several designs, techniques and different structural systems that mirror it in nature as well as architectural projects that have used wood to react to these systems and create methods to benefit from them.

In being intrigued particularly by Japanese Joinery I investigate the traditional patterns and methods associated with it and begin to analyse how far it can be incorporates into my project. I took elements forward from here to create my models, especially tessellation and symmetry. I was particularly inspired by the linking of pieces, as shingles, together with the integration of the joinery into the design. Trialling which shapes, patterns and scales work best when joined together then evolves into grappling with how to create angles and curvature through overlapping pieces rather than joining organically through grooves and notches.

The most important part of my project was in the detail of creating a shape that was versatile enough to form a curving structure when connected together and multiplied with minimum component parts.
In experimenting with different styles of wood, shapes and angled edges, my scale piece evolved from creating a curve on one axis as an arch to a multi-dimensional surface that has a rhythmic, organic quality.

Throughout the project I encountered many challenges. Manufacturing each scale through out machinery also proved unreliable, especially in the later stages of the project, however I was fortunate that my concept could come to life regardless of the volume of pieces I could create.

Using wood in this format also had an interesting design effect in that despite organising the scales in a uniform way, imperfections in each connection, when multiplied, created a final model that resembled reptile scales that I had originally been inspired by. My project I feel takes into account the natural, flexible characteristics of wood as a material; embracing imperfections in the system rather than attempting to defeat them.

By experimenting with different sequences, patterns, dimensions and shapes, this system has the potential to create structures that can be assembled, disassembled, transported and manipulated easily and creatively.

My journey to create this structure using wooden shingles that curves on two different planes creating an organic and sometimes imperfect surface with the relationship of each wooden piece to another was long and challenging but successful in my opinion. 

The aim of this project was to develop a sustainable building system, which uses repetition and symmetry to produce a large scale structure. To begin the development process, inspiration was taken from crystal structures to form a panel system with an infinite array.

A rectangular lattice can be simply formed by slotting together regular panels. By interchanging the direction of alternate panels, a continuous standing structure can be formed.

Furthermore, investigating the different possible scales the panel systems can be used for, and the dimensions which would be needed to achieve different forms, lead to the exploration of using different sized panels in the same system.

To design a system which works in many different scales, it become prudent, the test the system with a variety of plywood.

Plywood tests for every possible combination of plywood thickness between 0.8mm to 18mm were conducted, with the intention to design a proposal where the plywood thickness to increase with size of each panels.

For precision to be achieve, a laser cutter and CNC machine are required. This lead to the discovery of some rules:

• Laser cutting plywood is restricted for use on 0.8mm, 1.5mm and 4mm thickness.
• The 3mm drill bit on in CNC cutter, is limited to cut plywood sheets with thickness of 9mm, or smaller.
• The 6mm drill bit on in CNC cutter, is limited to cut gaps, no smaller than 6.1mm and sheets with thickness no greater than 9mm.

To scale up the system even further, the use of Cross-Laminated Timber (CLT) has been explored.

In construction, CLT is prefabricated to fit together making to very easy to construct, therefore very time efficient. Similarly, the panel system being designed requires these properties.

DIY CLT

In order to increase the stability of the overall system, a panel which fits in the z-axis, has been introduced to the system. This is achieved by extending a further slot in the panel on the x-axis, for the new panel can be slotted in without, changing how the system previously fit together.

Additionally, this third panel will then rest on the panels directly below it, to create a ‘floor’ level.

Addition of z-axis (floor) panel

Following the plywood tolerance tests, a slightly differing system for panels to slot together has been develop, to work for the more delicate, thinner plywood. The result is a lattice structure which, seems to be work effectively as a grille, or light filtering device, and therefore would work well positioned at a high level in the final proposal.

Roof system

The process of construction the large scale model included time using the laser cutter and CNC machine, as well as many manual construction hours. The entire process lasted approximately 10 days, including the preparation of files to upload to the cutting machines.

Click to view slideshow.

The result is a large (2.3m tall, 1.8mm width and depth), made entirely of timber panels which intercept each other to form an arrayed construction using repeated elements, with no additional fixtures required. The structure was built to allow uses to enter into it the look above to the roof system.