The Nature of Gridshell Form Finding

Grids, shells, and how they, in conjunction with the study of the natural world, can help us develop increasingly complex structural geometry.

Foreword

This post is the third installment of sort of trilogy, after Shapes, Fractals, Time & the Dimensions they Belong to, and Developing Space-Filling Fractals. While it’s not important to have read either of those posts to follow this one, I do think it adds a certain level of depth and continuity.

Regarding my previous entries, it can be difficult to see how any of this has to do with architecture. In fact I know a few people who think studying fractals is pointless.

Admittedly I often struggle to explain to people what fractals are, let alone how they can influence the way buildings look. However, I believe that this post really sheds light on how these kinds of studies may directly influence and enhance our understanding (and perhaps even the future) of our built environment.

On a separate note, I heard that a member of the architectural academia said “forget biomimicry, it doesn’t work.”

Firstly, I’m pretty sure Frei Otto would be rolling over in his grave.

Secondly, if someone thinks that biomimicry is useless, it’s because they don’t really understand what biomimicry is. And I think the same can be said regarding the study of fractals. They are closely related fields of study, and I wholeheartedly believe they are fertile grounds for architectural marvels to come.

7.0 Introduction to Shells

As far as classification goes, shells generally fall under the category of two-dimensional shapes. They are defined by a curved surface, where the material is thin in the direction perpendicular to the surface. However, assigning a dimension to certain shells can be tricky, since it kinda depends on how zoomed in you are.

A strainer is a good example of this – a two-dimensional gridshell. But if you zoom in, it is comprised of a series of woven, one-dimensional wires. And if you zoom in even further, you see that each wire is of course comprised of a certain volume of metal.

This is a property shared with many fractals, where their dimension can appear different depending on the level of magnification. And while there’s an infinite variety of possible shells, they are (for the most part) categorizable.

7.1 – Single Curved Surfaces

Analytic geometry is created in relation to Cartesian planes, using mathematical equations and a coordinate systems. Synthetic geometry is essentially free-form geometry (that isn’t defined by coordinates or equations), with the use of a variety of curves called splines. The following shapes were created via Synthetic geometry, where we’re calling our splines ‘u’ and ‘v.’

A-Barrel-Vault
Uniclastic: Barrel Vault (Cylindrical paraboloid)

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.

B-Conoid

Uniclastic: Conoid (Conical paraboloid)

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 surfaces. If made of paper, you could not flatten them without tearing, folding or crumpling them.

C-Dome.gif
Synclastic: Dome (Elliptic paraboloid)

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 without distortion,” a topic explored by Michael Stevens: https://www.youtube.com/watch?v=2lR7s1Y6Zig

D-Saddle.gif
Anticlastic: Saddle (Hyperbolic paraboloid)
This one was formed by 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.
Pringle
Kellogg’s potato and wheat-based stackable snack
Here is an example of a beautiful marriage of tensile and compressive potato and wheat-based anticlastic forces. Although I hear that Pringle cans are diabolically heinous to recycle, so they are the enemy.
11 Tensile and Compressive behaviour of shells.jpg
Structural Behaviour of Basic Shells [Source: IL 10 – Institute for Lightweight Structures and Conceptual Design]

7.3 – Translation vs Revolution

In terms of synthetic geometry, there’s more than one approach to generating anticlastic curvature:
E-Hyperbolic-Paraboloid-Saddle.gif
Hyperbolic Paraboloid: Straight line sweep variation

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 rails 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..

F-Hyperbolic-Paraboloid-Tower.gif
Ruled Surface & Surface of Revolution (Circular Hyperboloid)
The ruled surface was created by sliding a plane curve (a straight line) along another plane curve (a circle), while keeping the angle between them constant. The surfaces of revolution was simply made by revolving a plane curve around an axis. (Surface of translation also exist, and are similar to ruled surfaces, only the orientation of the curves is kept constant instead of the angle.)
 
Cylinder_-_hyperboloid_-_cone.gif
Hyperboloid Generation [Source:Wikipedia]

The hyperboloid has been a popular design choice for (especially nuclear cooling) towers. It has excellent tensile and compressive properties, and can be built with straight members. This makes it relatively cheap and easy to fabricate relative to it’s size and performance.

These towers are pretty cool acoustically as well: https://youtu.be/GXpItQpOISU?t=40s

 

8.0 Geodesic Curves

These are singly curved curves, although that does sound 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.

Geodesic Curves - Ribon.jpg
Conic Plank Lines (Source: The Geometry of Bending)

A ribbon is essentially a straight 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:

G-Geocurves.gif
Basic geodesic curves on a 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.

H-Geocurves.gif
Initial set of curves

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.

I-Geocurves.gif
Addition of secondary set of curves

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.

J-Geocurves.gif
Grid with independent sets

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.

8.2 – Basic Gridshells

The same principle is then applied to a series of surfaces with varied types of curvature.

K-Barrel
Uniclastic: Barrel Vault Geodesic Gridshell

First comes the shell (a barrel 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 necessitate single curvature.

L-Conoid
Uniclastic: Conoid Geodesic Gridshell

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.

M-Dome
Synclastic: Dome Geodesic Gridshell

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.

N-Saddle
Anticlastic: Saddle Geodesic Gridshell

Structurally speaking, things get much more interesting with anticlastic curvature. As previously stated, each member will behave differently based on their 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.

On another note:

While geodesic curves make it far more practical to fabricate shells, they are not a strict requirement. 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.)

6 - Alternate Grid
Alternate Gridshell Patterns [Source: IL 10 – Institute for Lightweight Structures and Conceptual Design]

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.

Biosphere Environment Museum - Canada
Biosphere Environment Museum – Canada

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.  

Centre Pompidou-Metz - France
Centre Pompidou-Metz – France

Gridshells with highly irregular geometry are far more challenging to fabricate. In this case, each and every piece had to be custom made 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.

O - Saddle 2
Hyperbolic Paraboloid: Straight Line Sweep Variation With Rotating Plank Line Grid

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.

Hyperbolic-Paraboloid-Plank-Lines
Hyperbolic Paraboloid: Flattened Plank Lines With Junctions

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 strikes me as cheating thought.. As if it’s not a genuine, bona fide gridshell.

Toledo-gridshell-20-Construction-process
Toledo Gridshell 2.0. Construction Process [source: Timber gridshells – Numerical simulation, design and construction of a full scale structure]

This is one of the original build method, where the gridshell is assembled flat, lifted into shape, then locked into place.

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 this phenomenon.

hyperbolic paraboloid soap bubble
Hyperbolic Paraboloid Soap Bubble [Source: Serfio Musmeci’s “Froms With No Name” and “Anti-Polyhedrons”]
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.

00---Minimal-Surface-Model
Kangaroo2 Physics: Surface Tension Simulation

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 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

Eden_Project_geodesic_domes_panorama.jpg
The Eden Project – United Kingdom

Here we have one of the most popular examples of minimal surface geometry in architecture. The shapes of these domes were 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):

Another powerful method of form finding has been to let gravity dictate the shapes of structures. In physics and geometry, 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 look similar to parabolic curves, but perform differently.

00---Haning-Model
Kangaroo2 Physics: Catenary Model Simulation

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.

While I would be remiss to not mention Antoni Gaudí on the subject of catenary structure, his work doesn’t particularly fall 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.

9.3 – Biomimicry in Architecture

There are a few different terms that refer to this practice, including biomimetics, bionomics or bionics. In principle they are all more or less the same thing; the practical application of discoveries derived from the study of the natural world (i.e. anything that was not caused or made my humans.) In a way, this is the fundamental essence of the scientific method: to learn by observation.
Biomimicry-Bird-Plane
Example of Biomimicry

Frei Otto is a fine example of ecological literacy at its finest. A profound curiosity of the natural world greatly informed his understanding of structural technology. This was all nourished by countless inquisitive and playful investigations into the realm of physics and biology. He even wrote a series of books on the way that the morphology of bird skulls and spiderwebs could be applied to architecture called Biology and Building. His ‘IL‘ series also highlights a deep admiration of the natural world.

Of course he’s the not the only architect renown their fascination of the universe and its secrets; Buckminster Fuller and Antoni Gaudí were also strong proponents of biomimicry, although they probably didn’t use the term (nor is the term important.)

Gaudí’s studies of nature translated into his use of ruled geometrical forms such as hyperbolic paraboloids, hyperboloids, helicoids etc. He suggested that there is no better structure than the trunk of a tree, or a human skeleton. Forms in biology tend to be both exceedingly practical and exceptionally beautiful, and Gaudí spent much of his life discovering how to adapt the language of nature to the structural forms of architecture.

Fractals were also an undisputed recurring theme in his work. This is especially apparent in his most renown piece of work, the Sagrada Familia. The varying complexity of geometry, as well as the particular richness of detail, at different scales is a property uniquely shared with fractal nature.

Antoni Gaudí and his legacy are unquestionably one of a kind, but I don’t think this is a coincidence. I believe the reality is that it is exceptionally difficult to peruse biomimicry, and especially fractal geometry, in a meaningful way in relation to architecture. For this reason there is an abundance of superficial appropriation of organic, and mathematical, structures without a fundamental understanding of their function. At its very worst, an architect’s approach comes down to: ‘I’ll say I got the structure from an animal. Everyone will buy one because of the romance of it.”

That being said, modern day engineers and architects continue to push this envelope, granted with varying levels of success. Although I believe that there is a certain level of inevitability when it comes to how architecture is influenced by natural forms. It has been said that, the more efficient structures and systems become, the more they resemble ones found in nature.

Euclid, the father of geometry, believed that nature itself was the physical manifestation of mathematical law. While this may seems like quite a striking statement, what is significant about it is the relationship between mathematics and the natural world. I like to think that this statement speaks less about the nature of the world and more about the nature of mathematics – that math is our way of expressing how the universe operates, or at least our attempt to do so. After all, Carl Sagan famously suggested that, in the event of extra terrestrial contact, we might use various universal principles and facts of mathematics and science to communicate.

Developing Space-Filling Fractals

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

 

Foreword

Following my last post on the “…first, second, and third dimensions, and why fractals don’t belong to any of them…“, this post is about documenting my journey as I delve deeper into the subject of fractals, mathematics, and geometry.
The study of fractals is an intensely vast topic. So much so that I’m convinced you could easily spend several lifetimes studying them. That being said, I chose to focus specifically on single-curve geometry. But, keep in mind that I’m only really scratching the surface of what there is to explore.

4.0 Classic Space-Filling

Inspired by Georg Cantor’s research on infinity near the end of the 19th century, mathematicians were interested in finding a mapping of a one-dimensional line into two-dimensional space – a curve that will pass through through every single point in a given space.
Jeffrey Ventrella writes that “a space-filling curve can be described as a continuous mapping from a lower-dimensional space into a higher-dimensional space.” In other words, an initial one-dimensional curve is developed to increase its length and curvature – the amount of space in occupies in two dimensions. And in the mathematical world, where a curve technically has no thickness and space is infinitely vast, this can be done indefinitely.

4.1 Early Examples

In 1890, Giuseppe Peano discovered the first of what would be called space-filing curves:

Peano-space-filling-Curve_-four-approximations_-version-A_1 4i.gif
4 Iterations of the Peano Curve
An initial ‘curve’ is drawn, then each element of the curve is replace by the whole thing. Here it is done four times, and it’s easy to imagine how you can keep doing this over and over again. One would think that if you kept doing this indefinitely, this one-dimensional curve would eventually fill all of two-dimensional space and become a surface. However it can’t, since it technically has no thickness. So it will be as close as you can get to a surface, without actually being a surface (I think.. I’m not that sure..)
A year later, David Hilbert followed with his slightly simpler space-filing curve:
Hilbert_curve 8i.gif
8 Iterations of the Hilbert Curve
In 1904, Helge von Koch describes a single complex continuous curve, generated with rudimentary geometry.
Von_Koch_curve 7i.gif
7 Iterations of the Koch Curve
Around 1967, NASA physicists John Heighway, Bruce Banks, and William Harter discovered what is now commonly known as the Dragon Curve.
Dragon_Curve_Unfolding 13i.gif
13 Iterations of the Dragon Curve

4.2 Later Examples

You may have noticed that some of these curves are better at filling space than others, and this is related to their dimensional measure. They fall under the category of fractals because they’re neither one-dimensional, nor two-dimensional, but sit somewhere in between. For these examples, their dimension is often defined by exactly how much space they fill when iterated infinitely.
While these are some of the earliest space-filling curves to be discovered, they are just a handful of the likely endless different variations that are possible. Jeffrey Ventrella spent over twenty-five years exploring fractal curves, and has illustrated over 200 hundred of them in his book ‘Brain-Filling Curves, A Fractal Bestiary.’ They are organised according to a taxonomy of fractal curve families, and are shown with a unique genetic code.
Incidentally, in an attempt to recreate one of the fractals I found in Jeffery Ventrella’s book, I accidentally created a slightly different fractal. As far as I’m concerned, I’ve created a new fractal and am unofficially naming it ‘Nicolino’s Quatrefoil.’ The following was created in Rhino and Grasshopper, in conjunction Anemone.
Nicolino-Quatrefoil_Animation i5.gif
5 Iterations of Nicolino’s Quatrefoil
You can find beautifully animated space-filling curves here:
(along with some other great videos by ‘3Blue1Brown’ discussing the nature of space-filling curves, fractals, infinite math, and more)

On A Strange Note:

It’s possible to iterate a version of the Hilbert Curve that (once repeated infinity) can fill three-dimensional space.
As an object, it seems perplexingly difficult to categorize. It is a single, one-dimensional, curve that is ‘bent’ in space following simple, repeating rules. Following the same logic as the original Hilbert Curve, we know that this can be done indefinitely, but this time it is transforming into a volume instead of a surface. (Ignoring the fact that it is represented with a thickness) It is a one-dimensional curve transforming into a three-dimensional volume, but is never a two-dimensional surface? As you keep iterating it, its dimension gradually increases from 1 to eventually 3, but will never, ever, ever be 2??
giphy.gif
Nevertheless this does actually support a statement I made in my last post suggesting “there is no ‘first’ or ‘second’ dimension. It’s a bit like pouring three cups of water into a vase and asking someone which cup is the first one. The question doesn’t even make sense…

5.0 Avant-Garde Space-Filling

In the case of the original space-filling curve, the goal was to fill all of infinite space. However the fundamental behaviour of these curves change quite drastically when we start to play with the rules used to generate them. For starters, they do not have to be so mathematically tidy, or geometrically pure. The following curves can be subdivided infinitely, making them true space-filling curves. But, what makes them special is the ability to control the space-filling process, whereas the original space-filling curves offer little to no artistic license.

5.1 The Traveling Salesman Problem

Let’s say that we change the criteria, from passing through every single point in space, to passing only through the ones we choose. This now becomes a well documented computational problem that has immediate ‘real world’ applications.
Our figurative traveling salesman wishes to travel the country selling his goods in as many cities as he can. In order to maximize his net profit, he must make his journey as short as possible, while of course still visiting every city on his list. His best possible route becomes exponentially more challenging to work out, as even just a handful of cities can generate thousands of permutations.
There are a variety of different strategies to tackle this problem, a few of which are described here:
The result is ultimately a single curve, filling a space in a uniquely controlled fashion. This method can be used to create single-lined drawings based on points extracted from Voronoi diagrams, a topic explored by Arjan Westerdiep:
Traveling Salesman Portrait.png
This illustration, commissioned by Bill Cook at University of Waterloo, is a solution to the Traveling Salesman Problem.

5.2 Differential Growth

If we let physics (rather than math) dictate the growth of the curve, the result becomes more organic and less controlled.
In this example Rhino is used with Grasshopper and Kangaroo 2. A curve is drawn on a plain, broken into segments, then gradually increased in length. As long as the curve is not allowed to cross itself (which is achieved here with ‘Collision Spheres’), the result is a curve that is pretty good at uniformly filling space.
Differential-Growth-With-Kangaroo-2.gif
Differential Growth with Rhino & Grasshopper – Kangaroo 2 – Planar
The geometry doesn’t even have to be bound by a planar surface; It can be done on any two-dimensional surface (or in three-dimensions (even higher spacial dimensions I guess..)).
Bunny-Differential-Growth.gif
Differential Growth with Rhino & Grasshopper – Kangaroo 2 – NonPlanar
Rotating-Stanford-Bunny.GIF
Differential Growth with Rhino & Grasshopper – Kangaroo 2 – Single-Curved Stanford Rabbit
Additionally, Anemone can be used in conjunction with Kangaroo 2 to continuously subdivide the curve as it grows. The result is much smoother, as well as far more organic.
Kangaroo & Anemone - Octo-Growth.gif
Differential Growth with Rhino & Grasshopper – Kangaroo 2 & Anemone – Octopus
Of course the process can also be reversed, allowing the curve to flow seamlessly from one space to another.
Kangaroo & Anemone - Batman Duck.gif
Differential Growth with Rhino & Grasshopper – Kangaroo 2 & Anemone – BatmanDuck
Here are far more complex examples of growth simulations exploring various rules and parameters:

6.0 Developing Fractal Curves

In the interest of creating something a little more tangible, it is possible to increase the dimension of these curves. Recording the progressive iterations of a space filling curve allow us to generate what is essentially a space-filling surface. This new surface has the unique quality of being able to fill a three-dimensional space of any shape and size, while being a single surface. It of course also shares the same qualities as its source curves, where it keep increasing in surface area (and can do so indefinitely).
Unrolling Surfaces.jpg
Surface Unrolling Study
If you were to keep gradually (but indefinitely) increasing the area of a surface this way in a finite space, the result will be a two-dimensional surface seamlessly transforming into a three-dimensional volume.

6.1 Dragon’s Feet

Here is an example of turning the dragon curve into a space-filling surface. Each iteration is recorded and offset in depth, all of which inform the generation of a surface that loosely flows through each of them. This was again achieved with Rhino and Grasshopper.
I don’t believe this geometry has a name beyond ‘the developing dragon curve’, so I’ve called it ‘Dragon’s Feet.’
Adding a little thickness to the model allow us to 3D print it.
3d Printed Dragon Curve.jpg
Developing Dragon Curve: Dragon’s Feet – 3D Print

6.2 Hilbert’s Curtain

Here is the Hilbert Curve going through the same process, which I am aptly naming ‘Hilbert’s Curtain.’
3D Printed Developing Hilbert Curve
Developing Hilbert Curve: Hilbert’s Curtain – 3D Print
3D Printing Space-Filling Curves with Henry Segerman at Numberphile:
‘Developing Fractal Curves’ by Geoffrey Irving & Henry Segerman:

6.3 Developing Whale Curve

Unsurprisingly this can also be done with differentially grown curve. The respective difference being that this method fills a specific space in a less controlled manner.
In this case with Kangaroo 2 is used to grow a curve into the shape of a whale. Like before, each iteration is used to inform a single-surface geometry.
Developing-Whale-Curve-b.gif
Iterative Steps of the Differentially Grown Whale Curve

3D print of the different recursive steps of a space-filling curve
Developing Whale Curve – 3D Print

Desire for Immortality

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.

DesireforImmortalityNight

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.

DesireforImmortalityCombined

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.

DesireforImmortalityGif

 

The Fractal Hourglass

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.

burning man -blog1.jpg

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.

burning man- blog2.jpg

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 Wishing Well

something caught in between dimensions – on its way to becoming more.

Summary

The Wishing Well is the physical manifestation, a snap-shot, of a creature caught in between dimensions – frozen in time. It is a digital entity that has been extracted from its home in the fractured planes of the mathematical realm; a differentially grown curve in bloom, organically filling space in the material world.

The notion of geometry in between dimensions is explored in a previous post: Shapes, Fractals, Time & the Dimensions they Belong to

 

Description

The piece will be built from the bottom-up. Starting with the profile of a differentially grown curve (a squiggly line), an initial layer will be set in pieces of 2 x 4 inch wooden studs (38 x 89 millimeter profile) laid flat, and anchored to the ground. Each subsequent layer will be built upon and fixed to the last, where each new layer is a slightly smoother version than the last. 210 layers will be used to reach a height of 26 feet (8 meters). The horizontal spaces in between each of the pieces will automatically generate hand and foot holes, making the structure easily climbable. The footprint of the build will be bound to a space 32 x 32 feet.

The design may utilize two layers, inner and out, that meet at the top to increase the structural integrity for the whole build. It will be lit from within, either from the ground with spotlights or with LED strip lights following patterns along the walls.

Different Recursive Steps of a Dragon Curve

Ambition

At the Wishing Well, visitors embark on a small journey, exploring the uniquely complex geometry of the structure before them. As they approach the foot of the well, it will stand towering above them, undulating organically across the landscape. The nature of the structure’s curves beckons visitors to explore the piece’s every nook and cranny. Moreover, its stature grants a certain degree of shelter to any traveller seeking refuge from the Playa’s extreme weather conditions. The well’s shape and scale allows natural, and artificial, light to interact in curious ways with the structure throughout the day and night. The horizontal gaps between every ‘brick’ in the wall allows light to filter through each layer, which in turn casts intriguing shadows across the desert. This perforation also allows Burners to easily, and relatively safely, scale the face of the build. Visitors will have the opportunity to grant a wish by writing it down on a tag and fixing it to the well’s interior.

171108 - Burning Man Timber Brick Laying Proposal View 2.jpg

 

Philosophy

If you had one magical (paradox free) wish, to do anything you like, what would it be?

Anything can be wished for at the Wishing Well, but a wish will not come true if it is deemed too greedy. Visitors must write their wish down on a tag and fix it to the inside of the well. They must choose wisely, as they are only allowed one. Additionally, they may choose to leave a single, precious, offering. However, if the offering does not burn, it will not be accepted. Visitors will also find that they must tread lightly on other people’s wishes and offerings.

The color of the tag and offering are important as they are associated with different meanings:

  • ► PINK – love
  • ► RED – happiness, joy, success, good luck, passion, vitality, celebration
  • ► ORANGE – change, adaptability, spontaneity, concentration
  • ► YELLOW – nourishment, warmth, clarity, empathy, being free from worldly cares
  • ► GREEN – growth, balance, healing, self-assurance, benevolence, patience
  • ► BLUE – conservation, healing, relaxation, exploration, trust, calmness
  • ► PURPLE – spiritual awareness, physical and mental healing
  • ► BLACK – profoundness,  stability, knowledge, trust, adaptability, spontaneity,
  • ► WHITE – mourning, righteousness, purity, confidence, intuition, spirits, courage

The Wishing Well is a physical manifestation of the wishes it holds. They are something caught in between – on their way to becoming more. I wish for guests to reflect on where they’ve been, where they are, where they are going, and where they wish to go.

171108 - Burning Man Timber Brick Laying Proposal View 1.jpg

Mixpinski’s Myriad

In mathematics a fractal is an abstract object used to describe and simulate naturally occurring objects. Artificially created fractals commonly exhibit similar patterns at increasingly small scales. It is also known as expanding symmetry or evolving symmetry.  Mandelbulb 3D allows us to explore fractals in 3D, creating a seamless amalgamation of maths, art and science.

Understanding how this geometry can become infinite and how it can be built within the constraints of the physical reality was part of the philosophy of my piece.

Mandelbulb 3d fractals:

 

161117 FRACTAL sheet 1

From these specific chosen 3d Fractals I noticed a clear correlation with the natural formation of crystalline structures, in particular Hopper crystals.

Hopper crystals form when there is more rapid growth at the outer edges of a face than at the centre. This results in what appears to be a hollowed out step lattice formation, as if someone had removed interior sections of the individual crystals. This missing part was never actually developed as the crystals grow so rapidly that there is never time for this to be developed. Hopper crystals are very similar to the cubic halite skeletal crystals formed from extreme supersaturation in salt lakes existing in nature. Hopper crystals can be found in rose quartz, gold, calcite, bismuth, salt and ice. I looked at the growth of these crystals to better understand the structural qualities.

Hopper Crystal Formation:

161117 FRACTAL sheet 3.1

From looking at the crystalline structure it became apparent that the connection between the tapered levels was very important to the structure and adaptability of the proposal. The versatility of this connection allows for flexibility and movement within the module. The connector can be placed on any material simply by adapting the end nodes width to factor for the material depth. By creating this modular junction I can join all the stepped timber elements of the proposal in such a way that they are all supporting each other.

Connection options:

161117 FRACTAL sheet 2

Hopper crystal growth is never as predicted due to outside influences such as movement and temperature change. These influences creates the beautiful images we see of their crystalline forms and without these the fractal crystal growth would be predictable and simplistic. It is with these outside interactions that the crystals have their own idiosyncrasies. By combining the hopper crystal growth with the organic forms created with the 3d fractal generator, I created a pavilion proposal. Using a stepped form and the junction designed above I could use the unpredictable growth lines to create an interesting pavilion which can be experienced in the same way that crystals would grow, naturally and not within their algorithmic form. Nature does not always conform to predictability. The pavilion expresses this individuality and in turn expresses the way in which we grow as individuals, adapting to our environments and moulded by our experiences.

This project is a physical exploration of crystal formation centred around the theme of fractals. It aims to combine one joint in order to create a crystalline structure. Inspired by the geometry from the crystalline growth the lattice structure provides sanctuary and calm in a sea of dust and at night mesmerising myriads of stepped lights will illuminate the playa.271117 render night.jpgThe proposed installation will be formed of a mixture of 2 x 4 timber with CNC curved plywood pieces incorporated into the structure. Each 2 x 4 will have a joint or a pocket in order for it to slot into and support the weight of the neighbouring beam or column. The project will appear out of the sand as an elegant stepped fractal structure which gives the proposal an ecclesiastical ambiance.

231117 close up night

The intertwined stepped lattice timber elements form congregation and celebratory spaces, whilst capturing special views of the playa. The stepped elements promote Burners to climb and crawl between the spaces created by the overlapped timbers. At night when you ascend through the individual spaces the lights will constantly change and oscillate. With the lights constantly changing and staggering further through the elements the stepped structure will be enhanced.  The project aims to play with the burners’ perception of depth where the lattice stepped geometry is staggered and rotated. At night this perception is further confused by  the LED coloured strips oscillating along the staggered stepped beams and columns. The burners can seek sanctuary in a space in which dimensionality and form is confused and adapted.

271117 renderday

 

 

 

 

The beauty of error

1

As our studio dipped into the complexity of fractals, it became easy to get lost. Suddenly, these geometries were everywhere. Trees, clouds, coastlines, our own bodies – all examples of fractals. Systems, that are made up of smaller self-simular parts until they reach infinity. Systems, that travel between dimension (more about it here https://wewanttolearn.wordpress.com/2017/10/18/shapes-fractals-time-the-dimensions-they-belong-to/). Wanting to understand these geometries better, I found a Fractal plugin for Grasshopper by albertovalis on Food4Rhino. Playing around with various parameters and GH components gave me interesting shapes, but which seemed far away from an architectural object. I then decided to give it a try and allow the program to randomly select elements by assigning different true/false patterns. Finally, an error happened and it was beautiful.

 

Error 101

Summary 

Error 101 is a visual representation of relationships between machines and humans. It illustrates what we can learn from each other (what does this mean?). The geometry was generated through a combination of fractal mathematics properties, parametric design tools and finally a computer error, which were all guided by human decisions.

Physical description

The artwork will be made out of ‘chaotically’ arranged ribbons that, together, form a tetrahedron. From far, the geometry will look well defined – a triangle or pyramid. As you get closer you notice the complexity. When you experience is physically, you find logic in the chaos. Inside the tetrahedron is a void.

Error 101 will be constructed using bent cross-laminated timber modules that are interlocked together with flitch plates. Their arrangement will allow the object to be self-supporting. The whole piece is 18’x18’x18’ (5.5 metres). Timber strips create the outer shell and are 25 inches wide (635 mm). Their surface will be treated to achieve a smooth finish to protect both the piece and visitors. Light strips will be fixed to edges of timber curves and turned on at night. Assembly will be completed on site.

Interactivity and Mission

Error 101 is left open to interpretation – everyone can have their personal take on the piece. Visual and emotional perception of Error 101 may change depending on how close you get to it. It may encourage visitors to think of it, as something that travels between dimensions, which is a liberating allegory of how one thing can become another and how the whole is just a collection of its parts. Just like water can be liquid, ice or vapor, Error 101 can be a triangle, pyramid or chaotic curves.

The structure is climbable and each of many unique curves can be treated as a nest. Occupying empty spaces on different levels may make burners feel like a part of the ‘chaos’, that has a space for everyone. Different curvature can suggest different positioning of a body that may influence visual as well as physical experience. Entering the structure’s core shifts the visitor’s focus away from the idea of a pyramid and allows them to focus on what’s within. Such study erases preconceptions and allows new ideas to be born. This notion is also enhanced with the use of lights at night.

Philosophy 

Error 101 is a product of human ability to perceive beauty, and computer’s power to process complex mathematics. Its development started with an attempt to try to understand fractal geometries that only became possible to study in the recent years due to the development of computer processing power. A continuous human-computer-human processes that involved both logic and error allowed for the piece to be born.

Error 101 is a common error in Internet browsing. A simple solution to it is clearing browsing history and cache. It may also appear in other spheres of digital world when software or a device is out of date. Burning Man participants are invited to clear their mind, update the ‘software’ and reset their system to become a new advanced version of themselves. The final steps of error 101 creation involved chance and error. The chaos led to something beautiful. We, as humans, can learn from this – learn to let go, to acknowledge and even appreciate mistakes, complexity of the world and our own selves. The geometry of an artwork is essentially a continuous strip that can be unrolled into one flat curve on the ground. This idea of continuity and interdependence is an allegory of a world’s structure.

The closer you get to Error 101, the more you can learn from it. A 2D triangle turns into 3D pyramid and then into a collection of overlaying shapes that are not truly from our dimension. With the speed of the modern world we tend to simplify things, which leads to inability to see details. Visitors are invited to come take time to study and appreciate the complexity of the Error, and to realize the beauty of a whole. From this, they may find that, in fact, all processes in our lives have a similar structure. Chaos generates order and order generates chaos.

2.jpg

IMG_0876.jpg