Plants, Math, Spirals, & the Value of the Golden Ratio

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 phullonleaf,’ and taxisarrangement) 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‘)).
Rational Numbers.gif
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.’
Low Spokes.jpg
Phyllotaxis Study – 1,000 Seed Spread: 45°, 36°, and 20°
Phyllotaxis Spoke Chart.jpg
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.
High Spokes
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.
Irrational Numbers.gif
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.
Expanded Irrational Numbers.jpg
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).
00 Pi Voronoi.jpg
Phyllotaxis Voronoi Diagram – Proximity to Closest Neighbour: Pi
Voronoi Proximity Legend.jpg
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.
e and root 2 Voronoi Diagrams.jpg
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°.
Phi.gif
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.
Golden Ration Spreads.jpg
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 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 might 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.
Mathematical Constants Phyllotaxis.jpg
Phyllotaxis Voronoi /Proximity Study: Various Known Mathematical Constants

Source:
The Silver Ratio & Metallic Means: [Link]

 

1.6 Why Spirals?

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 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.

The Nature of Gridshell Form Finding

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

Foreword

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

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

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

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

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

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

7.0 Introduction to Shells

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

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

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

7.1 – Single Curved Surfaces

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

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

These curves highlight each dimension of the two-dimensional surface. In this case only one of the two ‘curves’ is actually curved, making this shape developable. This means that if, for example, it was made of paper, you could flatten it completely.

B-Conoid

Uniclastic: Conoid (Conical paraboloid)

In this case, one of them grows in length, but the other still remains straight. Since one of the dimensions remains straight, it’s still a single curved surface – capable of being flattened without changing the area. Singly curved surfaced may also be referred to as uniclastic or monoclastic.

7.2 – Double Curved Surfaces

These can be classified as synclastic or anticlastic, and are non-developable surfaces. If made of paper, you could not flatten them without tearing, folding or crumpling them.

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

In this case, both curves happen to be identical, but what’s important is that both dimensions are curving in the same direction. In this orientation, the dome is also under compression everywhere.

The surface of the earth is double curved, synclastic – non-developable. “The surface of a sphere cannot be represented on a plane without distortion,” a topic explored by Michael Stevens: https://www.youtube.com/watch?v=2lR7s1Y6Zig

D-Saddle.gif
Anticlastic: Saddle (Hyperbolic paraboloid)
This one was formed by non-uniformly sweeping a convex parabola along a concave parabola. It’s internal structure will behave differently, depending on the curvature of the shell relative to the shape. Roof shells have compressive stresses along the convex curvature, and tensile stress along the concave curvature.
Pringle
Kellogg’s potato and wheat-based stackable snack
Here is an example of a beautiful marriage of tensile and compressive potato and wheat-based anticlastic forces. Although I hear that Pringle cans are diabolically heinous to recycle, so they are the enemy.
11 Tensile and Compressive behaviour of shells.jpg
Structural Behaviour of Basic Shells [Source: IL 10 – Institute for Lightweight Structures and Conceptual Design]

7.3 – Translation vs Revolution

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

This shape was achieved by sweeping a straight line over a straight path at one end, and another straight path at the other. This will work as long as both rails are not parallel. Although I find this shape perplexing; it’s double curvature that you can create with straight lines, yet non-developable, and I can’t explain it..

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

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

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

 

8.0 Geodesic Curves

These are singly curved curves, although that does sound confusing. A simple way to understand what geodesic curves are, is to give them a width. As previously explored, we know that curves can inhabit, and fill, two-dimensional space. However, you can’t really observe the twists and turns of a shape that has no thickness.

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

A ribbon is essentially a straight line with thickness, and when used to follow the curvature of a surface (as seen above), the result is a plank line. The term ‘plank line’ can be defined as a line with an given width (like a plank of wood) that passes over a surface and does not curve in the tangential plane, and whose width is always tangential to the surface.

Since one-dimensional curves do have an orientation in digital modeling, geodesic curves can be described as the one-dimensional counterpart to plank lines, and can benefit from the same definition.

The University of Southern California published a paper exploring the topic further: http://papers.cumincad.org/data/works/att/f197.content.pdf

8.1 – Basic Grid Setup

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

G-Geocurves.gif
Basic geodesic curves on a plane

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

H-Geocurves.gif
Initial set of curves

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

I-Geocurves.gif
Addition of secondary set of curves

After that, it’s simply a matter of playing with density, as well as adding an additional set of antagonistic curves. For practicality, each set share the same set of base points.

J-Geocurves.gif
Grid with independent sets

He’s an example of a grid where each set has their own set of anchors. While this does show the flexibility of a grid, I think it’s far more advantageous for them to share the same base points.

8.2 – Basic Gridshells

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

K-Barrel
Uniclastic: Barrel Vault Geodesic Gridshell

First comes the shell (a barrel vault in this case), then comes the grid. The symmetrical nature of this surface translates to a pretty regular (and also symmetrical) gridshell. The use of geodesic curves means that these gridshells can be fabricated using completely straight material, that only necessitate single curvature.

L-Conoid
Uniclastic: Conoid Geodesic Gridshell

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

M-Dome
Synclastic: Dome Geodesic Gridshell

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

N-Saddle
Anticlastic: Saddle Geodesic Gridshell

Structurally speaking, things get much more interesting with anticlastic curvature. As previously stated, each member will behave differently based on their relative curvature and orientation in relation to the surface. Depending on their location on a gridshell, plank lines can act partly in compression and partly in tension.

On another note:

While geodesic curves make it far more practical to fabricate shells, they are not a strict requirement. Using non-geodesic curves just means more time, money, and effort must go into the fabrication of each component. Furthermore, there’s no reason why you can’t use alternate grid patterns. In fact, you could use any pattern under the sun – any motif your heart desires (even tessellated puppies.)

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

Here are just a few of the endless possible pattern. They all have their advantages and disadvantages in terms of fabrication, as well as structural potential.

Biosphere Environment Museum - Canada
Biosphere Environment Museum – Canada

Gridshells with large amounts of triangulation, such as Buckminster Fuller’s geodesic spheres, typically perform incredibly well structurally. These structure are also highly efficient to manufacture, as their geometry is extremely repetitive.  

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

Gridshells with highly irregular geometry are far more challenging to fabricate. In this case, each and every piece had to be custom made to shape; I imagine it must have costed a lot of money, and been a logistical nightmare. Although it is an exceptionally stunning piece of architecture (and a magnificent feat of engineering.)

8.3 – Gridshell Construction

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

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

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

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

This is what you get by taking those plank lines and laying them flat. In this case both sets are the same because the shell happens to the identicall when flipped. Being able to use straight material means far less labour and waste, which translates to faster, and or cheaper, fabrication.

An especially crucial aspect of gridshells is the bracing. Without support in the form of tension ties, cable ties, ring beams, anchors etc., many of these shells can lay flat. This in and of itself is pretty interesting and does lends itself to unique construction challenges and opportunities. This isn’t always the case though, since sometimes it’s the geometry of the joints holding the shape together (like the geodesic spheres.) Sometimes the member are pre-bent (like Pompidou-Metz.) Although pre-bending the timber kinda strikes me as cheating thought.. As if it’s not a genuine, bona fide gridshell.

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

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

9.0 Form Finding

Having studied the basics makes exploring increasingly elaborate geometry more intuitive. In principal, most of the shells we’ve looked are known to perform well structurally, but there are strategies we can use to focus specifically on performance optimization.

9.0 – Minimal Surfaces

These are surfaces that are locally area-minimizing – surfaces that have the smallest possible area for a defined boundary. They necessarily have zero mean curvature, i.e. the sum of the principal curvatures at each point is zero. Soap bubbles are a great example of this phenomenon.

hyperbolic paraboloid soap bubble
Hyperbolic Paraboloid Soap Bubble [Source: Serfio Musmeci’s “Froms With No Name” and “Anti-Polyhedrons”]
Soap film inherently forms shapes with the least amount of area needed to occupy space – that minimize the amount of material needed to create an enclosure. Surface tension has physical properties that naturally relax the surface’s curvature.

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

We can simulate surface tension by using a network of curves derived from a given shape. Applying varies material properties to the mesh results in a shape that can behaves like stretchy fabric or soap. Reducing the rest length of each of these curves (while keeping the edges anchored) makes them pull on all of their neighbours, resulting in a locally minimal surface.

Here are a few more examples of minimal surfaces you can generate using different frames (although I’d like stress that the possibilities are extremely infinite.) The first and last iterations may or may not count, depending on which of the many definitions of minimal surfaces you use, since they deal with pressure. You can read about it in much greater detail here: https://tinyurl.com/ya4jfqb2

Eden_Project_geodesic_domes_panorama.jpg
The Eden Project – United Kingdom

Here we have one of the most popular examples of minimal surface geometry in architecture. The shapes of these domes were derived from a series of studies using clustered soap bubbles. The result is a series of enormous shells built with an impressively small amount of material.

Triply periodic minimal surfaces are also a pretty cool thing (surfaces that have a crystalline structure – that tessellate in three dimensions):

Another powerful method of form finding has been to let gravity dictate the shapes of structures. In physics and geometry, catenary (derived from the Latin word for chain) curves are found by letting a chain, rope or cable, that has been anchored at both end, hang under its own weight. They look similar to parabolic curves, but perform differently.

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

A net shown here in magenta has been anchored by the corners, then draped under simulated gravity. This creates a network of hanging curves that, when converted into a surface, and mirrored, ultimately forms a catenary shell. This geometry can be used to generate a gridshell that performs exceptionally well under compression, as long as the edges are reinforced and the corners are braced.

While I would be remiss to not mention Antoni Gaudí on the subject of catenary structure, his work doesn’t particularly fall under the category of gridshells. Instead I will proceed to gawk over some of the stunning work by Frei Otto.

Of course his work explored a great deal more than just catenary structures, but he is revered for his beautiful work on gridshells. He, along with the Institute for Lightweight Structures, have truly been pioneers on the front of theoretical structural engineering.

9.3 – Biomimicry in Architecture

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

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

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

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

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

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

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

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

Developing Space-Filling Fractals

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

 

Foreword

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

4.0 Classic Space-Filling

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

4.1 Early Examples

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

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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:
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8 Iterations of the Hilbert Curve
In 1904, Helge von Koch describes a single complex continuous curve, generated with rudimentary geometry.
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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.
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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.
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5 Iterations of Nicolino’s Quatrefoil
You can find beautifully animated space-filling curves here:
(along with some other great videos by ‘3Blue1Brown’ discussing the nature of space-filling curves, fractals, infinite math, and more)

On A Strange Note:

It’s possible to iterate a version of the Hilbert Curve that (once repeated infinity) can fill three-dimensional space.
As an object, it seems perplexingly difficult to categorize. It is a single, one-dimensional, curve that is ‘bent’ in space following simple, repeating rules. Following the same logic as the original Hilbert Curve, we know that this can be done indefinitely, but this time it is transforming into a volume instead of a surface. (Ignoring the fact that it is represented with a thickness) It is a one-dimensional curve transforming into a three-dimensional volume, but is never a two-dimensional surface? As you keep iterating it, its dimension gradually increases from 1 to eventually 3, but will never, ever, ever be 2??
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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:
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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.
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Differential Growth with Rhino & Grasshopper – Kangaroo 2 – Planar
The geometry doesn’t even have to be bound by a planar surface; It can be done on any two-dimensional surface (or in three-dimensions (even higher spacial dimensions I guess..)).
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Differential Growth with Rhino & Grasshopper – Kangaroo 2 – NonPlanar
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Differential Growth with Rhino & Grasshopper – Kangaroo 2 – Single-Curved Stanford Rabbit
Additionally, Anemone can be used in conjunction with Kangaroo 2 to continuously subdivide the curve as it grows. The result is much smoother, as well as far more organic.
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Differential Growth with Rhino & Grasshopper – Kangaroo 2 & Anemone – Octopus
Of course the process can also be reversed, allowing the curve to flow seamlessly from one space to another.
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Differential Growth with Rhino & Grasshopper – Kangaroo 2 & Anemone – BatmanDuck
Here are far more complex examples of growth simulations exploring various rules and parameters:

6.0 Developing Fractal Curves

In the interest of creating something a little more tangible, it is possible to increase the dimension of these curves. Recording the progressive iterations of a space filling curve allow us to generate what is essentially a space-filling surface. This new surface has the unique quality of being able to fill a three-dimensional space of any shape and size, while being a single surface. It of course also shares the same qualities as its source curves, where it keep increasing in surface area (and can do so indefinitely).
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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.
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Developing Dragon Curve: Dragon’s Feet – 3D Print

6.2 Hilbert’s Curtain

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

6.3 Developing Whale Curve

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

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

The Curves of Life

“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

The Curves Of Life

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.

252264because 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.

 

Relevant Material

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“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.

 

Further Reading:

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Biomimicry in Architecture by Michael Pawlyn

“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.

PNEU + PACK

All living organisms are composed of cells, and cells are fluid-filled spaces surrounded by an envelope of little material- cell membrane. Frei Otto described this kind of structure as pneus.

From first order,  peripheral conditions or the packing configuration spatially give rise to specific shapes we see on the second  and third order.

This applies to most biological instances.  On a larger scale, the formation of beehives is a translated example of the different orders of ‘pneu’.

Interested to see the impact of lattice configuration on the forms, I moved on to digital physics simulation with Kangaroo 2 (based on a script by David Stasiuk). The key parameters involved for each lattice configuration are:

Inflation pressure in spheres
Collision force between the spheres
Collision force of spheres and bounding box
Surface tension of spheres
Weight.

 

Physical exploration is also done to understand pneumatic behaviors and their parameters.

This followed by 3D pneumatic space packing. Spheres in different lattice configuration is inflated, and then taken apart to examine the deformation within. This process can be thought of as the growing process of seeds or pips in fruits such as pomegranates and citrus under hydrostatic pressure within its skin; and dissections of these fruits.

As the spheres take the peripheral conditions, the middles ones which are surrounded by spheres transformed into Rhombic dodecahedron, Trapezoid Rhombic dodecahedron and diamond respectively in Hex Grid, FCC Grid,  and Square Grid. The spheres at the boundary take the shape of the bounding box hence they are more fully inflated(there are more spaces in between spheres and bounding box for expansion).

   

Physical experimentation has been done on inflatables structures. The following shows some of the outcome on my own and during an Air workshop in conjunction with Playweek led by Will Mclean and Laylac Shahed.

To summarize, pneumatic structures are forms wholly or mainly stabalised by either
– Pressurised difference in gas. Eg. Air structure or aerated foam structures
– liquid/hydrostatic pressure. Eg. Plant cells
– Forces between materials in bulk. Eg. Beehive, Fruits seeds/pips

There is a distinct quality of unpredictability and playfulness that pneumatic structures could offer. The jiggly nature of inflatables, the unpredictability resulted from deformation by compression and its lightweightness are intriguing. I will call them as pneumatic behaviour. I will continually explore what pneumatic materials and assembly of them could offer spatially in Brief 02. Digital simulations proved to be helpful in expressing the dynamic behaviours of pneumatic structures too, which I intend to continue.

The Butterfly Egg

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

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

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

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

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

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

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Lycaenidae

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

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

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

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

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

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

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

Interior

—Tia

Elastic Bending Systems

In physics, elasticity (from Greek ἐλαστός “ductible”) is the ability of a body to resist a distorting influence or stress and to return to its original size and shape when the stress is removed.

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This can be explained looking closer at the components which form the cytoskeleton – the cellular structure – formed of elastic and semi-elastic arrangements of proteins, which are adaptable to the cell’s requirements. Not only do they hold the structural integrity of the cell, but they also also perform functions of communication, transport combined with other “plug-in” proteins, whilst elastically responding to external forces or stimuli.

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Elastic bending is used in both natural as well as man-made environments, expanding surfaces and volumes of various deployable structures.

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The force responsible for elastic bending can be described in terms of the amount of deformation (strain) resulting from a given stress, a ratio known as Young’s Modulus. Hooke’s Law adds that the force responsible with restoring the initial shape of a bent material is proportional to the amount of stretch.

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Using the formulas given by Young’s Modulus and Hooke’s Law, we can determine how much a certain material will bend when a force is applied to it.

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Deploying structures using bending is a space and construction time saving method of producing structural elements which mimic the behaviour of natural systems.

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After the structure is deployed, equilibrium  of forces is required in order to keep the structure open and usable.

This can either be achieved by combining bending elements with meshes,

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or by using the acting forces of bending elements against the reacting forces of other bending elements in both 2D and 3D.

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Using these principles of acting forces vs. reacting forces, an elastic module in equilibrium is created, which can be stacked using its geometry to achieve various configurations.

 

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Furthermore, using the same principles, a larger module is created containing groups of elements producing forces acting and reacting against each other, in order to achieve equilibrium.

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However, due to the elastic properties of the newly formed module, the structure can bend and morph shape without losing integrity or equilibrium, in order to form various shapes, or respond to or external factors or users requirements, similarly to the cytoskeleton initially studied.

 

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  • Mihai Chiriac