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.’
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Phyllotaxis Study – 1,000 Seed Spread: 45°, 36°, and 20°
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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.
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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.
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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.

Resonance cryptograph

A2-wide angle perspective.jpgJohann Wolfgang von Goethe says Architecture is frozen music. Albert Einstein believes the key to unlocking the universe is through the hidden geometry and mathematics.  This design seeks to unlock the geometry of Sound making sound visible through 3-dimensional volume and lights.

Johann Wolfgang von Goethe says Architecture is frozen music. Albert Einstein believes the key to unlocking the universe is through the hidden geometry and mathematics.

Sound is a hidden code when it unlocks allows us to perceive it as a set of geometrical patterns. The mechanic of sound is translated visually through frequency and amplitude represents itself with beautiful geometries as code from the universe. My design recreates Sound’s geometries into a physical symbolic Sanctuary for users to retreat their senses in the desert,to unravel meaning behind the symbol of Sound by deconstructing it and re-dressing it with physical form, making Sound visible.

 This design seeks to unlock the geometry of Sound making sound visible through 3-dimensional volume and lights.

 

The structure measures 13.77 feet in length &12.8 feet in height. The material for the structure would be paneled by birch plywood(4ft. x 2ft. panel).2-D dimensional geometry is translated into 3-Dimensional form by folding and joining edges.The sanctuary is made up of three mirroring layers, stacking vertically. The construction of the structure is to explore double curvature design with single curvature paneling and assembly. The ground storey encourages private space for reflection; individual sitting and resting area are carved inwards towards the air-well  ,in contrast, the upper storey is the communal area within the enclosure where users can access from a ladder. Pocket of windows are generated by the stacking and mirroring of sound vibration patterns.  Users enters into the enclosure and view the desert from within.

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Live feeding of Sound and the changing LED lights

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In the night, live feeding of sound is captured when in contact with the surfaces of the sanctuary. With a contact microphone attaches onto the surface, it captures the sound amplitude when a user touches or tap as sound travels through the surface as a medium. The device(computer coding with Arduino) then translates the amplitude variation (loudness) into changing colours of LED lights. The lights are attached on the rim of the panels.

resonance_cryptograph_process_4resonance_cryptograph_process_3resonance_cryptograph_process_2

Process
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Studies of Sound patterns through water

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Harmonograph

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Eigen vector

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Thin Film Interference

This is the very firs experiment I did in DS10, October 2012. Even if I was unable to find an architectural application, i still find the interference colour patterns to be beautiful images, so I’m posting them here, hoping they will inspire someone else.

Thin-film interference occurs when incident light waves reflected by the upper and lower boundaries of a thin film interfere with one another to form a new wave. Check out the wikepedia article: http://en.wikipedia.org/wiki/Thin-film_interference

Thin Film Interference Pattern

Thin Film Interference Pattern

Thin Film Interference Pattern

Thin Film Interference Pattern

Thin Film Interference Pattern

Thin Film Interference Pattern

Thin Film Interference Pattern

Thin Film Interference Pattern

Thin Film Interference Pattern

Thin Film Interference Pattern

Thin Film Interference Pattern

Thin Film Interference Pattern

Plankton Inspired Promotion Pavilion

” Pohl Architects have designed the Cocoon_FS for PlanktonTech, a German research institution that studies plankton. The form was inspired by a type of phytoplankton called diatoms, and is made of fibre-reinforced polymer panels. PlanktonTech will travel around the world and use Cocoon_FS to promote their work.”

“The Cocoon_FS pavilion was constructed from leaf-like panels of fiber-reinforced polymer. Fifteen original base modules were designed and a total of 220 modules were manufactured. Each panel fastens to the next to form a super strong, self-supporting dome. Its translucent shell admits light during the day and illuminates its surroundings at night.

The temporary featherweight structure weighs in at just 1650 pounds and measures under ten feet tall. Both exterior and interior walls carry the same variety of pores, ribs, minute spines, marginal ridges and elevations that characterize the silica cell wall of the slimy brown surface algae that inspired it. Researchers at PlanktonTech used microtechnology to transfer the richly patterned shells of the plankton to a 3D model. Those models were then analyzed and optimized using various computations to unlock biomechanical qualities and re purpose them for architectural design.

Algae is growing in popularity among biofuel enthusiasts, food developers, and entrepreneurs, but as far as we know, the Cocoon_FS is the first prefab to take its design cues from phytoplankton. The plankton-inspired building made its debut in Germany and will be erected at sites around the world in an effort to draw support, awe, and admiration for PlanktonTech’s ongoing investigation of plankton-based solutions.”

Via Inhabitat and Contemporist

 

 

 

Martian Landscape

Stunning images of the Martian Landscape taken by Mars Reconnaissance Orbiter since 2006. Thank you to Subtilitas for pointing it out and for The Big Picture for posting them.

Part of the Abalos Undae dune field. The sands appear blueish because of their basaltic composition, while the lighter areas are probably covered in dust. More, or see location on Google Mars. (NASA/JPL/University of Arizona) 

Scalloped sand dunes in the southern hemisphere of mars, displaying seasonal frost on the south-facing slopes, which highlights some of the regular patterns, as the frost forms only on parts of the ripples. More, or see location on Google Mars. (NASA/JPL/University of Arizona)