# Category: mathematics

## 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

*phyllotaxis*(from the Greek

*phullon*‘

**leaf**,’ and

*taxis*‘

**arrangement**) was coined around the 17th century by a naturalist called Charles Bonnet. Many notable botanists have explored the subject, such as Leonardo da Vinci, Johannes Kepler, and the Schimper brothers. In essence, it is

**the study of plant geometry**– the various strategies plants use to grow, and spread, their fruit, leaves, petals, seeds, etc.

### 1.1 Rational Numbers

**giving them each as much room as possible to grow, and propagate**.

**you have 360 degrees to choose from**. The first seed can go anywhere and becomes your reference point for ‘

**0**‘ degrees. To give your seeds plenty of room, the next one is placed on the opposite side, all the way at

**180°**. However the third seed comes back around another

**180°**, and is now touching the first, which is a total disaster (for the sake of the argument, plants lack sentience in this instance: they can’t make case-by-case decisions and must stick to one angle (the technical term is a ‘

*divergence angle*‘)).

*Phyllotaxis Study: 180° (see corn leaves), 90° (see mint leaves), and 72° (see gentiana petals)*

**90°**with your second seed, since you noticed free space on either side. This is great because you can place your third seed at

**180°**, and still have room for another seed at

**270°**. Bad news bears though, as you realise that all your subsequent seeds land in the same four locations. In fact, you quickly realise that any

**number that divides 360° evenly yields exactly that many ‘spokes.’**

*Phyllotaxis Study – 1,000 Seed Spread: 45*

**°**, 36**°**, and 20**°***Phyllotaxis Study – 1,000 Seed Spread: 8*

**°**, 5**°**, and 2**°**### 1.2 Irrational Numbers

**nal’ it is, the poorer the spread will be (**

*ratio***a number is rational if it can be expressed as the ratio of two integers**). Naturally this implies that a number can be irrational.

**Irrational numbers go on and on forever**, and never repeat.

**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

*(pi),*

**π***(Pythagoras’ constant), and*

**√2***(Euler’s number). Dividing your circle by*

**e***(360°/3.14159…) leaves you with an angle of roughly*

**π****114.592°**. Doing the same with

*and*

**√2***leave you with*

**e****254.558°**and

**132.437°**respectively.

*Phyllotaxis Growth Study: Pi, Square Root of 2, and Euler’s Number*

*is doing a much better job than*

**√2***, however the difference between*

**π***and*

**√2***appears far more subtle. Perhaps expanding these sequences will accentuate the differences between them.*

**e***Phyllotaxis Study – 1,000 Seed Spread: Pi, Square Root of 2, and Euler’s Number*

*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.*

**√2**### 1.3 Quantifiable irrationality

**some irrational numbers are actually more irrational than others**. For example,

*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).*

**π***Phyllotaxis Voronoi Diagram – Proximity to Closest Neighbour: Pi*

**colour code each cell based on proximity to nearest seed**. In this case, purple means the nearest neighbour is quite close by, and orange/red means the closet neighbour is relatively far away.

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

**is in fact more effective than**

*√2**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?*

**e****every single angle between 0° and 360°**to see what happens.

**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

**the most irrational of all**. This number is called

*(a.k.a. the Golden/Divine + Ratio/Mean/Proportion/Number/Section/Cut etc.), and is commonly written as*

**phi***(uppercase), or*

**Φ***(lowercase).*

**φ****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

*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*

**π***‘s continued fraction and still be quite far from its true value.*

**Φ***Source:*

*Infinite fractions and the most irrational number: [Link]*

*The Golden Ratio (why it is so irrational): [Link]*

*, which gives you an angle of roughly*

**Φ****137.5°**.

*Phyllotaxis Study: The Golden Ratio*

**Seeds always seem to pop up in spaces left behind by old ones, while still leaving space for new ones**.

*Phyllotaxis Voronoi Diagram – Proximity to Closest Neighbour – 1,000 Seed Spread: The Golden Ratio*

*‘s colour coded voronoi/proximity diagram with the one produced using*

**Φ***, or any other irrational number. What you’d find is that*

**√2****However**

*Φ*does do the better job of evenly spreading seeds.*(among with many other irrational numbers) is still pretty good.*

**√2**### 1.5 The Metallic Means & Other Constants

*(even if the range is tiny). The following video plots a range of only*

**Φ****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.

**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.

*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*

**Φ****there are countless irrational numbers that can define the growth of a plant in the form of spirals**. What you might not know is that there is such as thing as the

*Silver Ratio*, as well as the

*Bronze Ratio*. The truth is that there’s actually

**a vast variety of logarithmic spirals**that can be observed in nature.

*Phyllotaxis Voronoi /Proximity Study: Various Known Mathematical Constants*

*Source:*

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

### 1.6 Why Spirals?

**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.

**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.

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*.’

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

Uniclastic: Conoid (Conical paraboloid)

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

### 7.2 – Double Curved Surfaces

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

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

**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.

**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.

### 7.3 – Translation vs Revolution

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

**Ruled Surface & Surface of Revolution (Circular Hyperboloid)**

The 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.

# 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**.

*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:

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.

### 8.2 – Basic Gridshells

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**.

#### 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.)

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.)

### 8.3 – Gridshell Construction

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.

# 9.0 Form Finding

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

### 9.0 – Minimal Surfaces

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

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

*Kangaroo2*Physics: Surface Tension Simulation

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

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

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):

### 9.2 – Catenary Structures

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

*Kangaroo2*Physics: Catenary Model Simulation

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

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

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

### 9.3 – Biomimicry in Architecture

**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.