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‘)).
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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.’
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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.
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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.
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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).
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Phyllotaxis Voronoi Diagram – Proximity to Closest Neighbour: Pi
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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.
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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.

Fractured Cosmos

Truth is a personal conquest which one attains through a mystery.

The Burning Man festival has evolved from a simple gathering of people on a beach in San Francisco into a spectacular spectacle. Burners travel from all over the world to meet in the Black Rock desert of Nevada where they form a city of temporary structures and burn a huge towering figure of a man. The theme of the 2017 burning man event will be Radical Ritual, an attempt to reinvent ritual in our post-post-modern world disregarding assertions of belief and concentrating instead on the immediate experience of play. Fractured cosmos seeks to provide this year’s burners with an edible artwork experience in the playa, a crystalline structure made from hard boiled sugar candy.

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Fractured Cosmos draws the inspiration for its form from the Shri Yantra Mandala, a mystical diagram used in the Shri Vidya school of Hindu Tantra. The diagram, nine Isosceles triangles interlaced to form 43 smaller triangles, is said to be symbolic of the entire cosmos. The geometry of this symbol, usually depicted as a flat, two-dimensional construct, has been inclined, distorted and fractalized out of its two-dimensional plane to create a series of inclined planes for Burners to inhabit and play within.

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recursion-diagram

Those cultures which meditate using the Shri Yantra symbol believe life exists between two planes, that of Samsara, this earth plane, and Nirvana, a perfect heavenly plane. One transcends from Samsara to Nirvana when they have gained enough genuine insight into impermanence and non-self reality. This notion has been reinterpreted as a series of transparent colourful planes on the playa, casting colourful light onto the ground beneath our feet. Each of the planes will be made from coloured rock candy, offering burners the opportunity to perceive an altered sweeter perspective of the world, as well as a delicious treat if they’re brave enough to lick it.

Hard Boiled Candy Planes.png

The use of hard boiled candy as the structural material for the pavilion means the structure will be made by mixing sugar, liquid glucose and Creme de Tartar with water. This concoction will be boiled to a temperature of 145 degrees celsius before pouring into formwork to cure. Waterproof LED lights will be added once the liquid has cooled to a sufficient temperature, and these will be used to light the structure at night.

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The plinth of Fractured Cosmos must be strong enough to support the weight of the candy structure above plus the added weight of any burners, tie the structure down, capture any waste which falls off the structure as it is consumed and house the power source for the night time lighting. This has been designed using the same method of recursion as the candy structure to create a plinth using the mandala’s cosmic gate element as its inspiration. the final form generates a climbable podium for burners to ascend before inspecting and consuming.

mandala-diagrams

JAM.D G-Code

My study about a custom G-Code for FDM 3d printing geometries based on a central axis (not necessarily a straight line! – any curve would do). Rather than printing layer by layer horizontal sections that are uneven and inefficient in terms of travel time, the slices are consistent and always perpendicular to the central axes. Moreover the transition between layers – rather than being done from a single point through a vertical motion which is the traditional approach – is a continuous gradual motion upwards, the travel path resembling a spiral, thus improving efficiency.

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Fiery Lantern

Overview Render01

“Fiery Lanterns” is an expression of sustainability and simplicity, in a modern consumerist world when simple ancient systems are taken for granted. In the theme “Caravansary”, the installation aims to initiate a cultural exchange, encasing Burners and creating crossroads to connect world’s neighbourhoods. The installation shows the possibilities of analysing natural mathematical systems, which we commonly interact with in everyday life and reinterpreting them. Research to produce “Fiery Lanterns”, has allowed detailed prototyping of bending timber to create a memory structure. Testing it to destruction and pushing the understanding of its properties, utilising its strengths and minimalising its weaknesses.

The installation takes inspiration from natural repeating phyllotaxis spirals in sunflower heads. These interconnecting spirals are orientated by the mathematical rules of the golden ratio and governed by successive Fibonacci numbers, to create beautiful majestic forms. This is translated into the identical panels of the “Fiery Lantern”, simply stitched together; they twist and turn to create an ornate case to protect the flames within. This is communicated to the Burner by the symmetry in the design and how you interact with the lanterns. It gives a modern definition of the possibilities of encasement in wood and builds on what is capable using this system.

Panel Assembly Detail

“Fiery Lanterns” are made from four identical stitched panels, which are rotated around each lantern’s central axis.  The curves of the panels are all related to each other, allowing the lanterns to be tightly packed together. The exterior form purposely deceives the Burner, enticing them to explore within and encase themselves in the individual lanterns. The internal maze encourages the exchange and interaction of cultural ideas, to proceed to the pinnacle shimmering lantern.

Spirallehedron Packing

The lanterns are driven into the ground, creating a solid structure, with no predefined front, back or side. The hidden internal entrance, allows Burners to search for the center of the “Fiery Lantern”, crawling or relaxing in the shade, connecting with the playa so as to reach encasement. It does not create barriers in the vast landscape of the Black Rock City, but the lantern’s openings invite a glimpse of the vibrant treasure within. The Burner views the internal surprise and is encouraged to have their own sensory journey, weaving through the individual lanterns, taking their own experience and creating their personal connections. The pinnacle lantern, offers a perspective view over the playa, allowing the Burner reflection on the path below.

Fiery Lanterns spirals out of the harsh compact playa ground in the vast expanse of Black Rock City, surrounded by Burners and art installations. Burners can interact with the installation, explore the individual lanterns and find the Burners encased within. They seek to reach the pinnacle lantern, for reflection on the way forward. As the sun sets at the end of the festival, the internal flames in the lantern will be extinguished and the installation burnt.   

Fiery Lantern_Interior View to Pinnacle Lantern

The installation has evolved through a detailed research into the natural repeating phsyllotaxis spirals in sunflower heads. It is interesting to consider how each spiral is interconnected with the next and governed by the same set of mathematical rules, however changing the parameters changes the resultant architectural form. This is translated in the individuality of the Burning Man experience, the internal connections between the lanterns, offer the same experience, but generate differing personalised interactions with the Burners encased within.

Physical Model Construction_Page_2

Physical modelling demonstrated how the related curves in the components, allows for the lanterns to be tightly packed together. The repeating panels and simplistic design principle, enable open resourcing meaning once the Fiery Lanterns’ flame has been extinguished it can be resurrected at various scales, using the same laser cutting templates.

The scale of the lanterns, allows each component to be cut from a single sheet for plywood, meaning only repeating vertical seams are expressed. Each lantern is an intimate encasing, fitting one or two Burners. Burners must transverse through each lantern in order to reach the pinnacle, it encourages travellers to cross paths and initiate a cultural exchange. 12mm birch faced plywood is used, as the thickness allows the material to be flexible, but also retains its strength. The plywood becomes malleable once soaked in warm water and can be stitched into the panels. The assembly process and onsite installation is simple and uses minimal bolts.

The flame cannot be lit in the lantern until the ornate encasement has been completed. Construction commences through ground anchoring the secluded base lanterns. Soaking and stitching the panels for the internal maze, allows the form to evolve to enable the crowning of the pinnacle lantern. Once dry the Fiery Lantern is complete, the wood’s structural capabilities and sturdiness return, the flame can be lit and it can guide the souls of the playa.

Fiery Lantern_Shimmering in the Dusk Sunlight

The internal space provides a refuge in the vast expanse of the landscape, shielding from the winds and creating a portal of shade. However, it is not a solitary experience; the openings allow all Burners access to the space. You are able to see out, past or through the structure to the rest of Black Rock City.

Fiery Lantern_Night Time Specticle

The lack of a defined entrance means Fiery Lantern has a differing appearance from each angle. It encourages Burners to explore the proposal, to realise encasement. It draws them to the warmth and love of the flames within.

Fiery Lantern_Physical Model

Fiery Lantern PlanFiery Lantern_Section

The lack of a defined entrance means Fiery Lantern has a differing appearance from each angle. It encourages Burners to explore the proposal, to realise encasement. It draws them to the warmth and love of the flames within.

Infinity Stone

 

aka The Desert Diamond

A spectral construct unearthed by the shifting sands of the Black Rock Desert, the Infinity stone is left. An Architectural Cipher it lays on its side, open and yawning towards the sun.

The structure symbolizes both the illusion of material wealth and its realization – The diamond, is hollow, and mirrored. It forces us to instead of gazing at the stone, to enter it, to look past it, into ourselves and onto the horizon.

 

The Desert Diamond is a structure based on the morphed and architecturally interpreted geometry of the brilliant cut diamond.

A series of triangulated panels, the principal structure is composed of multiple bent acrylic panels, with mirror tint rolled across the surfaces. Then it is fixed together with cable ties to build flexible joints. A base of wooden ply is placed on the floor to fix the structure, and provide a solid base.

The mirror tint is inverted, so outside the structure one can see inside, and inside is an infinity of reflections, of people, each other – and the desert. Light will reflect and bounce, a multivlance of colour and fire will sparkle and burn into the memories of those who experience the Infinity Stone.

Near Unison | Dan Dodds | Kinetica Art Fair 2013

Near Unison, my project exploring harmonographic traces is currently being shown at Kinetica Art Fair. The exhibition is in Ambika P3, the exhibition space attached to the University of Westminster on Marylebone Road. For more information on the exhibition, and details about tickets and opening times please visit the Kinetica Art Fair website.

Kinetica Art Fair - Near Unison

The exhibit features a prototype of the interactive harmonograph swings that could form part of the larger installation proposed for Burning Man Festival, along with casts of the harmonographic traces left in sand, and photographic work documenting the process.

“The 5th Kinetica Art Fair returns February 28th – March 3rd 2013 at Ambika P3, as one of London’s annual landmark art exhibitions and a permanent fixture in the Art Fair calendar, renowned as the UK’s only art fair dedicated to kinetic, robotic, sound, light, time-based and new media art.

Kinetica is hosting the work of over 45 galleries and art organisations nationally and internationally, with representatives from UK, France, Russia, USA, Poland, Holland, Spain, Italy, Hungary, Indonesia and Japan, collectively showing over 400 works of art.

A huge interactive light sculpture from Dutch artist Titia Ex will greet visitors as they enter the impressive Ambika P3 venue, and giant 3D sculptures from Holotronica will hover above the main space of the Fair. Other highlights include an exoskeleton hybrid of mananimal-machine by Christiann Zwanniken; a giant three dimensional zoetrope by Greg Barsamian; and a life-size ‘Galloping Horse’ made of light by Remi Brun”

Kinetica Art Fair Press Release

Maya Fluids

Below are some digital experiments showing my first attempts to replicate the wax structures I created with hot wax in water. To achieve these I used maya fluids converted to meshes, then rendered. The difficulty is in understanding the fluid parameters which define it’s density, gravity, viscosity, friction, velocity, turbulence, dispersion and so on. Like with the physical experiments I have begun to add obstacles to see how the fluid reacts.

Digital wax flows 1 from Jakealsop on Vimeo.

Digital wax flows 4 from Jakealsop on Vimeo.

Digital wax flows 3 from Jakealsop on Vimeo.