## No Dig for Victory

3Dimensional Greenhouse

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# Tag: structure

## No Dig for Victory

## Nepenthes

## The Nature of Gridshell Form Finding

### Foreword

# 7.0 Introduction to Shells

### 7.1 – Single Curved Surfaces

Uniclastic: Conoid (Conical paraboloid)### 7.2 – Double Curved Surfaces

### 7.3 – Translation vs Revolution

**Ruled Surface & Surface of Revolution (Circular Hyperboloid)**
# 8.0 Geodesic Curves

*The Geometry of Bending*)
### 8.1 – Basic Grid Setup

Initial set of curves
### 8.2 – Basic Gridshells

Uniclastic: Barrel Vault Geodesic Gridshell
#### On another note:

# 9.0 Form Finding

### 9.0 – Minimal Surfaces

*Kangaroo2* Physics: Surface Tension Simulation
### 9.2 – Catenary Structures

*Kangaroo2* Physics: Catenary Model Simulation
### 9.3 – Biomimicry in Architecture

## Omnis Stellae

**Omnis Stellae – Redrawing your own constellation**

###### “Only in the darkness can you see the stars”

###### Martin Luther King

## The Curves of Life

## Relevant Material

**Further Reading:**

## Fractalized Gates – A journey of Self Discovery

## ‘Entwine’ – Submission for Burning Man 2016

## Hexagonal Patterned Spacial Definitions

## Students create installations for global engineering practice’s London office refurbishment

## Burning Man Festival | Bismuth Bivouac

3Dimensional Greenhouse

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

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**.

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.

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*.’

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.

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*.

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.

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

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.

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.

Structural Behaviour of Basic Shells [Source: IL 10 – Institute for Lightweight Structures and Conceptual Design]

In terms of synthetic geometry, there’s more than one approach to generating anticlastic curvature:

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..

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.)

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

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**.

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

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

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

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

Addition of secondary set of curvesAfter 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.

Grid with independent setsHe’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.

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

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.

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

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

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

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.)

Alternate Gridshell Patterns [Source: IL 10 – Institute for Lightweight Structures and Conceptual Design]
### 8.3 – Gridshell Construction

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 – CanadaGridshells 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 – FranceGridshells 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.)

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

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

Hyperbolic Paraboloid: Flattened Plank Lines With JunctionsThis 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.

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

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**.

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 [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.

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

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.

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.

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 by humans.) In a way, this is the fundamental essence of the scientific method: to learn by observation.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

With Love,

Maya

“An organism is so complex a thing, and growth so complex a phenomenon, that for growth to be so uniform and constant in all the parts as to keep the whole shape unchanged would indeed be an unlikely and an unusual circumstance. Rates vary, proportions change, and the whole configuration alters accordingly.” – D’Arcy Wentworth Thompson

*“This is the classic reference on how the golden ratio applies to spirals and helices in nature.” – Martin Gardner*

What makes this book particularly enjoyable to flip through is an abundance of beautiful hand drawings and diagrams. Sir Theodore Andrea Cook explores, in great detail, the nature of spirals in the structure of plants, animals, physiology, the periodic table, galaxies etc. – from tusks, to rare seashells, to exquisite architecture.

He writes, “*a staircase whose form and construction so vividly recalled a natural growth would, it appeared to me, be more probably the work of a man to whom biology and architecture were equally familiar than that of a builder of less wide attainments. It would, in fact, be likely that the design had come from some great artist and architect who had studied Nature for the sake of his art, and had deeply investigated the secrets of the one in order to employ them as the principles of the other.*”

Cook especially believes in a hands-on approach, as oppose to mathematic nation or scientific nomenclature – seeing and drawing curves is far more revealing than formulas.

“*because I believe very strongly that if a man can make a thing and see what he has made, he will understand it much better than if he read a score of books about it or studied a hundred diagrams and formulae. And I have pursued this method here, in defiance of all modern mathematical technicalities, because my main object is not mathematics, but the growth of natural objects and the beauty (either in Nature or in art) which is inherent in vitality.*”

Despite this, it is clear that Theodore Cook has a deep love of mathematics. He describes it at the beautifully precise instrument that allows humans to satisfy their need to catalog, label and define the innumerable facts of life. This ultimately leads him into profoundly fascinating investigations into the geometry of the natural world.

*“An organism is so complex a thing, and growth so complex a phenomenon, that for growth to be so uniform and constant in all the parts as to keep the whole shape unchanged would indeed be an unlikely and an unusual circumstance. Rates vary, proportions change, and the whole configuration alters accordingly.” **– D’Arcy Wentworth Thompson*

D’Arcy Wentworth Thompson wrote, on an extensive level, why living things and physical phenomena take the form that they do. By analysing mathematical and physical aspects of biological processes, he expresses correlations between biological forms and mechanical phenomena.

He puts emphasis on the roles of physical laws and mechanics as the fundamental determinants of form and structure of living organisms. D’Arcy describes how certain patterns of growth conform to the golden ratio, the Fibonacci sequence, as well as mathematics principles described by Vitruvius, Da Vinci, Dürer, Plato, Pythagoras, Archimedes, and more.

While his work does not reject natural selection, it holds ‘survival of the fittest’ as secondary to the origin of biological form. The shape of any structure is, to a large degree, imposed by what materials are used, and how. A simple analogy would be looking at it in terms of architects and engineers. They cannot create any shape building they want, they are confined by physical limits of the properties of the materials they use. The same is true to any living organism; the limits of what is possible are set by the laws of physics, and there can be no exception.

*“You could look at nature as being like a catalogue of products, and all of those have benefited from a 3.8 billion year research and development period. And given that level of investment, it makes sense to use it.” – Michael Pawlyn*

Michael Pawlyn, one of the leading advocates of biomimicry, describes nature as being a kind of source-book that will help facilitate our transition from the industrial age to the ecological age of mankind. He distinguishes three major aspects of the built environment that benefit from studying biological organisms:

The first being the quantity on resources that use, the second being the type of energy we consume and the third being how effectively we are using the energy that we are consuming.

Exemplary use of materials could often be seen in plants, as they use a minimal amount of material to create relatively large structures with high surface to material ratios. As observed by Julian Vincent, a professor in Biomimetics, “materials are expensive and shape is cheap” as opposed to technology where the inverse is often true.

Plants, and other organisms, are well know to use double curves, ribs, folding, vaulting, inflation, as well as a plethora of other techniques to create forms that demonstrate incredible efficiency.

The project is inspired by the interaction between people. It celebrates the female and male union and brings together people that already know each other or are completely strangers. Its interactive, it challenges its visitors, its playful.

It invites people to take part and be interactive with each other. It is called like that as it challenges people to reach the final gate and come together. It acts indeed like a mistletoe. Mysterious and magical. The moment you stand beneath the final gate you stand where all the magic happens. You feel the power and need to kiss the other person. Its design celebrates this union and offers a magical mutation.

It offers a labyrinth of complex, intriguing that seek to nourish individual experiences.

Black rock city seems to be the promised land, the promise of freedom, of self-expression, of immediacy and creativity and community. In a community like Burning Man you can assume the right to approach any random person and have an interesting interaction. You can overly express interest and curiosity. The Desert Mistletoe aims to give people a chance to be more expressed, more playful. Its for those people who want deeper connections, more meaningful interactions, less seriousness and more play.

Hugs and affection are a particularly significant aspect in which to expect more from strangers. We all need love, hugs and kisses are one of the best ways to deliver it. Take the risk to go in for a kiss. Of course some people will be hesitant. The project challenges people to express affection and admiration and relax. Hugs and kisses bring us together. At Burning Man, the endless parade of people flaunting their unusualness brings joy and excitement.

The unusual is both delightful and challenging. People love the unusual, the extraordinary and anything out of our everyday lives and routines. At Burning Man you do not need to sacrifice your wonderful weirdness. On the contrary you are challenged to explore your playful impulses and discover or express your freaky freedom.

The project is inspired specifically by the people of Burning Man.

**‘Burners respect the gift of being touched’ **

**Inspiration – The Symbol of Sri Yantra**

The Sri Yantra is conceived as a place of spiritual pilgrimage. It is a representation of the cosmos at the macrocosmic level and of the human body at the microcosmic level. It is a Journey. A spiritual journey from the stage of material existence to the union of the individual soul with the divine. The spiritual journey is taken as a pilgrimage in which every step is an ascent to the center. It represents a movement beyond one’s limited existence and every level is nearer to the goal. All the stages are within a gated frame, which is called the ‘earth citadel’ and forms an enclosed space.

The geometry

The symbol consists of nine interlocking triangles, centered around a bindu ( the central point). The five downward pointing triangles represent Shakti; the female principle. The four upright triangles represent Shiva; the male principle. The nine interlocking triangles form 43 small triangles each representing a particular aspect of existence.

**The Concept – Fractals & Koch Snowflake **

The proposed project is inspired by the concept of fractals – a never ending pattern, applied to a specific geometry. They are infinitely complex patterns created by repeating a simple process over and over, creating patterns that are self-similar across different scales. Fractals are images of dynamic systems that are driven by recursion.

The idea is based on the Koch Snowflake, a mathematical curve which has a finite area bounded by an infinitely long line.

For the project a script was created in Grasshopper and different patterns were created based on the triangles extracted from the Sri Yantra symbol.

Below: Acrylic model based on the fractal pattern that was created above

The model below is based on the fractal pattern created by

- the triangles representing the Female principle
- the triangles representing the Male principle

Together they represent, like the Sri Yantra symbol, the union of female and male. A celebration of our Creation.

Developing the design and the creation of a journey through a structure

The initial idea was to extract some patterns and create something that will look like gates. Gates that could provide a journey to their visitors.

Further development of the gates was undertaken. The final concept consists of 4 gates which intersect with each other in order to provide a journey. There are 4 different designs for each gate. Each gate has a dense pattern in order to create something elegant and ornamental. The pattern is based on the Koch snowflake concept and has been applied to multiple directions. The gates have different heights and dimensions with the last one reaching up to 10m height. This was designed in such a way to provide a climax of the visitor journey at the final gate/stage.

**The design is all about a journey**. A personal journey with different experiences. A welcoming structure, inviting the visitors to interact with each other. It says a story, it represents a story and it invites you to create one.

Experiment model in order to understand at what points the gates could intersect. At this point, the gates are connecting only on the top.

This has as a result to have an ’empty’ structure on the sides.

All the gates are connected with each other not only at the top but also on the sides. To achieve best results and better connections, the 3 gates are duplicated and connected with each other. The second gate will be connected with the third one only at the bottom. The 4rth gate has three sides creating a triangular central space. The gates have been rotated along the 2 other sides allowing for its visitors to enter from different directions. This also creates an enclosed space, like the Sri Yantra symbol.

The proposed installation aims to be part of this beautiful journey of self-discovery. It takes part of expecting more from strangers, while noticing the weirdness of others and encourages everybody to express themselves. With expecting more of strangers it increases the likelihood that the people you meet will become a part of that sometimes elusive network of connections we call community.

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.

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.

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.The 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.

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 inspiration for this research came from the Asian artist Ren Ri, who uses bees in order to generate his sculptural work. He predefines the space for the bees to work with, and allows for a time period for the honeycombs to take shape.

There are three types of surface division that manage to fill up all the area with prime geometric space – triangular (S3), square (S4) and hexagonal (S6). Other types of surface division, either leave gaps between the prime elements, which need to be filled by secondary shapes, or are confined to irregular shapes.

Research shows that the most efficient way of dividing a surface is through a minimum number of achievable line intersections, or a maximum number of membranes. In either case, the hexagonal division fits the case. This type of organization is a second degree iteration from the triangular division. It is formed by identifying and connecting the triangular cell centroids.

Such as in the case of soap-bubble theory, these cells expand, tending to fill up all the surface area around them, and finally joining through communicating membranes.

From a structural point of view, the best integration is the triangular one, because of the way each element (beam) reacts to the variation of the adjacent elements.

By converting the elemental intersection in the hexagonal division from a single triple intersection to a triple double intersection, the structure would gain sufficient structural resistance. This can be done through two methods – translation or rotation. Translation implies moving the elements away from the initial state in order to open up a triangular gap at the existing intersection. This method results in uneven shapes. In the case of rotation, the elements are adjusted around each middle point until a sufficient structural component is created. It is through rotation that the shape is maintained to a relative hexagonal aspect, due to the unique transformation method.

Pursuing the opportunity to test the system through a 1:1 scale project, I was offered the chance to design a bar installation for a private event at the Saatchi Gallery. The project has been a success and represents a stage test for the system.

Moving further, the attempt was to implement dynamic force analysis to the design, through variation of the elemental thickness. The first test was a bridge design. The structure was anchored on 2 sides, and had a span of 5m.

The next testing phase includes domed structures, replicating modular structures and double curved instances.

At the end of a marathon day during which 23 students presented project work to **BuroHappold Engineering**’s top executives, four students from Diploma Studio 10 (DS10) at the University of Westminster were selected to build temporary installations for the global engineering practice’s soon to be refurbished London offices. The presentations were overseen by their tutors **Arthur Mamou-Mani **and** Toby Burgess**, and **Neil Billett, Andrew Best, Emma Greenough** **and James Solly** from BuroHappold.

The winners of the competition were:

**Garis Iu**(71 Newman Street window) with**The Meander**, an 12 meters long flowing cluster of laser-cut curved origami filtering views on street level towards the reception area.**Diana Raican**(17 Newman Street window) with**The Colliding Cubes**, a dramatic 5 meters wide wall suspended above the street and assembled with friction-based component dissolving through one another, parametrically designed to filter light and views.**Joe Leach**(17 Newman Street Staircase) with**The Falling Leaves**, an innovative laser-cut curved truss system assembled around a series of nodes holding a beautiful array of wooden leaves.**Charlotte Yates**(17 Newman Street separating screen) with**The Jitterbug,**a kinetic installation inspired by Buckminster Fuller and made from punch-pressed aluminium icosahedron opening and closing depending on space requirements.

The projects celebrate 40 years of innovative structural solutions from BuroHappold and the practice’s commitment to supporting education in the fields of architecture, engineering and digital fabrication and will have a lifespan of a year before the next DS10 intake follow the same process and vie for an exhibition space.

We would like to thank **David Scott** and **Edward Lancaster** from the University of Westminster’s Fabrication Laboratory for all their support.

Here are images of the winning designs. More news to come on the expected launch date.

The student work will join the permanent installation “Wooden Waves” designed by DS10 tutor Arthur Mamou-Mani. More information on this project may be found on http://mamou-mani.com/BuroHappold.

Help us reach our Kickstarter target! http://kck.st/1ESCVFb

Inspired by the geometry from the crystalline growth pattern of the element Bismuth (Bi), the Bismuth Bivouac is a playful pavilion that celebrates the orthogonal geometries that can exist in natural Bismuth crystals to form an intriguing cubic structure, with spiralling disruptions on each face that are governed by the golden ratio. From a distance, the structure appears as a seemingly solid cube, but upon closer inspection, the internal spaces can be explored and utilised.The beautiful iridescent colours of crystal are to be translated into the proposal through coloured LED strip lighting, built into the simple dimensional lumber structure of the pavilion, so at night the Bismuth Bivouac gives has the same visually mesmerizing, colourful effect of the bismuth crystals in nature. The project aims to play with the participants perception of depth and scale in this mirroring structure – from afar, the structure will appear as a dense cube that sits on the playa, but as the participants move towards the structure, they will begin to be able to see through parts of the structure due to the stepped nature of the geometry and holes formed from spiral disruptions. The structure provides sheltered from harsh desert sun, but also provides a plaything for the sun to casts its shadows during the day, and for people to cast their own shadows with their own illuminations at night.

**Development**

**Construction**

**Visualizations**