# Category: Growth

## 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*‘)).

**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.’**

### 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.

*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***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****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.

**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°**.

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

*‘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.

*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 Curves of Life

“An organism is so complex a thing, and growth so complex a phenomenon, that for growth to be so uniform and constant in all the parts as to keep the whole shape unchanged would indeed be an unlikely and an unusual circumstance. Rates vary, proportions change, and the whole configuration alters accordingly.” – D’Arcy Wentworth Thompson

*“This is the classic reference on how the golden ratio applies to spirals and helices in nature.” – Martin Gardner*

What makes this book particularly enjoyable to flip through is an abundance of beautiful hand drawings and diagrams. Sir Theodore Andrea Cook explores, in great detail, the nature of spirals in the structure of plants, animals, physiology, the periodic table, galaxies etc. – from tusks, to rare seashells, to exquisite architecture.

He writes, “*a staircase whose form and construction so vividly recalled a natural growth would, it appeared to me, be more probably the work of a man to whom biology and architecture were equally familiar than that of a builder of less wide attainments. It would, in fact, be likely that the design had come from some great artist and architect who had studied Nature for the sake of his art, and had deeply investigated the secrets of the one in order to employ them as the principles of the other.*”

Cook especially believes in a hands-on approach, as oppose to mathematic nation or scientific nomenclature – seeing and drawing curves is far more revealing than formulas.

“*because I believe very strongly that if a man can make a thing and see what he has made, he will understand it much better than if he read a score of books about it or studied a hundred diagrams and formulae. And I have pursued this method here, in defiance of all modern mathematical technicalities, because my main object is not mathematics, but the growth of natural objects and the beauty (either in Nature or in art) which is inherent in vitality.*”

Despite this, it is clear that Theodore Cook has a deep love of mathematics. He describes it at the beautifully precise instrument that allows humans to satisfy their need to catalog, label and define the innumerable facts of life. This ultimately leads him into profoundly fascinating investigations into the geometry of the natural world.

## Relevant Material

*“An organism is so complex a thing, and growth so complex a phenomenon, that for growth to be so uniform and constant in all the parts as to keep the whole shape unchanged would indeed be an unlikely and an unusual circumstance. Rates vary, proportions change, and the whole configuration alters accordingly.” **– D’Arcy Wentworth Thompson*

D’Arcy Wentworth Thompson wrote, on an extensive level, why living things and physical phenomena take the form that they do. By analysing mathematical and physical aspects of biological processes, he expresses correlations between biological forms and mechanical phenomena.

He puts emphasis on the roles of physical laws and mechanics as the fundamental determinants of form and structure of living organisms. D’Arcy describes how certain patterns of growth conform to the golden ratio, the Fibonacci sequence, as well as mathematics principles described by Vitruvius, Da Vinci, Dürer, Plato, Pythagoras, Archimedes, and more.

While his work does not reject natural selection, it holds ‘survival of the fittest’ as secondary to the origin of biological form. The shape of any structure is, to a large degree, imposed by what materials are used, and how. A simple analogy would be looking at it in terms of architects and engineers. They cannot create any shape building they want, they are confined by physical limits of the properties of the materials they use. The same is true to any living organism; the limits of what is possible are set by the laws of physics, and there can be no exception.

**Further Reading:**

*“You could look at nature as being like a catalogue of products, and all of those have benefited from a 3.8 billion year research and development period. And given that level of investment, it makes sense to use it.” – Michael Pawlyn*

Michael Pawlyn, one of the leading advocates of biomimicry, describes nature as being a kind of source-book that will help facilitate our transition from the industrial age to the ecological age of mankind. He distinguishes three major aspects of the built environment that benefit from studying biological organisms:

The first being the quantity on resources that use, the second being the type of energy we consume and the third being how effectively we are using the energy that we are consuming.

Exemplary use of materials could often be seen in plants, as they use a minimal amount of material to create relatively large structures with high surface to material ratios. As observed by Julian Vincent, a professor in Biomimetics, “materials are expensive and shape is cheap” as opposed to technology where the inverse is often true.

Plants, and other organisms, are well know to use double curves, ribs, folding, vaulting, inflation, as well as a plethora of other techniques to create forms that demonstrate incredible efficiency.

## Hexagonal Patterned Spacial Definitions

The inspiration for this research came from the Asian artist Ren Ri, who uses bees in order to generate his sculptural work. He predefines the space for the bees to work with, and allows for a time period for the honeycombs to take shape.

There are three types of surface division that manage to fill up all the area with prime geometric space – triangular (S3), square (S4) and hexagonal (S6). Other types of surface division, either leave gaps between the prime elements, which need to be filled by secondary shapes, or are confined to irregular shapes.

Research shows that the most efficient way of dividing a surface is through a minimum number of achievable line intersections, or a maximum number of membranes. In either case, the hexagonal division fits the case. This type of organization is a second degree iteration from the triangular division. It is formed by identifying and connecting the triangular cell centroids.

Such as in the case of soap-bubble theory, these cells expand, tending to fill up all the surface area around them, and finally joining through communicating membranes.

From a structural point of view, the best integration is the triangular one, because of the way each element (beam) reacts to the variation of the adjacent elements.

By converting the elemental intersection in the hexagonal division from a single triple intersection to a triple double intersection, the structure would gain sufficient structural resistance. This can be done through two methods – translation or rotation. Translation implies moving the elements away from the initial state in order to open up a triangular gap at the existing intersection. This method results in uneven shapes. In the case of rotation, the elements are adjusted around each middle point until a sufficient structural component is created. It is through rotation that the shape is maintained to a relative hexagonal aspect, due to the unique transformation method.

Pursuing the opportunity to test the system through a 1:1 scale project, I was offered the chance to design a bar installation for a private event at the Saatchi Gallery. The project has been a success and represents a stage test for the system.

Moving further, the attempt was to implement dynamic force analysis to the design, through variation of the elemental thickness. The first test was a bridge design. The structure was anchored on 2 sides, and had a span of 5m.

The next testing phase includes domed structures, replicating modular structures and double curved instances.

## Lotus Hypars

##### Lotus Hypars – A study of hyperbolic bamboo structures

The Lotus Hypars symbolise the “Caravansary” trading centre. The structure is assembled as the centre for exchange after journeying across land and water to a resting point, Burning Man. Hammocks offer a space for the festivals unique style of trading to be discussed and carried out. The tangible nature of the Lotus also creates a playfulness in an otherwise formal system of resources exchange. The lightweight structure evolves from the horizontal lines of the desert and forms a hyperbolic shelter. The user can inhabit not only underneath the structure, but also the petal shaped hammocks. Here, individuals can exchange stories, supplies and treasures.

In Buddhism, the Lotus flower is symbolic of fortune. It grows in muddy water, and it is this environment that gives forth the flower’s first and most literal meaning: rising and blooming above the murk to achieve enlightenment. The Lotus Hypar story has evolved from the same principles. In the harsh desert environment, man can create beauty. The folded geometries are playfully excited by human participation. A twist, a fold and a push.

The structure is assembled using bamboo sticks that are arranged in a reciprocal formation. These canes are then bound using high strength elastic bands. This allows for the flat cells to twist and take on new shapes. The Lotus Hypar is formed by a repetitive series of folds and the result forms petals. These are symbolic of the Lotus flower. The cells are covered with a white semi-elastic membrane that adds to the strength of the structure and the petal geometries become more visible. These are also the hammocks that can be inhabited by the Burning Man users.

In order to test the structural performance of the proposal, I constructed a series of 1:1 scale models. This was done using 6m and 3m bamboo canes (35mm diameter). By testing a small segment of the full proposal, it is easier to determine the success of the final proposal.

## 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

## Urban Aquaponics, Internet and Food

**“My garden is sending tweets!” Eric Maundu**

Just came across this amazing video in which Eric Maundu talks about his start-up “**Kijani Grows**” (“Kijani” is Swahili for green), a small startup that designs and sells custom aquaponics systems for growing food using cheap technology including arduino boards. Toby and I often talk about “closed loop systems”, this is a great example of one.

*“The land in West Oakland where Eric Maundu is trying to farm is covered with freeways, roads, light rail and parking lots so there’s not much arable land and the soil is contaminated. So Maundu doesn’t use soil. Instead he’s growing plants using fish and circulating water. It’s called aquaponics- a gardening system that combines hydroponics (water-based planting) and aquaculture (fish farming). It’s been hailed as the future of farming: it uses less water (up to 90% less than traditional gardening), doesn’t attract soil-based bugs and produces two types of produce (both plants and fish). **Aquaponics has become popular in recent years among urban gardeners and DIY tinkerers, but Maundu- who is trained in industrial robotics- has taken the agricultural craft one step further and made his gardens smart. Using sensors (to detect water level, pH and temperature), microprocessors (mostly the open-source Arduino microcontroller), relay cards, clouds and social media networks (Twitter and Facebook), Maundu has programmed his gardens to tweet when there’s a problem (e.g. not enough water) or when there’s news (e.g. an over-abundance of food to share).*

*Maundu himself ran from agriculture in his native Kenya- where he saw it as a struggle for land, water and resources. This changed when he realized he could farm without soil and with little water via aquaponics and that he could apply his robotics background to farming. Today he runs Kijani Grows (“Kijani” is Swahili for green), a small startup that designs and sells custom aquaponics systems for growing food and attempts to explore new frontiers of computer-controlled gardening. Maundu believes that by putting gardens online, especially in places like West Oakland (where his solar-powered gardens are totally off the grid), it’s the only way to make sure that farming remains viable to the next generation of urban youth.”*