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

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.

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The beauty of error

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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.

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Shapes, Fractals, Time &, the Dimensions they Belong to

First, second and third dimensions, and why fractals don’t belong to any of them, as well as what happens when you get into higher dimensions.

Foreword

In this post, I’m going to try my best to explain the first, second, and third dimensions, and why fractals don’t belong to any of them, as well as what happens when you get into higher dimensions. But before getting into the nitty-gritty of the subject, I think it’s worth prefacing this post with a short note on the nature of mathematics itself:

Alain Badiou said that mathematics is a rigorous aesthetic; it tells us nothing of real being, but forges a fiction of intelligible consistency. That being said, I think it’s interesting to think about whether or not mathematics were invented or discovered – whether or not numbers exist outside of the human mind.

While I don’t have an answer to this question (and there are at least three different schools of thought on the subject), I do think it’s important to keep in mind that we only use math as a tool to measure and represent ‘real world’ things. In other words, our knowledge of mathematics has its limitations as far as understanding the space-time continuum goes.

 

1.0 Traditional Dimensions

In physics and mathematics, dimensions are used to define the Cartesian plains. The measure of a mathematical space is based on the number of variables require to define it. The dimension of an object is defined by how many coordinates are required to specify a point on it.

It’s important to note that 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.

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Excerpt from ‘minutephysics’
We usually arbitrarily pick a dimension and calling it the ‘first’ one.

 

1.1 – Zero Dimensions

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Something of zero dimensions give us a point. While a point can inhabit (and be defined in) higher dimensions, the point itself has a dimension of zero; you cannot move anywhere on a point.

1.2 – One Dimension

R1

A line or a curve gives us a one-dimensional object, and is typically bound by two zero-dimensional things.
Only one coordinate is required to define a point on the curve.
Similarly to the point, a curve can inhabit higher dimension (i.e. you can plot a curve in three dimensions), but as an object, it only possesses one dimension.
Another way to think about it is: if you were to walk along this curve, you could only go forwards or backward – you’d only have access to one dimension, even though you’d be technically moving through three dimensions.

 

1.3 – Two Dimensions

 R2

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&yy&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 defined on the surface of a sphere with latitude and longitude.

 

1.4 – Three Dimensions

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A volume gives us a three-dimensional shape, and can be bound by two-dimensional shapes (surfaces).

Shapes in three dimensions are most commonly represented in relation to an x, y and z axis. If a person were to swim in a body of water, their position could be defined by no less than three coordinates – their latitude, longitude and depth. Traveling through this body of water grants access to three dimensions.

 

2.0 Fractal Dimensions

Fractals can be generally classified as shapes with a non-integer dimension (a dimension that is not a whole number). They may or may not be self-similar, but are typically measured by their properties at different scales.

Felix Hausdorff and Abram Besicovitch demonstrated that, though a line has a dimension of one and a square a dimension of two, many curves fit in-between dimensions due to the varying amounts of information they contain. These dimensions between whole numbers are known as Hausdorff-Besicovitch dimensions.

 

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.

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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.

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

 

2.2 – Between the Second & Third Dimensions

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Developing Koch Snowflake
The same logic applies when exploring fractals plotted in three dimensions:

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

A volume gives us a three-dimensional shape where a point could be defined by no less than three coordinates.

While these models live in three dimensions, they do not quite have access to all of them. You cannot define a point on them with two coordinates: they are more than a surface and less than a volume.

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Fourth Iteration Menger Sponge
The Menger Sponge for example has (mathematically) a volume of zero, but an infinite surface area.

 

2.3 – Calculating Fractal Dimensions

The following are three methods of calculating Hausdorff-Besicovitch dimension:
• The exactly self-similar method for calculating dimensions of mathematically generated repeating patterns.
• The Richardson method for calculating a dimensional slope.
• The box-counting method for determining the ratios of a fractal’s area or volume.

 

On another note:

In theory, higher (non-integer) dimensional fractals are possible.
As far as I’m concerned however, they’re not particularly good for anything in a three-dimensional world. You are more than welcome to prove me wrong though.

 

3.0 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.
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Excerpt from ‘Seeker’

 

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.

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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.

Cube-Rotation

 

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.

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3D representation of eight cubes folding in 4D space to form a Hypercube

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.

 

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3.3 – The Fifth and Sixth Dimensions

On the temporal side of things, adding the ability to move ‘left & right’ and ‘up & down’ in time gives us the fifth and sixth dimensions.

(For example: x, y, z, t1, t2, t3)

This is a space where one can move through time based on probability and permutations of what could have been, is, was, or will be on alternate timelines. For any one point in this space, there are six coordinates that describe its position.

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In spacial dimensions, it is theoretically possible to fold four-dimensional objects with a fifth dimension. However, it becomes increasingly difficult for us to visualise what is happening to the shapes that we’re folding.
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In theory, objects can keep being folded together into higher and higher spacial dimensions indefinitely. (R1, R2, R3,R4,R5, R6, Retc.)

There’s a terrific explanation of what happens to platonic solids and regular polytopes in higher dimensions on Numberphile: https://youtu.be/2s4TqVAbfz4

 

3.4 – Even Higher Dimensions

If we can take a point and move it through space and time, including all the futures and pasts possible, for that point, we can then move along a number line where the laws of gravity are different, the speed of light has changed.

Dimensions seven though ten are different universes with different possibilities, and impossibilities, and even different laws of physics. These grasp all the possibilities and permutations of how each universe operates, and the whole of reality with all the permutations they’re in, throughout all of time and space. The highest dimension is the encompassment of all of those universes, possibilities, choices, times, places all into a single ‘thing.’

These ten time-space dimensions belong to something called Super-string Theory, which is what physicists are using to help us understand how the universe works.

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Cross section of the quintic Calabi–Yau manifold
There may very well be a link between temporal dimensions and spacial dimensions. For all I know, they are actually the same thing, but thinking about it for too long makes my head hurt. If the topic interests you, there is a philosophical approach to the nature of time called ‘eternalism’, where one may find answers to these questions. Other dimensional models include M-Theory, which suggests there are eleven dimensions.

While we don’t have experimental or observational evidence to confirm whether or not any of these additional dimensions really exist, theoretical physicists continue to use these studies to help us learn more about how the universe works. Like how gravity affects time, or the higher dimensions affect quantum theory.