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

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

1.0 Phyllotaxis Spirals

The term phyllotaxis (from the Greek phullonleaf,’ and taxisarrangement) was coined around the 17th century by a naturalist called Charles Bonnet. Many notable botanists have explored the subject, such as Leonardo da Vinci, Johannes Kepler, and the Schimper brothers. In essence, it is the study of plant geometry – the various strategies plants use to grow, and spread, their fruit, leaves, petals, seeds, etc.

1.1 Rational Numbers

Let’s say that you’re a flower. As a flower, you want to give each of your seeds the greatest chance of success. This typically means giving them each as much room as possible to grow, and propagate.
Starting from a given center point, you have 360 degrees to choose from. The first seed can go anywhere and becomes your reference point for ‘0‘ degrees. To give your seeds plenty of room, the next one is placed on the opposite side, all the way at 180°. However the third seed comes back around another 180°, and is now touching the first, which is a total disaster (for the sake of the argument, plants lack sentience in this instance: they can’t make case-by-case decisions and must stick to one angle (the technical term is a ‘divergence angle‘)).
Rational Numbers.gif
Phyllotaxis Study: 180° (see corn leaves), 90° (see mint leaves),  and 72° (see gentiana petals)
Next time you only go to 90° with your second seed, since you noticed free space on either side. This is great because you can place your third seed at 180°, and still have room for another seed at 270°. Bad news bears though, as you realise that all your subsequent seeds land in the same four locations. In fact, you quickly realise that any number that divides 360° evenly yields exactly that many ‘spokes.’
Low Spokes.jpg
Phyllotaxis Study – 1,000 Seed Spread: 45°, 36°, and 20°
Phyllotaxis Spoke Chart.jpg
Note: This is technically true with numbers as high as 120, 180, or even 360(a spoke every 1°.) However the space between seeds in a spoke gradually becomes greater than the space between spokes themselves, leaving you with one big spiral instead.
High Spokes
Phyllotaxis Study – 1,000 Seed Spread: 8°, 5°, and 2°

1.2 Irrational Numbers

These ‘spokes’ are the result of the periodic nature of a circle. When defining an angle for this experiment, the more ‘rational’ it is, the poorer the spread will be (a number is rational if it can be expressed as the ratio of two integers). Naturally this implies that a number can be irrational.
Sal Khan has a great series of short videos going over the difference between the two [Link]. For our purposes, the important take-aways are:
-Between any two rational numbers, there is at least on irrational number.
Irrational numbers go on and on forever, and never repeat.
You go back to being a flower.
Since you’ve just learned that an angle defined by a rational number gives you a lousy distribution, you decide to see what happens when you use an angle defined by an irrational number. Luckily for you, some of the most famous numbers in mathematics are irrational, like π (pi), √2 (Pythagoras’ constant), and e (Euler’s number). Dividing your circle by π (360°/3.14159…) leaves you with an angle of roughly 114.592°. Doing the same with √2 and e leave you with 254.558° and 132.437° respectively.
Irrational Numbers.gif
Phyllotaxis Growth Study: Pi, Square Root of 2, and Euler’s Number
Great success. These angles are already doing a much better job of dispersing your seeds. It’s quite clear to you that √2 is doing a much better job than π, however the difference between √2 and e appears far more subtle. Perhaps expanding these sequences will accentuate the differences between them.
Expanded Irrational Numbers.jpg
Phyllotaxis Study – 1,000 Seed Spread: Pi, Square Root of 2, and Euler’s Number
It’s not blatantly obvious, but √2 appears to be producing a slightly better spread. The next question you might ask yourself is then: is it possible to measure the difference between the them? How can you prove which one really is the best? What about Theodorus’, Bernstein’s, or Sierpiński’s constants? There are in fact an infinite amount of mathematical constants to choose from, most of which do not even have names.

1.3 Quantifiable irrationality

Numbers can either be rational or irrational. However some irrational numbers are actually more irrational than others. For example, π is technically irrational (it does go on and on forever), but it’s not exceptionally irrational. This is because it’s approximated quite well with fractions – it’s pretty close to 3+1⁄7 or 22⁄7. It’s also why if you look at the phyllotaxis pattern of π, you’ll find that there are 3 spirals that morph into 22 (I have no idea how or why this is. It’s pretty rad though).
00 Pi Voronoi.jpg
Phyllotaxis Voronoi Diagram – Proximity to Closest Neighbour: Pi
Voronoi Proximity Legend.jpg
Generating a voronoi diagram with your phyllotaxis patterns is a pretty neat way of indicating exactly how much real estate each of your seeds is getting. Furthermore, you can colour code each cell based on proximity to nearest seed. In this case, purple means the nearest neighbour is quite close by, and orange/red means the closet neighbour is relatively far away.
e and root 2 Voronoi Diagrams.jpg
Phyllotaxis Voronoi Diagram – Proximity to Closest Neighbour: Square Root of 2, and Euler’s Number
Congratulations! You can now empirically prove that √2 is in fact more effective than e at spreading seeds (e‘s spread has more purple, blue, and cyan, as well as less yellow (meaning more seeds have less space)). But this begs the question: how then, can you find the most irrational number? Is there even such a thing?
You could just check every single angle between 0° and 360° to see what happens.
This first thing you (by which ‘you,’ I mean ‘I’) notice is: holy cats, that’s a lot of options to choose from; how the hell are you suppose to know where to start?
The second thing you notice is that the pattern is actually oscillating between spokes and spirals, which makes total sense! What you’re effectively seeing is every possible rational angle (in order), while hitting the irrational one in between. Unfortunately you’re still not closer to picking the most irrational one, and there are far too many to compare one by one.

1.4 Phi

Fortunately you don’t have to lose any sleep over this, because there is actually a number that has been mathematically proven to be the most irrational of all. This number is called phi (a.k.a. the Golden/Divine + Ratio/Mean/Proportion/Number/Section/Cut etc.), and is commonly written as Φ (uppercase), or φ (lowercase).
It is the most irrational number because it is the hardest to approximate with fractions. Any number can be represented in the form of something called a continued fraction. Rational numbers have finite continued fractions, whereas irrational numbers have ones that go on forever. You’ve already learned that π is not very irrational, as it’s value is approximated pretty well quite early on in its continued fraction (even if it does keep going forever). On the other hand, you can go far further in Φ‘s continued fraction and still be quite far from its true value.

Source:
Infinite fractions and the most irrational number: [Link]
The Golden Ratio (why it is so irrational): [Link]

Since you’re (by which ‘you’re,’ I mean I’m) a flower (by which ‘a flower,’ I mean ‘an architecture student’), and not a number theorist, it’s less important to you why it’s so irrational, and more so just that it is so. So then, you plot your seeds using Φ, which gives you an angle of roughly 137.5°.
Phi.gif
Phyllotaxis Study: The Golden Ratio
It seems to you that this angle does a an excellent job of distributing seeds evenly. Seeds always seem to pop up in spaces left behind by old ones, while still leaving space for new ones.
Golden Ration Spreads.jpg
Phyllotaxis Voronoi Diagram – Proximity to Closest Neighbour – 1,000 Seed Spread: The Golden Ratio
Expanding the this pattern, as well as the generation of a voronoi diagram, further supports your observations. You could compare Φ‘s colour coded voronoi/proximity diagram with the one produced using √2, or any other irrational number. What you’d find that Φ does do the better job of evenly spreading seeds. However √2 (among with many other irrational numbers) is still pretty good.

1.5 The Metallic Means & Other Constants

If you were to plot a range of angles, along with their respective voronoi/proximity diagrams, you can see there are plenty of irrational numbers that are comparable to Φ (even if the range is tiny). The following video plots a range of only 1.8°, but sees six decent candidates. If the remaining 358.2° are anything like this, then there could easily well over ten thousand irrational numbers to choose from.
It’s worth noting that this is technically not how plants grow. Rather than being added to the outside, new seeds grow from the middle and push everything else outwards. This also happens to by why phyllotaxis is a radial expansion by nature. In many cases the same is true for the growth of leaves, petals, and more.
It’s often falsely claimed that the Φ shows up everywhere in nature. Yes, it can be found in lots of plants, and other facets of nature, but not as much as some people might have you believe. You’ve seen that there are countless irrational numbers that can define the growth of a plant in the form of spirals. What you might not know is that there is such as thing as the Silver Ratio, as well as the Bronze Ratio. The truth is that there’s actually a vast variety of logarithmic spirals that can be observed in nature.
Mathematical Constants Phyllotaxis.jpg
Phyllotaxis Voronoi /Proximity Study: Various Known Mathematical Constants

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

 

1.6 Why Spirals?

A huge variety of plants have been observed to exhibit spirals in their growth (~80% of the 250,000+ different species (some plants even grow leaves at 90° and 180° increments)). These patterns facilitate photosynthesis, give leaves maximum exposure to sunlight and rain, help moisture spiral efficiently roots, and or maximize exposure for insect pollination. These are just a few of the ways plants benefit from spiral geometry.
Some of these patterns may be physical phenomenons, defined by their surroundings, as well as various rules of growth. They may also be results of natural selection – of long series of genetic deviations that have stood the test of time. For most cases, the answer is likely a combination of these two things.
In some of the cases, you could make an compelling arrangement suggesting that these spirals don’t even exist. This quickly becomes a pretty deep philosophical question. If you put a series of points in a row, one by one, when does it become a line? How close do they have to be? How many do you have to have? The answer is kinda slippery, and subjective. A line is mathematically defined by an infinite sum of points, but the brain is pretty good at seeing patterns (even ones that don’t exist).

M.C. Escher said that we adore chaos because we love to produce order. Alain Badiou also said that mathematics is a rigorous aesthetic; it tells us nothing of real being, but forges a fiction of intelligible consistency.

The Tragedy of Planned Obsolescence

Technology today is designed to fail. Products are made so that you will buy a new one after a pre-determined time. This is called planned obsolescence and is a widely accepted commercial concept within industrial companies.

The Phoebus Kartel  was a cartel of, among others, Osram, Philips and General Electricfrom December 23, 1924 until 1939 that controlled the manufacture and sale of light bulbs. It decided that it would limit the lifetime of a lightbulb to 1000 hours. Before this arbitrary and profit-driven decision, light bulbs could last for a very long time, a solid proof for that is the Livermore’s Centennial Lightbulb which shines since 1890. The 1000 hours rule was the beginning of an imposed large-scale planned obsolescence.

Above: The Livermore’s Centennial Lightbulb’s webcam

After the great depression, Bernard London thought that imposing planned obsolescence by law would bring prosperity to Americans.

The american designer Brook Stevens gave many conferences on the advantage of planned obsolescence. His products would always look newer, better than the existing one. By his definition, planned obsolescence was “Instilling in the buyer the desire to own something a little newer, a little better, a little sooner than is necessary.”

Above: The toastalator by Brook Stevens

Without planned obsolescence, shopping malls would probably not exits and economic growth would not be as crucial as it it today to the economy.  In essence, economic growth does not attempt to make human life better, it just tries to grow for the sake of it. This growth is based on debt and on consuming products that are not necessary. As the economist and system theorist Kenneth Boulding once said: “Someone who believes that an economy that constantly grows on a planet that is finite is either mad or an economist, the problem is that we are all economists now.”

The Waste Makers, published in 1960 by Vance Packard is the first book on the topic.

Apple, largest public company in the U.S., gave a clear notice to its reseller when the IPOD battery would fail: “buy a new ipod“.  Apple was sued for that by consumers, the case was called Wesley vs. Apple. Apple lost the case and was forced to extend the warranty on the battery. Apple has no environmental policy for its products and tries to sell as many products as possible, not products that will last.

Image courtesy of Stay Free Magazine.

Epson adds microchips in some of their printers that counts the amount  of prints and breaks the printer after reaching a pre-determined printer. In fact, some freewares help you to reset the count so that you can use your printers more.

Electronic products that could have lasted much longer end up in illegal dump site in countries such as Ghana and Nigeria (have a look at the Agbogbloshie dump site on this BBC documentary).

Above: kid looking for copper on the Agbogbloshie illegal E-Waste dump site, Ghana

The idea of creating “Open-Source” buildings from simple materials that can be made and improved by anyone and based on home-grown or widely accessible products is DS10’s answer to the tragedy of planned obsolescence. Similarly to open source software that can always be updated and maintained by the end user, the makers will not be at the sole mercy of a proprietary vendor. We will also look into temples, timeless monuments for spirituality and best counter example for modernist buildings, a theory which emerged around the same time as the Phoebus Kartel.

Sources:

-This post is based on the documentary “The Light Bulb Conspiracy” by Cosima Dannoritzen.

-http://www.apfelkraut.org/2011/03/the-untold-story-of-planned-obsolescence/

-http://quiet-environmentalist.com/is-the-earth-doomed-due-to-planned-obsolescence/

-http://www.amazon.com/Made-Break-Technology-Obsolescence-America/dp/0674022033

-http://www.amazon.co.uk/Planned-Obsolescence-Publishing-Technology-Academy/dp/0814727883

Algae Growth

Vertical growth/closed loop production has been developed by biofuel companies to produce algae faster and more efficiently than open pond growth. With vertical growing, algae are placed in clear plastic bags/tubes, so they can be exposed to sunlight on all sides. The extra sun exposure increases the productivity rate of the algae, which in turn increases oil production. The algae are also protected from contamination due the closed envionment of their growth.

This small scale experiment shows the growth of algae in a clear plastic tube of rainwater exposed to sunlight. An air pump is used to circulate the algae solution and to provide ample ‘dirty’ air including carbon dioxide. The rainwater provides nutrients such as nitrogen and phosphorus also required for algae growth. It can be seen that algae grows rapidly – in a matter of days – unlike seasonal crops.

 

 

Random Linear Growth – Hoopsnake

This example shows an animation of my ‘work-in-progress’ Grasshopper definition that uses Hoopsnake to recursively perform a ‘copy by mirror’ function on a geometric form. The two examples are based on a cube and a tetrahedron. The growth is linear; expanding by one module with each step. The position of each new module is determined by a new randomly selected face of the preceding module.

I would like to develop the definition so that it doesn’t self intersect, so any comments with ideas on how to achieve this would be appreciated!

Grasshopper and Hoopsnake

This video shows a summary of the potential using grasshopper combined with Hoopsnake.
The intention is to be able to multiply the component to a large amount of them by controlling the angles of rotation and building a mega-structure out of it.
As grasshopper is not very good on closed loop systems the Hoopsnake plug in comes in to repeat the definition over and over again in order to produce the outcome. The video indicates how you do it. Simply by double clicking on the Hoopsnake component in the definition and then adjust the preferable angles and click loop.
You can repeat the operation as many times as you want with different variation of angles and once the satisfied result is there you can bake the meshes into Rhino and then use them to render.
The possibilities are numerous and pretty exciting results can occur.