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..)).
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Differential Growth with Rhino & Grasshopper – Kangaroo 2 – NonPlanar
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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

Omnis Stellae

Omnis Stellae – Redrawing your own constellation

“Only in the darkness can you see the stars”
Martin Luther King

 

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

Example of Stellation diagram on a platonic polygon

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

Omnis Stellae – Daytime interior render view

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

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

Omnis Stellae – Daytime exterior render view

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

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

Omnis Stellae – Daytime interior render view from the ground

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

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

 

Omnis Stellae – Nightime exterior render view

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

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

Omnis Stellae – Daytime interior render view from above

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

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

Omnis Stellae – Daytime and Nightime

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

Omnis Stellae – Daytime interior render view

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

With Love,

Maya

 

 

 

Resonance cryptograph

A2-wide angle perspective.jpgJohann Wolfgang von Goethe says Architecture is frozen music. Albert Einstein believes the key to unlocking the universe is through the hidden geometry and mathematics.  This design seeks to unlock the geometry of Sound making sound visible through 3-dimensional volume and lights.

Johann Wolfgang von Goethe says Architecture is frozen music. Albert Einstein believes the key to unlocking the universe is through the hidden geometry and mathematics.

Sound is a hidden code when it unlocks allows us to perceive it as a set of geometrical patterns. The mechanic of sound is translated visually through frequency and amplitude represents itself with beautiful geometries as code from the universe. My design recreates Sound’s geometries into a physical symbolic Sanctuary for users to retreat their senses in the desert,to unravel meaning behind the symbol of Sound by deconstructing it and re-dressing it with physical form, making Sound visible.

 This design seeks to unlock the geometry of Sound making sound visible through 3-dimensional volume and lights.

 

The structure measures 13.77 feet in length &12.8 feet in height. The material for the structure would be paneled by birch plywood(4ft. x 2ft. panel).2-D dimensional geometry is translated into 3-Dimensional form by folding and joining edges.The sanctuary is made up of three mirroring layers, stacking vertically. The construction of the structure is to explore double curvature design with single curvature paneling and assembly. The ground storey encourages private space for reflection; individual sitting and resting area are carved inwards towards the air-well  ,in contrast, the upper storey is the communal area within the enclosure where users can access from a ladder. Pocket of windows are generated by the stacking and mirroring of sound vibration patterns.  Users enters into the enclosure and view the desert from within.

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Live feeding of Sound and the changing LED lights

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In the night, live feeding of sound is captured when in contact with the surfaces of the sanctuary. With a contact microphone attaches onto the surface, it captures the sound amplitude when a user touches or tap as sound travels through the surface as a medium. The device(computer coding with Arduino) then translates the amplitude variation (loudness) into changing colours of LED lights. The lights are attached on the rim of the panels.

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Process
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Studies of Sound patterns through water

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Harmonograph

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Eigen vector

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3 Days left to help us with the Tangential Dreams Crowdfunding Campaign

Hello WeWantToLearn community. We’re going to Burning Man in less than a month!

Our project this year will be a physical manifestation of our collective dreams and is called Tangential Dreams.  It is a seven meters high temporary timber tower displaying inspiring messages from around the world, written on a multitude of swirling “tangents”.

We need your help to realise our project! There is only three days left to collect the missing £5,000 on our crowdfunding campaign to finance the many expenses associated with the creation of such an ambitious project.

Please click on the image below or use the following shortlink to share/help – everything helps: http://kck.st/28KlbPk 🙂

 

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MamouMani_TangentialDreams (15)

The project is a climbable sinuous tower made from off-the-shelf timber and digitally designed via algorithmic rules. One thousand “tangent” and light wooden pieces, stenciled with inspiring sentences, are strongly held in position by a helicoid sub-structure rotating along a central spine which also forms a safe staircase to climb on. Each one of the poetic branches faces a different angle, based on the tangent vectors of a sweeping sine curve. In line with this year’s theme, the piece is reminiscent of Leonardo’s Vitruvian man’s movement, helicoid inventions such as the “aerial screw” helicopter and Chambord castle helicoid staircase as well as his deep, systematic, understanding of the rules behind form to create art. From a wave to a flame all the way to a giant desert cactus, the complex simplicity of the art piece will trigger many interpretations, many dreams.

The art piece attempts to maximize an inexpensive material by using the output of an algorithm – (the value of the piece being the mathematics behind it, as well as the experience, not the materials being used). The computer outputs information to locate the column, sub-structure and tangents.  We believe digital tools in design are giving rise to a new Renaissance, in which highly sophisticated designs, mimicking natural processes by integrating structural and environmental feedback, can be achieved at a very low cost. We worked very closely with our structural engineer format, sharing our algorithms, to give structural integrity to the piece and resist the strong climbing and wind loads. There are now three “legs” to our proposal, each rotated from each other at 60 degrees angles around a central solid spine, to ensure the stability of the piece, similarly to a tripod. The tangents are not just a decoration, they act as a spiky balustrade to prevent people from falling.

We have a fantastic team for the project:  Philip Olivier, Eira Mooney, Maialen Calleja, Aaron Porterfield, Sebastian Morales, Antony Dobrzensky, Laura Nica, Karina Pitis, Hamish Macpherson, Jon Goodbun, Yannick Yamanga, Matthew Springer ,Josh NG ,Lola Chaine, Dror BenHay, Peter Wang, Charlotte Chambers, Michael DiCarlo, Sandy Kwan.

 

We want our structure to have an intangible aspect, a magical side, one that is beyond matter and geometry. We want to connect our art with every each of you and make you part of our own BIG DREAM, building Tangential Dreams.
We want our structure to have an intangible aspect, a magical side, one that is beyond matter and geometry. We want to connect our art with every each of you and make you part of our own BIG DREAM, building Tangential Dreams.

 

We use physical modelling as a way to understand how the pieces fit together, the best assembly sequence as well as the structural integrity of the project. It takes time, material, money to create a truly original project.
We use physical modelling as a way to understand how the pieces fit together, the best assembly sequence as well as the structural integrity of the project. It takes time, material, money to create a truly original project.

 

Gif Animation of the assembly process. the project will take two weeks to pre-cut and assemble together with volunteers. We need your help for all the expenses.
Gif Animation of the assembly process. the project will take two weeks to pre-cut and assemble together with volunteers. We need your help for all the expenses.

 

 

Exciting rewards to thank you for your supports! from top left to bottom right: Pendants, Earrings, T-Shirts, Tangents, Vase, Ceiling Panels, 3D Printed Smoke Stool, Full Physical Model.
Exciting rewards to thank you for your supports! from top left to bottom right: Pendants, Earrings, T-Shirts, Tangents, Vase, Ceiling Panels, 3D Printed Smoke Stool, Full Physical Model.

 

 

‘Entwine’ – Submission for Burning Man 2016

Final Day Render

INSTALLATION SUBMISSION TO BURNING MAN 2016 – ‘Entwine’

Entwine is a timber frame structure which has been developed through rigorous physical and digital testing to ensure a safe climbing frame for all to enjoy. When exploring Entwine, the vast expanse of the playa is framed through beautiful intertwining curved plywood beams. Burners can view the event from glorious vantage points nestled amidst multiple communal spaces that encourage interaction and play.

The structure predominantly consists of strips of curved plywood which have been connected together using pioneering construction techniques, specifically the utilisation of conflicting forces, similar to those apparent in ‘Tensegrital’ design. Drawing inspiration from Leonardo Da Vinci and his various experimentations with physical form, ‘Entwine’ is a marvel of geometry. The piece is formed from an arrangement of 19 octahedral components, each consisting of six beams, which are paired and positioned upon one of three axis. These three elements represent the unity of man, nature and the universe that surrounds us.

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Each modular component is tessellated to form an octahedral space frame structure. The rigidity resulting from this tessellation is in direct contrast to the curving structural beams which exude an organic aesthetic. As Burners view Entwine from different aspects, a remarkable array of different patterns and forms are revealed, many bearing resemblance to sacred geometry, specifically the Flower of Life, which was a significant study within Leonardo Da Vinci’s work.

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Entwine is unorthodox in its composition, and this is a contributing factor to what makes it so unique: Each module is constructed through tensioning layers of ¼ inch thick plywood, which are then mechanically fixed together when a desired radius has been reached. By laminating the plywood in this manner, each component retains its curvature but remains in compression. These conflicting forces are integral to the design of Entwine: Each octahedral module is constructed from these compressed plywood elements, and are held together with tensioning ropes creating a structure of isolated components in compression within a net of continuous tension.MODEL PHOTOGRAPHSMODEL PHOTOGRAPHS 2The form of the structure is based on the octahedron, which is a Platonic solid composed of eight equilateral triangles; four of which meet at each vertex. One of the eight triangles acts as a base for the structure. This results in one edge creating a small cantilever, whilst the counter edge can be anchored to the ground. As previously studied by Buckminster Fuller, the geometry of an octahedron is particularly good at forming space frames with a strong cantilevers.

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Entwine Construction Proposal

The participatory aspect of the installation voids the role of the ‘spectator’ and creates more active engagement. In many of Leonardo Da Vinci’s paintings, his subjects are framed by surreal, dreamlike landscapes. This is reflected within Entwine: As Burners become part of the installation, they are framed by the awe inspiring backdrop of Black Rock Desert: In many ways Entwine becomes the artist, the playa the canvas, and Burners the subjects.

“the artist is not a special sort of person, but every person is a special sort of artist.”

This is not only true in the sense of physical involvement but during the construction the ‘spectator’ becomes involved in making strategic decisions in the realisation of the work of art. The development, design and construction of the project embodies the principles of self-reliance and self-expression, whilst a proposal that is safe, interactive and beautiful will be gifted to the community at Burning Man.

Entwine’s curving form will be illuminated using LED spot lights to enhance the organic patterning existent within the structure. This allows the full form of the structure to be fully visible.

The Butterfly Egg

Geometry can be found on the smallest of scales, as is proven by the beautiful work of the butterfly in creating her eggs. The butterflies’ metamorphosis is a recognised story, but few know about the start of the journey. The egg from which the caterpillar emerges is in itself a magnificently beautiful object.

Geometry can be found on the smallest of scales, as is proven by the beautiful work of the butterfly in creating her eggs. The butterflies’ metamorphosis is a recognised story, but few know about the start of the journey. The egg from which the caterpillar emerges is in itself a magnificently beautiful object. The tiny eggs, barely visible to the naked eye, serve as home for the developing larva as well as their first meal.

White Royal [Pratapa deva relata] HuDie's Microphotography
White Royal [Pratapa deva relata] HuDie’s Microphotography
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Clockwise: Hesperidae, Nymphalidae, Satyridae, Pieridae

Each kind of butterfly has its unique egg design, creating a myriad of beautiful variations.

These are some of the typical shapes that each family produce.

But it is the Lycaenidae family that have the most geometrical and intricate eggs.

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Lycaenidae

Other eggs
Lycaenidae eggs from left to right: Acacia Blue [Surendra vivarna amisena], Aberrant Oakblue [Arhopala abseus], Miletus [Miletus biggsii], Malayan [Megisba malaya sikkima]. HuDie’s Microphotography
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Biomimetics, or biomimicry is an exciting concept that suggests that every field and industry has something to learn from the natural world. The story of evolution is full of problems that have been innovatively solved.

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There are thousands of species of butterfly, each with their unique egg design. 3A truncated icosahedron for a frame, the opposite of a football. Instead of panels pushed out, they are pulled in.

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Fractals are commonly occurring in nature, and can be described as a never-ending pattern on different scales. People are subconsciously familiar with fractals, so are inherently more relaxed when surrounded by them.

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3D Printing is a relatively new technology that is set to change our world. Innovations in the uses of 3D printers, combined with falling costs, means that they could be a ubiquitous tool in every home and industry. 3D printers and scanners are already used a great deal in everything from the biomedical field to art studios, and experiments are currently being done to construct entire homes. This technology is in its infancy, and it is exactly for this reason that every effort should be taken to research its potential. It is common to use 3D printers in architecture to show small working models, I would like to now use it to make a large and complex structure at full scale.

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This research will underpin the design of a sculptural installation in which people can interact with live butterflies. With the ever-declining numbers of butterflies worldwide and in the UK, conservation and education are paramount.

The link between butterflies and humans in our ecosystem is one that is vital and should be conserved and celebrated.

I can imagine an ethereal space filled with dappled light where people can come for contemplation and perhaps their own personal metamorphosis.

Interior

—Tia