## No Dig for Victory

3Dimensional Greenhouse

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# Category: Resources

## No Dig for Victory

## Intersecting Geo-Dome Arcology

## The Corn-Crete House System

## The Palm Arms

## Mimicking the Geometry

## Palm Wine

## Alternative Fuel

## Future Proposal

## Reciprocal Fern Fronds

## Fern Parameters

## Reciprocal Testing

## Reciprocal Testing – Flat Component

## Reciprocal Testing – Large Component

## Ferntastic Azolla

## Transcendental Staircase

## Bend Active

The pre-fabricated modules can then be replicated and scaled, to suit various habitable typologies a community would need or can be used individually as a deployable shelter for the homeless/emergency relief.
## Plants, Math, Spirals, & the Value of the Golden Ratio

# 1.0 Phyllotaxis Spirals

### 1.1 Rational Numbers

*Phyllotaxis Study: 180° (see corn leaves), 90° (see mint leaves), and 72° (see gentiana petals)*
*Phyllotaxis Study – 1,000 Seed Spread: 45***°**, 36**°**, and 20**°**
*Phyllotaxis Study – 1,000 Seed Spread: 8***°**, 5**°**, and 2**°**
### 1.2 Irrational Numbers

*Phyllotaxis Growth Study: Pi, Square Root of 2, and Euler’s Number*
*Phyllotaxis Study – 1,000 Seed Spread: Pi, Square Root of 2, and Euler’s Number*
### 1.3 Quantifiable irrationality

*Phyllotaxis Voronoi Diagram – Proximity to Closest Neighbour: Pi*
*Phyllotaxis Voronoi Diagram – Proximity to Closest Neighbour: Square Root of 2, and Euler’s Number*
### 1.4 Phi

*Phyllotaxis Study: The Golden Ratio*
*Phyllotaxis Voronoi Diagram – Proximity to Closest Neighbour – 1,000 Seed Spread: The Golden Ratio*
### 1.5 The Metallic Means & Other Constants

*Phyllotaxis Voronoi /Proximity Study: Various Known Mathematical Constants*
### 1.6 Why Spirals?

## The Nature of Gridshell Form Finding

### Foreword

# 7.0 Introduction to Shells

### 7.1 – Single Curved Surfaces

Uniclastic: Barrel Vault (Cylindrical paraboloid)

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

Synclastic: Dome (Elliptic paraboloid)
Anticlastic: Saddle (Hyperbolic paraboloid)
Kellogg’s potato and wheat-based stackable snack
Structural Behaviour of Basic Shells [Source: IL 10 – Institute for Lightweight Structures and Conceptual Design]
### 7.3 – Translation vs Revolution

Hyperbolic Paraboloid: Straight line sweep variation
**Ruled Surface & Surface of Revolution (Circular Hyperboloid)**
Hyperboloid Generation [Source:Wikipedia]
# 8.0 Geodesic Curves

Conic Plank Lines (Source: *The Geometry of Bending*)
### 8.1 – Basic Grid Setup

Basic geodesic curves on a plane
Initial set of curves
Addition of secondary set of curves
Grid with independent sets
### 8.2 – Basic Gridshells

Uniclastic: Barrel Vault Geodesic Gridshell
Uniclastic: Conoid Geodesic Gridshell
Synclastic: Dome Geodesic Gridshell
Anticlastic: Saddle Geodesic Gridshell
#### On another note:

Alternate Gridshell Patterns [Source: IL 10 – Institute for Lightweight Structures and Conceptual Design]
Biosphere Environment Museum – Canada
Centre Pompidou-Metz – France
### 8.3 – Gridshell Construction

Hyperbolic Paraboloid: Straight Line Sweep Variation With Rotating Plank Line Grid
Hyperbolic Paraboloid: Flattened Plank Lines With Junctions
Toledo Gridshell 2.0. Construction Process [source: Timber gridshells – Numerical simulation, design and construction of a full scale structure]
# 9.0 Form Finding

### 9.0 – Minimal Surfaces

*Kangaroo2* Physics: Surface Tension Simulation
The Eden Project – United Kingdom
### 9.2 – Catenary Structures

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

Example of Biomimicry
## Developing Space-Filling Fractals

### Foreword

# 4.0 Classic Space-Filling

### 4.1 Early Examples

4 Iterations of the Peano Curve
8 Iterations of the Hilbert Curve
7 Iterations of the Koch Curve
13 Iterations of the Dragon Curve
### 4.2 Later Examples

5 Iterations of Nicolino’s Quatrefoil
#### On A Strange Note:

# 5.0 Avant-Garde Space-Filling

### 5.1 The Traveling Salesman Problem

This illustration, commissioned by Bill Cook at University of Waterloo, is a solution to the Traveling Salesman Problem.
### 5.2 Differential Growth

Differential Growth with* Rhino* & *Grasshopper – Kangaroo 2 – *Planar
Differential Growth with* Rhino* & *Grasshopper – Kangaroo 2 – *NonPlanar
Differential Growth with* Rhino* & *Grasshopper – Kangaroo 2 – *Single-Curved Stanford Rabbit
Differential Growth with* Rhino* & *Grasshopper – Kangaroo 2 & Anemone – *Octopus
Differential Growth with* Rhino* & *Grasshopper – Kangaroo 2 & Anemone – *BatmanDuck
# 6.0 Developing Fractal Curves

Surface Unrolling Study
### 6.1 Dragon’s Feet

Developing Dragon Curve: Dragon’s Feet – 3D Print
### 6.2 Hilbert’s Curtain

Developing Hilbert Curve: Hilbert’s Curtain – 3D Print
### 6.3 Developing Whale Curve

Iterative Steps of the Differentially Grown Whale Curve
Developing Whale Curve – 3D Print

3Dimensional Greenhouse

A modular approach to **integrate nature into the city** through a **highly spatially and materially efficient structure.**

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.

Palm trees are angiosperms, which means flowering plants. They are monocots which means their seeds produce a single, leaf-like cotyledon when they sprout. This makes palms closely related to grasses and bamboo.

This mature palm shows how the pattern originally seen in the young plant, forms a distinct mathematic pattern known as ‘Phyllotaxis’. This is a pattern with reoccurs throughout nature and is based on the Fibonacci sequence. In order to try to understand the use and formation of the palm fibre, the overall formation of the palm stem needed to be mathematically explored.

However, redrawing the cross-section of the base of the palm plants allows a better understanding of the arrangement of the palm plant.

This exercise allows models to be made to recreate the patterns found in palm plants. By engineering plywood components, the basic shape of the palm geometry can be made into a physical model.

This was pushed further by curving the plywood components to make extruded palm structure models

The arrayed components can then be altered so that the base of the models form regular polygon shapes. Doing this allows the potential for the structures to be tesselated. Using different numbers of components mean the structure can then be tested for strength.

There are hundreds of used for palm fruits, this the plant producing materials which range from durable, to flexible to edible. One of the more interesting ones if the production of palm wine using the sap from the tree. Within 2 hours of the wine tapping process, the wine may reach up to 4%, by the following day the palm wine will become over fermented. Some prefer to drink the beverage at this point due to the higher alcohol content. The wine immediately begins fermenting, both from natural yeast in the air and from the remnants of wine left in the containers to add flavour. Ogogoro described a ‘local gin’, is a much stronger spirit made from Raffia palm tree sap. After extraction, the sap is boiled to form steam, which is then condensed and collected for consumption. Ogogoro is not synthetic ethanol but it is tapped from a natural source and then distilled.

To understand the fermentation process more clear, the process of fermenting sugar to make wine has been undertaken.

The distillation of the wine can be used to make bio-ethanol. This production of this fuel can act as a sustainable alternative to fossil fuel energy, which is overused and damaging to our environment.

The developed structure, as well as the production of palm wine and bio-ethanol, can be collaborated to develop a programme, which provides sustainable energy, within a space that is inviting and exciting.

The production of bio-fuel releases a lot of carbon dioxide. In order to ensure the process does not impact the environment, this needs to occur inside a closed system, so the CO2 does not enter the atmosphere. This can be done by using the properties of a Solar Updraft Tower. Carbon dioxide released from the fermentation and distillation processes can be received by palm trees for increased photosynthesis, while the excess oxygen from the trees provides fresh air for visitors.

The fermentation process can be controlled within an isolated area of the model.

The Distillation process, which requires a store of water for cooling, can also be conducted in an isolated area of the model, with apparatus incorporated into the structure.

The final proposal will be a combination of all three forms

The fern is one of the basic examples of fractals. Fractals are infinitely complex patterns that are self-similar across different scales, created by repeating a simple process over and over in a loop. The Barnsley fern (Example here) shows how graphically beautiful structures can be built from repetitive uses of mathematical formulas.

Due to the fractal nature of the fern fronds, the perimeter of the laser cutting took a long time. By simplifying this, I began joining fronds to each other and the large perimeter allowed for enough friction for the fronds to adhere to the adjacent one. I explored this through a series of 4 different frond types (X Axes on matrix below), angles of rotation (Y2, Y3) and distance between each leaf (Y4).

With the study of many different arrangements of fronds and distances between each leaf in the frond, I was then able to select those that slotted in to the adjacent ones best and began arranging them with more components.

The arching nature of each individual leaf meant the configuration was only stable once the fitting in of each component had passed the node of the arch. By flattening each component into rectangular members, the friction that allows the components to adhere to each other would be constant throughout the length of the individual part. This means they could now be placed more or less fitted in to the other component, as desired.

I then scaled up the component and attempted to array these as done with the smaller components above. Each component measured 600 mm length-wise and consisted of 5 members (3 facing one way and 2 facing the other, with a gap between them matching the width of each member). They originated from a central “stem” and attached to this by using glue and nails as to allow for easy manufacturing.

Simultaneously, I also became intrigued by a small aquatic fern called Azolla which I thought would be worth exploring too.

What is interesting about this little plant is that it holds the world record in biomass producer – doubling in size from 3-10 days. It is all thanks to its symbiotic relationship with the nitrogen fixing cyanobacterium, Anabaena. This superorganism provides a micro-climate in exchange for nitrate fertilizer.They remain together during the fern’s reproductive cycle. They also have a complimentary photosynthesis, using light from most of the visible spectrum.

The initial aspiration of the project was to produce a method of simplifying the construction of the 5 regular platonic solids (Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron) using only bend active timber and simple bolted connections, eliminating the need for complex nodal connections as seen in geodesic dome construction and other compound angle connections.

Bend active timber being the main research topic, the structural capabilities and bending radii of plywood were physically tested incorporating the several thicknesses and crucially the direction of the bend either being parallel or perpendicular to the grain, resulting in an informative results matrix.

Digital explorations were undertaken for each of the platonic solids creating various sized volumetric frame structures. The resultant sizes were due to each component being constrained to fit on a standard 2440x1220mm plywood sheet and the resultant bending radius

Following physical investigations of each, the cube was taken forward as it was; more efficient in terms of material usage, easier method of assembly comparatively and unintentionally produced a deployable mechanism similar to the famous Hoberman’s sphere.

The system utilizes identical components meaning the fabrication process can be quickly and easily performed using a CNC machine, with assembling being intuitive, not requiring different parts or specialised assembly instructions. The components were cut using my own CNC and were then simply assembled by hand using bolted connections to create the skeletal frame. Assembly was extremely quick, from flat components to finished volumetric module taking only 20 minutes.

The pre-fabricated modules can then be replicated and scaled, to suit various habitable typologies a community would need or can be used individually as a deployable shelter for the homeless/emergency relief.

The natural world is brimming with ratios, and spirals, that have been captivating mathematicians for centuries.

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.

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*‘)).

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.’**

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.

These ‘spokes’ are the result of the periodic nature of a circle. When defining an angle for this experiment, the more ‘*ratio*nal’ 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.

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.

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.

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).

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.

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.

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°**.

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**.

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.

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.

*Source:*

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Uniclastic: Conoid (Conical paraboloid)

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

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

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

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

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.

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.

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

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

The ruled surface was created by sliding a plane curve (a straight line) along another plane curve (a circle), while keeping the angle between them constant. The surfaces of revolution was simply made by revolving a plane curve around an axis. (Surface of translation also exist, and are similar to ruled surfaces, only the orientation of the curves is kept constant instead of the angle.)

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

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

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

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

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

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

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

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.

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.

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.

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.

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

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

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

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

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

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

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.

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.

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.)

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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.

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.

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

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:

In 1904, Helge von Koch describes a single complex continuous curve, generated with rudimentary geometry.

Around 1967, NASA physicists John Heighway, Bruce Banks, and William Harter discovered what is now commonly known as the Dragon Curve.

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*.

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)

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??**

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…“

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.

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:

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.

Of course the process can also be reversed, allowing the curve to flow seamlessly from one space to another.

Here are far more complex examples of growth simulations exploring various rules and parameters:

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).

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**.

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.

Here is the Hilbert Curve going through the same process, which I am aptly naming ‘*Hilbert’s Curtain*.’

3D Printing Space-Filling Curves with Henry Segerman at Numberphile:

‘Developing Fractal Curves’ by Geoffrey Irving & Henry Segerman:

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.