The breathing pods are bio-machines focused on creating a natural solution to the issue of pollution within large cities. Continuing the aim of City Tree walls placed around London: the pods offer an architectural and natural rendition inspired by AI generated Midjourney images. A single pod is a semi-enclosed space created from structural wooden logs placed in a circular motion: depending on one another to hold themselves up. Moss growth in its natural habitat and wind velocity created through the gaps within the logs make 1 breathing pod equal to the air filtration of up to 600 urban trees during high wind levels.
The main aspects of the Corn-Crete House system are the use of space, material efficiency and relationship to site. The way space is shaped influences human behaviour. According to a research paper done by KAYVAN MADANI NEJAD in 2007 the curvilinearity of interior design directly affects the way people feel inside them. It concluded that the more curvilinear a space is the more comfortable, safe, relaxed and friendly it feels. My project builds upon this argument. Research also shows that the concrete industry is a major environment pollutant. Cement is the most damaging ingredient. I am proposing a new system which will be using less concrete & less cement thanks to: 1) corn residues partially replacing aggregate making the structure lighter and more porous 2) casting around inflatables resulting in curvilinear architecture suitable for compression which requires less tensile strength.
Azolla is a minuscule floating plant that forms part of a genus of species of aquatic ferns, also known as Mosquito Fern. It holds the world record in biomass producer – doubling in 2-3 days. The secret behind this plant is itssymbiotic relationship with nitrogen fixing cyanobacterium Anabaena making it a superorganism. The Azolla Provides a microclimate for the cyanobacteria in exchange for nitrate fertiliser. Azolla is the only known case where a symbiotic relationship endures during the fern’s reproductive cycle and is passed on to the next generation. They also have a complimentary photosynthesis, using light from most of the visible spectrum and their growth is accelerated with elevated CO2 and Nitrogen.
Azolla is capable of producingnatural biofertiliser, bioplastics because of its sugar contents and biofuel because of the large amount of lipids. Its growth requirements can accommodate many climates too, allowing it to be classified as a weed in many countries. I was able to study the necessary m2 of growing Azolla to sequester the same amount as my yearly CO2 emissions, resulting in 57% of a football field equivalent of growing Azolla to make me carbon neutral.
Why is this useful? Climate change will inevitably bring more adverse climate conditions that will put many world wide crops at risk and, as a consequence, will affect our lives. A crop that produces biomass at the speed of Azolla provides at advantage in flexibility: a soya bean can take months to grow until ready to be harvested, Azolla can be harvested twice a week. This plant has the potential to be used in the larger agricultural sector and diminish the Greenhouse Gas Emissions of one of the most pollutant sectors.
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REAL LIFE ACTION
I contacted the Azolla Research Group at the University of Utrechtand they kindly accepted to give us a tour of their research facilities, providing us with an in-depth insight into the aquatic fern. I also decided to approach the Floating Farm with a proposal of using Azolla in their dairy process. They agreed to explore this and I put them in contact with the research team in the University of Utrecht, who are now cooperating with the dairy farm’s team in decreasing the carbon emissions of the cows on the farm.
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ARCHITECTURAL PROPOSAL
Floating Azolla District Masterplan
The Floating Azolla District consists on a proposed community that emphasises a circular economy with a focus on sustainable agriculture in Rotterdam. It builds on to the existing Floating Farm found in the M4H area. It is formed of three areas:
1) Azolla – Dwellings combining a series of residential units for the increasing number of young entrepreneurs in the RID with three central cores growing stacked trays of Azolla as in vertical farming.
2) The Floating Farm which continues to produce dairy products and a Bamboo growing area to maintain the upkeep of floating platforms and construction of new dwellings. Floating rice paddies are grown in the warmer months in a closely monitored system of permaculture.
3) A production facility which concentrates on research and development into Azolla as well as retrieving the water fern’s byproducts such as bioplastics extracted from the sugars; biofuel, from the lipids; and bamboo plywood lumber for the construction of the expanding Floating District.
Dwellings
Floating Farm
Bamboo Growing Pods
Closed System in the Floating Azolla District
Floating Azolla District – Dwellings
Floating Azolla District – Dwellings
This section concentrates on the detail construction of the Azolla-Dwellings. These floating units are designed to be used as a combination of co-housing for entrepreneurs working in the Rotterdam Innovation District, where the Floating Farm is located, and indoor Azolla growing facilities which is then used further along in the masterplan. The growing areas are built on a series of building components that provide support for trays of Azolla to be grown in a vertical farming manner and provide support for the floor plates as well as anchoring for the entire dwelling.
Dwellings Construction SequenceExploded Axo – Dwellings
The materials are a combination of local bamboo grown on a series of floating platforms that prevent the cold winter winds from affecting the overall masterplan and pallets sourced from neighbouring industrial facilities. Using the reciprocal building system developed in Brief 1, a series of stacked components are linked to form the vertical farming support for the Azolla. This system is then extended to support the floors for the dwellings.
Azolla Vertical Farming Support
Similar to an aperture ring on a camera, this mechanism uses the varying tide to automatically collect the Azollafrom the vertical farming trays to then be used throughout the Masterplan. By displacing 2.5% of the area for each tray every tidal change, this mechanism collects 50% of the harvest every 10 days allowing for a continuous growth of Azolla.
Azolla Collection MechanismAzolla Vertical Farming Trays and Floor SupportAzolla Vertical Farming Trays and Floor Support
The dwellings’ facade is a result of a careful analysis of harmful and beneficial solar radiation. By setting an initial average temperature to monitory, the facade will block sun that naturally would drive the temperature above the chosen one and the beneficial would bring the temperature up. This shading serves a buffer zone that surrounds the internal living spaces and is used to grow vegetables for the residents.
Thermal Responsive Facade Options.
Semi-public spaces are located on the ground floor (open plan kitchen and living) and bedrooms are located on the first floor, surround a central spiral staircase for circulation.
Ground Floor Plan
First Floor Plan
Exterior View
Interior View
The same building system based on reciprocal structures is coated in azolla bioplastic preventing the wood from rotting and making the form waterproof. These are used as underwater columns which allow the dwellings and platforms to float. Each ‘column’ can support a load of 2011Kg.
Support Floating Columns
SITE LOCATION
Floating Farm Rotterdam
Based on relationship between the University of Utrecht and the Floating Farm taking place outside the initially academic intention of the visit, I decided to use the Floating Farm as a site and a starting point for my proposal. The floating farm is intended to stand out and create an awareness of the possibility or idea of living on water and taking ownership of one’s food production, which seems to match the potential uses and benefits of Azolla. The researchers at the University of Utrecht expressed their need of getting the advantages of this plant to a wider public and this remained in my mind, possibly being the main reason behind my approach to the Floating Farm.
The Floating Farm sits in the Merwehaven area or M4H in the Port of Rotterdam. Highlighted below are the natural site conditions that determined the placement of the masterplan parts according to their function. Bamboo growing pods are placed southwards of the port to block the winter wind while allowing the summer winds from the west to navigate through.
Site Study
In 2007, Rotterdam announced its ambition to become 100% climate-proof by 2025 despite having 80% of its land underwater, therefore it was important to look at the flood risk and tidal change. The Merwehaven area in Rotterdam seems to have an average tidal change of 2 metres which I thought could be taken advantage of in a mechanical system mentioned previously.
Tidal Changes in the Floating Farm RotterdamFlood Risk & Tidal changes in Rotterdam
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BUILDING SYSTEM & MATERIAL RESEARCH
The reciprocal building system used in the construction of the dwellings began by looking into the Fern plant and its’ form. All ferns are Pinnate – central axis and smaller side branches – considered a primitive condition. The veins never coalesce and are known to be ‘free’. The leaves that are broadly ovate or triangular tend to be born at right angles to the sunlight.
Reciprocal Fern Building System
I then decided to model a leaf digitally, attempting to simulate the fractal nature found in a fern frond and the leaves to 3 degrees of fractals. I then simplified the fern frond to 2 levels to allow for easier laser cutting and structural stability. The large perimeter meant, therefore, there was a large amount of surface area for friction so I explored different configurations and tested their intersections.
Iteration 1 Component Study
I then selected the fern frond intersection I found to show the best stability out of the tested ones shown previously. By arraying them further, they began to curve. When pressure is applied to the top of the arch, the intersections are strengthened and the piece appears to gain structural integrity.
Iteration 2 Double Curvature Study
When a full revolution is completed, the components appear to gain their maximum structural integrity. Since I had decided to digitally model the fern frond,I was able to decrease the distance between the individual leaves in the centre of each frond through grasshopper. By doing this, the intersections connecting a frond with another were less tight in the centre than on the extremities of each frond,allowing for double curvature.
Iteration 2 Double Curvature Study – Stress TestIteration 2 Double Curvature Study – Stress Test Findings
I continued to iterate the leave by decreasing any arching on the leaf and finding the minimum component, the smallest possible component in the system. By arraying a component formed of 3 ‘leaves’ on one hand and 2 on the other, I would be able to grow the system in one direction as before due to the reciprocal organisation and in the other direction by staggering the adjacent component. The stress tests of this arrangement showed a phased failure of the ‘column’. Instead of breaking at once, row by row of components failed with time, outwards-inwards.
Iterations 3, 4, 5 Minimum Building Component
Iteration 5 Minimum Building Component – Stress TestIteration 5 Minimum Building Component – Stress Test Findings
I extracted the minimum possible component from the previous iterations and attempted to merge the system with firstly, 3d printed PLA bioplastic components and then with an algae bioplastic produced at home. I became interested in the idea of being able to coat the wood in an algae bioplastic substituting the need for any epoxy for waterproofing. The stress tests for this component showed a surprising total of 956 kg-force for it to fail.
Agar Bioplastic Test
Here, I began combining different quantities of vegetable glycerine, agar agar(extracted from red algae and used for cooking) and water. By changing the ratios of agar and glycerine I was able to create 2 different bioplastics: one being brittle and the other flexible. See above for the flexible sample and below for the brittle sample. Both samples appeared to fail under the same 7 Kg-force.
Gum Arabic is a natural adhesive grown on the Senegalia Senegal tree. This tree grows in 6 years and only requires 100-200ml of water a year, this is a tree which has evolved to survive in the desert.
How can a third world country like Sudan can use a natural adhesive to act as a binder? How can we use the natural terrain as a framework for creating complex and controllable design?
Below the images illustrate how the construction process works.
I have designed a time-based construction programme. It begins with a setting out plan where string is used to determine where an industrial hoover (which usually transfers sand) sucks out sand and it is spewed out elsewhere. When my mix has been added to these cone shape voids the hoovering process is repeated but this time a thick layer of the Gum Arabic, Clay and Sand mix. The mixture is then lightly misted with salt water which causes the Gum Arabic to act as the binder. The Desert’s scorching sun then does the rest to solidify the material. Excavation around the land enables a structure which stands upright.
After exploring this method myself I discovered some interesting variations depending on whether I suck the sand first or pour it
I explored these forms digitally.
If you are wondering how I got to this point, well I will jump back to the beginning.
It started in Kew Gardens London, where I chose to study a plant and look into the early stages of bio-mimicry. I chose to study the Lotus Pod (Nelumbo Nucifera) found in Asia.
I wanted to find consistency between the holes of the flowers. Therefore I purchased 40 flower heads and begun experiments to study the arrangement of holes and the parameters within the plant. After experimenting I discovered that flower heads sized between 50-60mm have a gradient like effect where the largest hole is 3.5 times larger than the smallest. I therefore used Frei Ottos sand draining technique to explore what forms can be achieved with the arrangement of holes being that of the Lotus Pod.
After designing and building a smart box I began a matrix study.
I then Explored the parameters of each of these and found out that this sand grain drains at 30 degrees.
From this point I went on to look at how to solidify sand in its current form and that is when I discovered the properties of the Gum Arabic and began to explore. I had began to mix the mixture with sand and clay.
I then explored a site based on where Gum Arabic is produced and where sand and clay is in abundance. Therefore leaving me with Al-Fashir Sudan.
I then Explored the construction techniques using the gravity. Using the terrain as a natural formwork which can be moulded.
I then continued to design a construction process which requires less labour and would achieve high quality design attributes. Which is where I began with the hoovering process.
Through extensive research into the construction of grid shells, as well as differential geometry, I present a design solution for a complex grid structure inspired by the highly symmetrical and optimised physical properties of a triply periodic minimal surface. The proposal implements the asymptotic design method of Eike Schling and his team at Technical University of Munich.
‘Minimal Matters’ utilises the several geometric benefits of an asymptotic curve network to optimise cost and fabrication. From differential geometry, it is determined asymptotic curves are not curved in the surface normal direction. As opposed to traditional gridshells, this means they can be formed from straight, planar strips perpendicular to the surface. In combination with 90° intersections that appear on all minimal surfaces (soap films) this method offers a simple and affordable construction method. Asymptotic curves have a vanishing normal curvature, and thus only exist on anticlastic surface-regions.
Asymptotic curves can be plotted on any anticlastic surface using differential geometry.
On minimal surfaces, the deviation angle α is always 45 (due to the bisecting property of asymptotic curves and principle curvature lines). Both principle curvature networks and asymptotic curve networks consist of two families of curves that follow a direction field. The designer can only pick a starting point, but cannot alter their path.
(a) Planes of principle curvature are where the curvature takes its maximum and minimum values. They are always perpendicular, and intersect the tangent plane.
(b) Surface geometry at a generic point on a minimal surface. At any point there are two orthogonal principal directions (Blue), along which the curves on the surface are most convex and concave. Their curvature is quantified by the inverse of the radii (R1 and R2) of circles fitted to the sectional curves along these directions. Exactly between these principal directions are the asymptotic directions (orange), along which the surface curves least.
(c) The direction and magnitude for these directions vary between points on a surface.
(d) Starting from point, lines can be drawn to connect points along the paths of principal and asymptotic directions on the respective surface.
Gyroid TPMS
The next step is to create the asymptotic curve network for the Gyroid minimal surface; chosen from my research into Triply Periodic Minimal Surfaces.
As the designer, I can merely pick a starting point on an anticlastic surface from which two asymptotic paths will originate. It is crucial to understand the behaviour of asymptotic curves and its dependency on the Gaussian curvature of the surface.
Through rotational symmetry, it is resolved to only require six unique strips for the complete grid structure (Seven including the repeated perimeter piece).
The node to node distance, measured along the asymptotic curves, is the only variable information needed to draw the flat and straight strips. They are then cut flat and bent and twisted into an asymptotic support structure.
Plywood Prototype: 600mm cubed
Eight fundamental units complete the cubit unit cell of a Gyroid surface. Due to the scale of the proposal, I have introduced two layers of lamellas. This is to ensure each layer is sufficiently slender to be easily bent and twisted into its target geometry, whilst providing enough stiffness to resist buckling under compression loads.
‘Minimal Matters’ aims to create an explorative, meditative and interactive experience for visitors. It is a strained grid shell utilising the geometrical benefits of an asymptotic curve network; digitally designed via algorithmic rules to minimise material, cost, and construction time.
A minimal surface is the surface of minimal area between any given boundaries. In nature such shapes result from an equilibrium of homogeneous tension, e.g. in a soap film.
Minimal surfaces have a constant mean curvature of zero, i.e. the sum of the principal curvatures at each point is zero. Particularly fascinating are minimal surfaces that have a crystalline structure, in the sense of repeating themselves in three dimensions, in other words being triply periodic.
Many triply periodic minimal surfaces are known. The first examples of TPMS were the surfaces described by Schwarz in 1865, followed by a surface described by his student Neovius in 1883. In 1970 Alan Schoen, a then NASA scientist, described 12 more TPMS, and in 1989 H. Karcher proved their existence.
My research into grid structures with the goal of simplifying fabrication through repetitive elements prompted an exploration of TPMS. The highly symmetrical and optimised physical properties of a TPMS, in particular the Gyroid surface, inspired my studio proposal, Minimal Matters.
Gyroid: left: Fundamental region, middle: Surface patch, right: Cubic unit cell
Evolution of a Gyroid Surface
The gyroid is an infinitely connected periodic minimal surface discovered by Schoen in 1970. It has three-fold rotational symmetry but no embedded straight lines or mirror symmetries.
The boundary of the surface patch is based on the six faces of a cube. Eight of the surface patch forms the cubic unit cell of a Gyroid.
For every patch formed by the six edges, only three of them is connected with the surrounding patches.
Note that the cube faces are not symmetry planes. There is a C3 symmetry axis along the cube diagonal from the upper right corner when repeating the cubic unit cell.
Curiously, like some other triply periodic minimal surfaces, the gyroid surface can be trigonometrically approximated by a short equation:
cos(x)sin(y)+cos(y)sin(z)+cos(z)sin(x)=0
Using Grasshopper and the ‘Iso Surface’ component of Millipede, many TPMS can be generated by finding the result of it’s implicit equation.
Standard F(x,y,z) functions of minimal surfaces are defined to determine the shapes within a bounding box. The resulting points form a mesh that describes the geometry.
TPMS Grasshopper Definition
A cube of points are constructed via a domain and fed into a function. Inputs of standard minimal surfaces are used as the equation.
The resulting function values are plugged into Millipede’s Isosurface component.
The bounding box sets up the restrictions for the geometry.
Xres, Yres, Zres [Integer]: The resolution of the three dimensional grid.
Isovalue: The ‘IsoValue’ input generates the surface in shells, with zero being the outermost shell, and moving inward.
Merge: If true the resulting mesh will have its coinciding vertices fused and will look smoother (continuous, not faceted)
Triply Periodic Minimal Surfaces generated by their implicit equations
The above diagrams show Triply Periodic Minimal surfaces generated from their implicit mathematical equations. The functions are plotted with a domain of negative and positive Pi. By adjusting the domain to 0.5, the surface patch can be generated.
Many TPMS can best be understood and constructed in terms of fundamental regions (or surface patches) bounded by mirror symmetry planes. For example, the fundamental region formed in the kaleidoscopic cell of a Schwarz P surface is a quadrilateral in a tetrahedron, which 1 /48 of a cube (shown below left). Four of which create the surface patch. The right image shows a cubic unit cell, comprising eight of the surface patch.
Schwarz P: left: Fundamental region, middle: Surface patch, right: Cubic unit cell Evolution of a Schwarz P Surface
Schoen’s batwing surface has the quadrilateral tetrahedron (1/48 of a cube) as it’s kaleidoscopic cell, with a C2 symmetry axis. As shown in the evolution diagram below, the appearance of two fundamental regions is the source of the name ‘batwing’. Twelve of the fundamental regions form the cubic unit cell; however this is still only 1/8 of the complete minimal surface lattice cell.
Schoen’s PA Batwing Surface: left: kaleidoscopic cell, middle: Fundamental region, right: Cubic unit cellEvolution of a Schoen’s Pa (Batwing) Surface
The natural world is brimming with ratios, and spirals, that have been captivating mathematicians for centuries.
1.0 Phyllotaxis Spirals
The term phyllotaxis (from the Greek phullon ‘leaf,’ and taxis ‘arrangement) was coined around the 17th century by a naturalist called Charles Bonnet. Many notable botanists have explored the subject, such as Leonardo da Vinci, Johannes Kepler, and the Schimper brothers. In essence, it is the study of plant geometry – the various strategies plants use to grow, and spread, their fruit, leaves, petals, seeds, etc.
1.1 Rational Numbers
Let’s say that you’re a flower. As a flower, you want to give each of your seeds the greatest chance of success. This typically means giving them each as much room as possible to grow, and propagate.
Starting from a given center point, you have 360 degrees to choose from. The first seed can go anywhere and becomes your reference point for ‘0‘ degrees. To give your seeds plenty of room, the next one is placed on the opposite side, all the way at 180°. However the third seed comes back around another 180°, and is now touching the first, which is a total disaster (for the sake of the argument, plants lack sentience in this instance: they can’t make case-by-case decisions and must stick to one angle (the technical term is a ‘divergence angle‘)).
Phyllotaxis Study: 180° (see corn leaves), 90° (see mint leaves), and 72° (see gentiana petals)
Next time you only go to 90° with your second seed, since you noticed free space on either side. This is great because you can place your third seed at 180°, and still have room for another seed at 270°. Bad news bears though, as you realise that all your subsequent seeds land in the same four locations. In fact, you quickly realise that any number that divides 360° evenly yields exactly that many ‘spokes.’
Phyllotaxis Study – 1,000 Seed Spread: 45°, 36°, and 20°
Note: This is technically true with numbers as high as 120, 180, or even 360(a spoke every 1°.) However the space between seeds in a spoke gradually becomes greater than the space between spokes themselves, leaving you with one big spiral instead.
Phyllotaxis Study – 1,000 Seed Spread: 8°, 5°, and 2°
1.2 Irrational Numbers
These ‘spokes’ are the result of the periodic nature of a circle. When defining an angle for this experiment, the more ‘rational’ it is, the poorer the spread will be (a number is rational if it can be expressed as the ratio of two integers). Naturally this implies that a number can be irrational.
Sal Khan has a great series of short videos going over the difference between the two [Link]. For our purposes, the important take-aways are:
-Between any two rational numbers, there is at least on irrational number.
–Irrational numbers go on and on forever, and never repeat.
You go back to being a flower.
Since you’ve just learned that an angle defined by a rational number gives you a lousy distribution, you decide to see what happens when you use an angle defined by an irrational number. Luckily for you, some of the most famous numbers in mathematics are irrational, like π (pi), √2 (Pythagoras’ constant), and e (Euler’s number). Dividing your circle by π (360°/3.14159…) leaves you with an angle of roughly 114.592°. Doing the same with √2 and e leave you with 254.558° and 132.437° respectively.
Phyllotaxis Growth Study: Pi, Square Root of 2, and Euler’s Number
Great success. These angles are already doing a much better job of dispersing your seeds. It’s quite clear to you that √2 is doing a much better job than π, however the difference between √2 and e appears far more subtle. Perhaps expanding these sequences will accentuate the differences between them.
Phyllotaxis Study – 1,000 Seed Spread: Pi, Square Root of 2, and Euler’s Number
It’s not blatantly obvious, but √2 appears to be producing a slightly better spread. The next question you might ask yourself is then: is it possible to measure the difference between the them? How can you prove which one really is the best? What about Theodorus’, Bernstein’s, or Sierpiński’s constants? There are in fact an infinite amount of mathematical constants to choose from, most of which do not even have names.
1.3 Quantifiable irrationality
Numbers can either be rational or irrational. However some irrational numbers are actually more irrational than others. For example, π is technically irrational (it does go on and on forever), but it’s not exceptionally irrational. This is because it’s approximated quite well with fractions – it’s pretty close to 3+1⁄7 or 22⁄7. It’s also why if you look at the phyllotaxis pattern of π, you’ll find that there are 3 spirals that morph into 22 (I have no idea how or why this is. It’s pretty rad though).
Phyllotaxis Voronoi Diagram – Proximity to Closest Neighbour: Pi
Generating a voronoi diagram with your phyllotaxis patterns is a pretty neat way of indicating exactly how much real estate each of your seeds is getting. Furthermore, you can colour code each cell based on proximity to nearest seed. In this case, purple means the nearest neighbour is quite close by, and orange/red means the closet neighbour is relatively far away.
Phyllotaxis Voronoi Diagram – Proximity to Closest Neighbour: Square Root of 2, and Euler’s Number
Congratulations! You can now empirically prove that √2 is in fact more effective than e at spreading seeds (e‘s spread has more purple, blue, and cyan, as well as less yellow (meaning more seeds have less space)). But this begs the question: how then, can you find the most irrational number? Is there even such a thing?
You could just check every single angle between 0° and 360° to see what happens.
This first thing you (by which ‘you,’ I mean ‘I’) notice is: holy cats, that’s a lot of options to choose from; how the hell are you suppose to know where to start?
The second thing you notice is that the pattern is actually oscillating between spokes and spirals, which makes total sense! What you’re effectively seeing is every possible rational angle (in order), while hitting the irrational one in between. Unfortunately you’re still not closer to picking the most irrational one, and there are far too many to compare one by one.
1.4 Phi
Fortunately you don’t have to lose any sleep over this, because there is actually a number that has been mathematically proven to be the most irrational of all. This number is called phi (a.k.a. the Golden/Divine + Ratio/Mean/Proportion/Number/Section/Cut etc.), and is commonly written as Φ (uppercase), or φ (lowercase).
It is the most irrational number because it is the hardest to approximate with fractions. Any number can be represented in the form of something called a continued fraction. Rational numbers have finite continued fractions, whereas irrational numbers have ones that go on forever. You’ve already learned that π is not very irrational, as it’s value is approximated pretty well quite early on in its continued fraction (even if it does keep going forever). On the other hand, you can go far further in Φ‘s continued fraction and still be quite far from its true value.
Source: Infinite fractions and the most irrational number: [Link] The Golden Ratio (why it is so irrational): [Link]
Since you’re (by which ‘you’re,’ I mean I’m) a flower (by which ‘a flower,’ I mean ‘an architecture student’), and not a number theorist, it’s less important to you why it’s so irrational, and more so just that it is so. So then, you plot your seeds using Φ, which gives you an angle of roughly 137.5°.
Phyllotaxis Study: The Golden Ratio
It seems to you that this angle does a an excellent job of distributing seeds evenly. Seeds always seem to pop up in spaces left behind by old ones, while still leaving space for new ones.
Phyllotaxis Voronoi Diagram – Proximity to Closest Neighbour – 1,000 Seed Spread: The Golden Ratio
Expanding the this pattern, as well as the generation of a voronoi diagram, further supports your observations. You could compare Φ‘s colour coded voronoi/proximity diagram with the one produced using √2, or any other irrational number. What you’d find is that Φ does do the better job of evenly spreading seeds. However √2 (among with many other irrational numbers) is still pretty good.
1.5 The Metallic Means & Other Constants
If you were to plot a range of angles, along with their respective voronoi/proximity diagrams, you can see there are plenty of irrational numbers that are comparable to Φ (even if the range is tiny). The following video plots a range of only 1.8°, but sees six decent candidates. If the remaining 358.2° are anything like this, then there could easily well over ten thousand irrational numbers to choose from.
It’s worth noting that this is technically not how plants grow. Rather than being added to the outside, new seeds grow from the middle and push everything else outwards. This also happens to by why phyllotaxis is a radial expansion by nature. In many cases the same is true for the growth of leaves, petals, and more.
It’s often falsely claimed that the Φ shows up everywhere in nature. Yes, it can be found in lots of plants, and other facets of nature, but not as much as some people mi
ght have you believe. You’ve seen that there are countless irrational numbers that can define the growth of a plant in the form of spirals. What you might not know is that there is such as thing as the Silver Ratio, as well as the Bronze Ratio. The truth is that there’s actually a vast variety of logarithmic spirals that can be observed in nature.
Phyllotaxis Voronoi /Proximity Study: Various Known Mathematical Constants
A huge variety of plants have been observed to exhibit spirals in their growth (~80% of the 250,000+ different species (some plants even grow leaves at 90° and 180° increments)). These patterns facilitate photosynthesis, give leaves maximum exposure to sunlight and rain, help moisture spiral efficiently towards roots, and or maximize exposure for insect pollination. These are just a few of the ways plants benefit from spiral geometry.
Some of these patterns may be physical phenomenons, defined by their surroundings, as well as various rules of growth. They may also be results of natural selection – of long series of genetic deviations that have stood the test of time. For most cases, the answer is likely a combination of these two things.
In some of the cases, you could make an compelling arrangement suggesting that these spirals don’t even exist. This quickly becomes a pretty deep philosophical question. If you put a series of points in a row, one by one, when does it become a line? How close do they have to be? How many do you have to have? The answer is kinda slippery, and subjective. A line is mathematically defined by an infinite sum of points, but the brain is pretty good at seeing patterns (even ones that don’t exist).
M.C. Escher said that we adore chaos because we love to produce order. Alain Badiou also said that mathematics is a rigorous aesthetic; it tells us nothing of real being, but forges a fiction of intelligible consistency.
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 directlyinfluence 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.’
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.
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.
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
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]
7.3 – Translation vs Revolution
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..
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.)
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 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.
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.
For simplicity, here’s a basic grid set up on a flat plane:
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.
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.
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.
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.
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.
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.
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.
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.)
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
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
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.
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: 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 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 [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.
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
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.
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 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.
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:
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:
8 Iterations of the Hilbert Curve
In 1904, Helge von Koch describes a single complex continuous curve, generated with rudimentary geometry.
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.
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.
5 Iterations of Nicolino’s Quatrefoil
You can find beautifully animated space-filling curves here:
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??
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:
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.
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..)).
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.
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).
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.
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.
Iterative Steps of the Differentially Grown Whale Curve
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