nature by numbers reflection essay

International edition
Australia edition
Europe edition

Nature by Numbers

In this beautiful video, "Nature by Numbers," filmmaker Cristobal Vila presents a series of animations illustrating various mathematic principles, beginning with a breathtaking animation of the Fibonacci Sequence before moving on to the Golden Ratio, the Angle Ratio, the Delaunay Triangulation and Voronoi Tessellations. The words are scary-sounding, but the math is beautiful and the film serves to remind us of the intimate relationship between nature and math.

Music, "Often a Bird," by Belgian composer Wim Mertens [ Download this piece from iTunes ].

This movie was inspired by numbers, geometry and nature.

You can learn more about the videographer, Cristóbal Vila, at etereaestudios . That website has more information about the film, including a fascinating step-by-step explanation of the theory behind the film (English is side-by-side con Español), along with stills, screenshots, tutorials and workshops.

GrrlScientist

Comments (…)

Nature by numbers: The hidden beauty of mathematics

L iving with a mathematician this year has made me realise the unsung contribution mathematics makes when it comes to providing us with the reasoning to better appreciate the beauty of nature. I’d always thought fractal symmetry, which cropped up in my physical chemistry lectures, was solely a chemical concept. You can probably imagine my surprise when I realised that this characteristic actually stemmed from mathematics. It came as an even greater shock to discover that many natural phenomena are, in fact, fractal to some degree. The Fibonacci sequence, which you may think exists only in the pages of a Dan Brown novel, is also visible in some of nature’s most exquisite structures. So, just how many of us are aware of the way in which mathematics provides us with the reasoning to be able to praise the intrinsic beauty of nature? This is exactly what I hope to achieve in this article; to show you how mathematics, something some of us may have dreaded at school, actually explains a lot of the things we see around us.

Fractal symmetry is when the same pattern is seen at increasingly small scales. In fractal symmetry, you find the same pattern within the pattern, which is why this can also be referred to as self-similarity. The best example to think of is a tree. The trunk of a tree separates into branches which then separate into smaller branches and then twigs, and these get smaller and smaller. In this way, we see a repetition. Another example is the Romanesco broccoli, but my favourite would have to be the Lotus effect. The Lotus leaves have a rough surface with micro- and nano-structures including waxy crystals that contribute to making the surface superhydrophobic. This means that any water droplets on the lotus leaves are strongly repelled and slide off the surface. While doing so, they take up the dust particles from the leaves in order to reduce the surface tension, resulting in the cleaning of the lotus leaves. This is an example of self-cleaning in nature and it is the fractal symmetry of the waxy crystals on the surface of the leaves that provides the enhanced hydrophobicity which makes this possible.

Fractal symmetry is when the same pattern is seen at increasingly small scales. In fractal symmetry, you find the same pattern within the pattern, which is why this can also be referred to as self-similarity

Another type of symmetry I wish to discuss is the so-called wallpaper symmetry. This is the mathematical classification of a two-dimensional repetitive pattern inspired by honeycomb structures. Besides often being seen in architecture and other arts such as textiles, this structure has found great use in the field of chemical catalysis. One of the finest examples of the use of catalysis is in catalytic converters used to turn pollutant gases such as nitrogen oxides and carbon monoxides into nitrogen dioxide and carbon dioxide gases, which are safer alternatives. The support for the precious rhodium or platinum metal catalysts used is cordierite monolith. The metal is dispersed on the honeycomb structure of the support which provides a larger surface area to optimise the flow of gases over the catalyst. This is a fine example of symmetry observed in nature that has now been employed on a large industrial scale and is something used by many of us every day.

Fractal and wallpaper symmetry are the two types I wanted to discuss. However, this article would be incomplete without a nod to the spirals that are too often seen in nature. Some of these spirals arise due to the golden ratio of 1.618[…] which is the most irrational number we can get. Put simply, it is the furthest away we can be from a fraction. In this way, the golden ratio gives the best spiral with no gaps. Hence, flower petals and pinecones are guided by the golden ratio, which is related to the Fibonacci sequence. In the Fibonacci sequence, each number is the sum of the two numbers preceding it. What we find is that if we take the ratio of any two numbers from the Fibonacci sequence, we get values very close to the golden ratio. In nature, the flowers and the shells are not genetically, or in any other way, programmed to abide by the mathematics of the Fibonacci sequence. This is purely a result of evolutionary design. Petals and seeds find that the golden ratio offers the best packing with minimum gaps. I find it absolutely amazing that a series of numbers on a piece of paper can explain why many elements in nature have chosen to adopt this particular configuration.

This article would be incomplete without a nod to the spirals that are too often seen in nature. Some of these spirals arise due to the golden ratio of 1.618[…] which is the most irrational number we can get

In addition to mathematics, you could not have avoided noticing the not-so-subtle mention of chemistry in this article. As I researched more into maths, I could easily find myself making connections with chemistry and much of the material I have studied to date. Thinking back to one of the conversations I’ve had with my flatmates in the kitchen, it’s clear that mathematics provides the key to untie the knots in many fields both within and outside STEM. As a chemist, I certainly see how fundamental a role mathematics plays. In this article, I hope I have enabled you to at least begin to appreciate just how much a bunch of numbers can explain to us about the universe.

Nature by Numbers

A short movie inspired on numbers, geometry and nature

Project completed in March, 2010

Artists and architects have used since ancient times many geometrical and mathematical properties: we could take some examples simply by observing the refined use of the proportions by architects from Ancient Egypt, Greece and Rome or other Renaissance artists like Michelangelo, Da Vinci or Raphael.

But what is more surprising for me is that many of these properties and mathematical developments are also present in nature. We could find countless cases, but I wanted to refer only three of them on this short animation: The Fibonacci Series and Spiral / The Golden and Angle Ratios / The Delaunay Triangulation and Voronoi Tessellations.

The concepts behind Nature by Numbers

This section is meant to be a complement to the animation, in order to better understand the theoretical basis that you can find behind the sequences. It was also, more or less, the appearance of the screenplay in the days that I was planning this project.

The animation begins by presenting a series of numbers. This is a very famous and recognized sequence since many centuries ago in the Western World thanks to Leonardo of Pisa , a thirteenth century Italian mathematician, also called Fibonacci. So it is known as Fibonacci Sequence , even although it had been described much earlier by Indian mathematicians.

This is an infinite sequence of natural numbers where the first value is 0, the next is 1 and, from there, each amount is obtained by adding the previous two.

This is the next thing to be shown on the animation, appearing just after the first values on the succession: the process of building one of these spirals.

We will create first a few squares that correspond to each value on the sequence: 1×1 – 1×1 – 2×2 – 3×3 – 5×5 – 8×8, etc. And they are arranged in the way how we see in the diagram at left.

Then we draw a quarter circle arc (90°) within each little square and we can easily see how it builds step by step the Fibonacci Spiral, looking at right graphic.

I have introduced a small optical correction in the animation in order to get the resulting curve more like a true Golden Spiral (more harmonious and balanced), as explained on this plate . It’s something similar to what happens when we try to approach to an ellipse by drawing an oval using circular segments: the result is not the same as a true ellipse. And it’s something evident.

IMPORTANT NOTE: while watching the animation conveys the idea that the Fibonacci spiral (or the Golden Spiral, it doesn’t matter) is on the origin of the shape of a Nautilus , this isn’t absolutely right.

It’s funny because if you perform this search at Google Images: “ spiral + nautilus ” you will see how many images suggest that this shell is really based on the construction system described above.

But this isn’t correct, as it’s outlined below, on the “Making of” section.

The truth is that this is something I discovered when I had completely finished the screenplay for this project and I was too lazy to change. Therefore I must confess that I did a kind of cheat with this animation. Or you could explain in a more “genteel” way, saying that I have taken an artistic license 😅

Once it has appeared the Nautilus we advance to the second part of the animation. It introduces the concept of Golden Ratio by constructing a Golden Rectangle. We start from a simple square to get that and use a classic method that requires only a ruler and drawing compass. See the complete process on the following series of illustrations:

This is very special rectangle known since ancient times. It fulfills this ratio, also known as the Golden Ratio or Divine Proportion: the ratio of the sum of the quantities (a+b) to the larger quantity (a) is equal to the ratio of the larger quantity (a) to the smaller one (b).

The result of this ratio (ie the division of a by b) is an irrational number known as Phi —not to be confused with Pi— and an approximate value of 1.61803399…

Formerly was not conceived as a true “unit” but as a simple relationship of proportionality between two segments. And we find in many works created by the mankind in art and architecture, from the Babylonian and Assyrian civilizations to our days, passing through ancient Greece or the Renaissance.

JUST A CURIOSITY: it isn’t evident on the animation, but there is a deep connection between the Fibonacci Sequence and Golden Ratio.

You have an example at right (we will see another one): if we divide each value in the Fibonacci Series by the previous, the result tends to Phi. The higher the value, the greater the approximation (consider that Phi, like any irrational number, has infinite decimals).

We are going one step further on the animation by introducing a new concept, maybe less known but equally important, the Golden Angle . That is, the angular proportional relationship between two circular segments:

And this value is deeply present in nature. This is the next concept we see on the animation: how to configure the structure formed by the sunflower seeds.

Look at the figures below:

We add a first red seed.
Turn 137.5º
Add a second green color seed and make the previous traveling to the center.
Turn other 137.5º
Add a third ocher seed and make the previous traveling to the center, to stay side by side with the first one.
Turn other 137.5º…

…and so on, seed after seed, we will obtain gradually a kind of distributions like the ones you have in the following figures.

This leads to the characteristic structure in which all seeds are arranged into a sunflower, which is as compact as possible. We have always said: nature is wise :-)

ANOTHER CURIOSITY: Do you remember we had commented that there had a deep connection between the Fibonacci Sequence and Golden Ratio? Well, next we have another meeting point between both concepts. Look at the following images of a sunflower:

Well, if you look at one of the typologies, for example the one in green, and you go to the illustration above right you can check that there is a certain number of spirals like this, specifically 55 spirals. Coincidentally a number that is within the Fibonacci Sequence ;-)

And we have more examples in the two upper panels, cyan and orange, they are also arranged following values that are within the sequence: 34 and 21 spirals.

In principle, all the sunflowers in the world show a number of spirals that are within the Fibonacci Sequence. You could go out to the countryside and look for a plantation to be sure or you can also use this image of a real sunflower.

Finally we reached the third segment of the animation in which we work with a concept that is a little less known than the others: the Voronoi Tessellations , also called Dirichlet Tessellation.

I discovered this issue thanks to Hector Garcia’s personal site , which I visit almost daily (and despite being a blog dedicated to Japanese culture and everything that is related to that country, also delights us from time to time with other interesting topics, like this one about Delaunay and Voronoi ).

These geometric formations are based on a distribution pattern that is easily recognisable in many natural structures, like the wings of some insects or these small capillary ramifications in some plant’s leaves.

It is also widely used to optimise the distribution systems based on areas of influence, at the time to decide, for example, where to install phone antennas, or where to build the different delegations for a pizza chain.

Let me show you a very intuitive way to understand how it forms a Voronoi Tiling:

Above right we added a third green point, generating two new bisectors that intersect with the first.

If we continue adding points to generate succesive bisectors, with their intersections, will lead to a series of polygons —Voronoi Tiles— around a set of “control points”. Thus, the perimeter of each one of these tiles is equidistant to neighboring points and defines their area of influence.

All these segments that interconnect the points form a triangular structure called Delaunay Triangulation . In the illustration below you can see the process as we continue adding points:

We can find interactive sites on the internet (like this ) to draw points, move them, and check how the structure becomes updated in real time.

In fact, if we have a series of random dots scattered in the plane, the best way of finding the correct Voronoi Tessellation for this set is using the Delaunay triangulation. And in fact, this is precisely the idea shown on the animation: first the Delaunay Triangulation and then, subsequently, the Voronoi Tessellation.

But to draw a correct Delaunay Triangulation is necessary to meet the so-called “Delaunay Condition”. This means that: a network of triangles could be considered Delaunay Triangulation if all circumcircles of all triangles of the network are “empty”.

Notice that actually, given a certain number of points in the plane there is no single way to draw triangles, there are many. But only one possible triangulation meets this condition. It is very simple: we draw a triangle using 3 points only if the circumcircle created using these 3 points is “empty” (not enclosing any other dot).

You see that in the graph below, from Wikipedia:

We could rotate 90 degrees each side of the triangle using the the midpoint after defining the Delaunay Triangulation (top left), to construct the Voronoi Tiling (top right). This is exactly what the animation shows just before that the camera pulls back to show us the structure of our dragonfly wing.

We could also use the centers of each circle, marked in red, as they describe the vertices of Voronoi Tilings.

Of course, I am pretty sure of one thing: if we take a real dragonfly, and we analyze their wings with the help of a magnifying glass or microscope ( example ), we find exceptions and deviations. But it is clear the similarity of both structures.

Making of Nature by Numbers

Building a “true” golden spiral.

The genuine ancient Golden Spiral should be created drawing circular arcs connecting the opposite corners of squares in the Fibonacci multisquare tiling. In fact this should be considered an “approximate” Golden Spiral. As you can see here, the red spiral used on this animation has a subttle corrections to made it more harmonious, with a continuous curvature, since the circular arcs give us some discontinuity perception. Anyway this is too another rough approximation to a True Golden Logarithmic Spiral using the Illustrator Bezier Curves.

On the next picture we can see again the difference between the golden spiral and that other one present on nautilus shell:

Nautilus shell: 3D model process

The shell modeling was a great challenge. On the next gallery you will find a small step by step, with descriptions on each image:

And here is a close-up render showing the final result of that modelling process:

Building the wing structure

The wing skeleton was planned and designed using Illustrator, looking at some Flickr photograph references. All the geometrical structures were easily solved there and then exported to Cinema4D where I converted all to tubes, exported to Modo to integrate with the overall dragonfly body geometry and finally come back to Cinema4D again to proceed with texture and animation work. Yes: kind of convoluted pipeline, but it worked great for me :-)

Sunflower 3D model

A small curiosity: those small florets around the central seeds are, actually, the REAL flowers. The sunflower itself is a “head”, a so called “composite flower”.

Dragonfly 3D model

As many other models on this project, the dragonfly was modelled in Modo and then exported to Cinema 4D to be textured, rigged and animated. Here are some screen-captures from Modo:

Procedural animation

This project, Nature by Numbers, was a milestone in my day to day with Cinema4D. Working on it I learned a lot of techniques within the field of what we could call “procedural animation” (or even better: “not key-framed animation”) thanks to the use of Xpresso, a kind of visual programming system by graphical nodes. Read notes for each image once is enlarged:

fibonacci, golden ratio, phyllotaxis, voronoi, tessellations, euler, pi, color theory…

Cristóbal Vila – Zaragoza, Aragón, SPAIN

Modified date:

Website Powered by WordPress.com .

Already have a WordPress.com account? Log in now.
Subscribe Subscribed
Copy shortlink
Report this content
View post in Reader
Manage subscriptions
Collapse this bar

Healing Forest

Forest Walks, Nature Art, Mindfulness Activities, Healing with Nature

Nature by numbers | nature meditations on life and self.

The quest for understanding the self and search for the meaning of life is as old as memory. No one can give us the correct answer because for each one of us, the path to the answer is different. But clues are all around in nature because understanding Self is linked to understanding life . In this post, we look for answers in nature by numbers and see where they lead us. Many people are afraid of Math because no one helped them make friends with numbers. Nobody introduced them to the wonder and wisdom that is hidden in the language of the Universe.

After all the positive feedback for our Healing forest learning program and requests for more nature games, meditations and mindfulness activities we have come up with this interesting walk format. We hope these mindfulness exercises help you create new learning and a new respect for yourself. Because nature and numbers are a part of you, just as you are a part of them. But first, here’s an inspiring film by Cristobal Vila.

NATURE BY NUMBERS FILM

NUMBERS IN NATURE & MEDITATION

Let us take a slow and gentle walk in nature. Our aim is to observe and reflect. Walk with a few like minded friends or go alone. Carry a pen and paper to note down your insights and experiences. And be open to try something new. ( A free download of all nature meditations in this article is given at the very end.)

Math is like love; a simple idea, but it can get complicated.

9 Numbers in Nature: We begin the walk with an exercise to start seeing numbers in nature. Participants have to find the numbers 1 to 9 in their surroundings. Everyone can quickly strike off 1 and 2 as we all have one nose and two ears. The other numbers have to be found outside of the human body. For example a flower with 5 petals, an ant with 6 legs. All participants are given a time limit of 10 minutes to find as many of the remaining numbers as possible. The exercise is to be done individually or in pairs. The group reassembles at the end of the time to work together as a team and find any missing number that no one could find.

8 Patterns in nature: Nature is full of patterns that have astounded mathematicians and poets alike for centuries. One such pattern is called the Fibonacci. The Fibonacci sequence starts like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and so on forever. Each number is the sum of the two numbers that come before it. It’s a simple pattern, but it appears to be a kind of built-in numbering system to the cosmos. The numbers in the pattern can be found in our own DNA as well the spirals of the Galaxy. The numbers of the Fibonacci sequence are very commonly seen in petals of flowers . Examples include the lily, which has three petals, buttercups, which have five, the chicory’s 21, the daisy’s 34. These are all numbers from the Fibonacci sequence.

Nature Meditation: The aim of our exercise is to find interesting patterns in nature and take pictures. Try and see if you can collect a similar pattern in two different objects of nature.

7 Colours: Every Colour that you see is a number. Light travels as a wave and each colour in the spectrum has a specific wavelength and frequency. Our visual sense is not only able to gauge and see different colours but also associates certain emotions with them subconsciously.

Nature Meditation: The aim of our next exercise is to spot all the 7 rainbow colours during your nature walk. Each individual makes a list of at-least 7 different colours they can observe during the walk. The aim is also to spend a little time with each separate colour and become aware of how the colour makes us feel. At the end of the exercise people who are drawn to the same colour can group together to see whether they share other common personality traits.

6 Geometry: Have you ever marvelled at the beauty and shape of a spiral sea-shell? Geometry is all about shapes and their properties. Lines, curves and shapes that can be drawn on paper make up plane geometry, while 3 dimensional objects are part of solid geometry. The spiral curve is one of the many examples in nature that give us a hint of the underlying simplicity which gives rise to the complexities in nature. The study of geometry allows us to become aware of the larger design of Nature. Here’s a short film on the curve called ‘life’. Watch it patiently to find a beautiful lesson for the ups and downs of life.

Nature Meditation: Creating a spiral. This can be done individually or in a group. The intention is to create a beautiful spiral with objects found in nature. Make it as big as you can. Each person starts from the same center point and creates one arm of a spiral radiating outwards. After working on it for 10 minutes, the creator stands on the outer edge of their spiral arm and starts to walk back to the center slowly and mindfully following the path of their spiral. The last person to reach the center wins. (You cannot pause and have to continue walking inward as slow as you can).*Don’t forget to erase your spiral and disperse everything back in nature, before you leave.

To get uplifting new ideas from nature – once a month, join our free newsletter . Discover amazing activities and fascinating forests from around the world.

5 Senses are our window to the world. Every person perceives the world differently based on how each of their different senses have developed. Staying in the city does take a toll on our overall sense perceptions where some senses can be overloaded like our sight and some underdeveloped like our smell. Being in nature allows us to relax our senses and sharpen them so that our experience of the world can become richer. Nature Meditation: Walk or sit silently in the forest. Focus on any one sense at a time for a short period of 2-3 minutes. Make a note of all the unique things you can observe and sense. Repeat the exercise with another of your senses. Notice how each sense reveals something new about the nature around us. The aim of this exercise is to bring us into the present moment and stop our thoughts from leaping into the past or future.

4 Breath of 4: In this exercise we focus on our breath to bring our mind to the present moment and take the help of numbers to build mindfulness. We use counting to stay focused on the breath. This is a very good exercise for creating alertness and concentration. The exercise is known as Box Breathing where we breathe in 4 stages. Inhale – Hold – Exhale – Hold , all with equal durations, from 2 to 5 seconds. For example, inhale for 3 seconds, hold your breath for 3 seconds, exhale for 3 seconds, hold your breath for 3 seconds. And repeat that. Do it any time you can for a few minutes This meditation makes us realise that numbers live with us as part of our lives and we can always turn to them for focus, attention and peace of mind.

3 Triangles: One of the most interesting things about triangles is that if you sum up the three angles within any triangle, they will always total 180 degrees . The Triangle is a favorite among architects. As a geometrical object, the triangle is the strongest shape, capable of holding its shape, having a strong base, and providing immense support.

Nature Meditation: Just like the 3 sides of a triangle, there are 3 stages that connect all life on our planet. An interesting exercise with number 3 is to observe the 3 different stages of life. Birth. Maturity. Death. Find and take pictures of objects in different stages. Reflect on how everything is changing from one stage to another.

π Circles: Take a circle. Any circle in the world. Measure it’s length, all the way around the circle. Then measure it across, from one edge to the other edge. Now divide the two lengths. You will always get the same number. 3.14… This number is called Pi and is often written using the greek symbol π. What’s strange about Pi is that the division is never complete. You can go on dividing without reaching an end. π has been calculated to over two quadrillion decimal places and still there is no pattern to the digits. Nature Meditation: Spot the circles. Count the maximum number of circular objects one can notice in your surrounding nature within a time frame of 5 minutes. If you can, try and measure the circumference and diameter of any circle and divide them to find your own Pi.

2 Opposites: Nature is made up of opposites. Day and night. Left and right. Sound and silence. Hot and cold. To observe the two opposing sides of nature is to understand our own true nature. Nature Meditation: For this exercise participants divide into 2 equal groups ‘Positives’ and ’Negatives’. Members of the 2 groups walk in separate directions and each individual (depending on their group) makes a list of 5 positives or 5 negatives they observe in nature. The groups reassemble after 5 minutes and make pairs between the positives and negatives group. Each pair then tries to see how many matching opposites do they have between their 2 lists. For e.g.: If one person wrote ‘light’ and the other person wrote ‘shadow’ then it is considered a successful match. With this exercise we observe that perception of life – positive or negative is based on our mind. And the mind can be trained to choose. * Some wise people might raise a doubt and say that in nature there are no positives and negatives. Everything just is. They are right. Agree with them and tell them it’s just a game.

1 Oneness: Have you ever wondered why we only count in multiples of 10 ? A counting base of 10 is natural probably because we have 10 fingers. In ancient societies, a base 10 system wasn’t always used. The Sumerians used a base 60 system. This is why we count time in bases of 60 (60 minutes in an hour, 60 seconds in a minute). Machines are built using switches, so it is natural for them to count only off (0) and on (1).This system is called binary. There can be many other number systems, but one thing connects them all. Numbers are concepts which only have meaning when they relate to each other. In a sense, it is this relationship that gives each number a specific meaning in the larger number system. For e.g 5 is related to 10,15,20,25 in a certain way. Here’s another chain of relationships: 1,2,6,24, 120….Can you figure out the next 2 numbers in this chain? Nature Meditation: Participants take some time out to observe the many relationships that exist in nature. Each person comes up with a chain of nature relations. For e.g. Sun – Plant – Flower – Bee. The aim is to see who can come up with the longest chain of relationships… and perhaps to realize that we are all part of multiple chain of relationships, which give our life true meaning.

0The number Zero is widely seen as one of the greatest innovations in human history. Zero is both a number and a concept meaning the absence of any quantity. With the help of Zero we can do complicated equations and perform calculus. It is also at the heart of the language of computers which speak in 0’s and 1’s. In philosophical terms it represents nothingness or emptiness out of which all existence arises. We end our nature walk by taking a few minutes to walk in silence and reflect on the concept of zero. Meditating on zero is a meditation in humility. It is to become aware that in the vastness of the Universe – both in space and time, our small individuality amounts to nothingness. And yet, just like the importance of zero, one can realize how significant even the most insignificant thing in the world can be.

MEDITATIONS ON NATURE | FREE DOWNLOAD

Download link of 2 posters for mindfulness meditations on nature by numbers.

Please share these 10 beautiful meditations on nature by numbers with friends who might find it interesting.

END NOTE: To get amazing new ideas each month, check out our Uplifting Newsletter . For more walks and many other activities, try our Nature Calm course.

Healing Forest is a volunteer driven project that aims to bring people and forests closer to each other through creativity and mindfulness. Our aim is simple. Helping people heal. Helping forests heal .

Spread A Little Calm. Our World Needs It.

Click to share on Facebook (Opens in new window)
Click to share on WhatsApp (Opens in new window)
Click to share on Twitter (Opens in new window)
Click to share on Pinterest (Opens in new window)
Click to share on Reddit (Opens in new window)
Click to share on Pocket (Opens in new window)
Click to share on Telegram (Opens in new window)
Click to email a link to a friend (Opens in new window)

4 Comments

Posted on September 25, 2019 by Healing Forest

Category: Healing Forest

Tags: Forest Bathing , Forest meditations , forest therapy , healing arts , Healing forest walks , healing meditation , meditations , meditations on nature , mindful meditations , mindfulness , nature , Nature Art , nature by numbers , nature connection activities , nature meditation , nature meditations , nature walk , nature walks , shinrin yoku , what is nature by numbers all about

4 Comments on “ Nature By Numbers | Nature Meditations on Life and Self ”

Pingback: Nature By Numbers | Nature Meditations on Life and Self. – MindBodySimple

Share your story Cancel reply

Create your own future.

753,463 visits
Forest Bathing – What, How, Where? A Beginner’s Guide
Mind Craft: Learn Life’s Most Useful Skills
Mindfulness Activities, Exercises For Groups | 7 Nature Ideas
Nature’s Design: 5 Tips For Important Decisions

150+ Ways To Find Your Calm

Join Our Monthly Mailer

753,463 Views
Forest Bathing
Mindful Activities

Website Built with WordPress.com .

Already have a WordPress.com account? Log in now.
Subscribe Subscribed
Copy shortlink
Report this content
View post in Reader
Manage subscriptions
Collapse this bar

Winter 2020
The Mandelbrot Set

Nature by Numbers

Today, I would like to share one of my favorite things: a short film by the very talented Cristóbal Vila called Nature by Numbers . The film melds the beautiful composition “Often a Bird” by minimalist composer Wim Mertens with some wonderfully imaged examples of mathematics in the natural world. I highly recommend that you watch the film in HD—it is utterly worth it.

As a work of art, I think that the film stands very well on its own. Even to a person who knows nothing about mathematics, the link between mathematics and nature is made very clear, even if the mathematics itself is obscure to the viewer. That being said, I think that there is value in going a step deeper. In the interests of fostering that deeper understanding, I would like to examine some of the mathematics on display in the film.

The Fibonacci Sequence

The film begins by introducing the Fibonacci sequence. The Fibonacci sequence is a fascinating gem of mathematics. The sequence itself, which can be described fairly simply, shows up in many branches of mathematics, including number theory, analysis, and geometry. The sequence was first described in relation to the following problem:

If a pair of [newborn] rabbits is placed in an enclosed area, how many rabbits will be born there if we assume that every month a pair of rabbits produces another pair, and that rabbits begin to bear young two months after their birth? 1 http://www.math.temple.edu/~reich/Fib/fibo.html

The rabbit population can be summarized in the following table:

In each column, we see the sequence \(1,1,2,3,5,\ldots\) This sequence, which appears in the film, is the Fibonacci sequence. In the table above, the value in each cell could be calculated using information from other cells. For instance, if we wanted to know how many newborns there would be in May, we would look at the number of mature pairs of rabbits in May, which is in turn found by adding the number of one month old and mature rabbits from April. That is, bot the rabbits that were mature in April and the rabbits that were one month old in April will bear young in May.

This can be generalized a little. In any cell after February, the value will be equal to the sum of the two cells directly above. For instance, the number of newborn rabbits in August will be equal to the sum of newborn rabbits in July and June.

Removing the context of rabbits, we can talk about the sequence in purely mathematical terms. We start with two terms: 0 and 1 (these are the zeroth and first terms, respectively). In order to get the next term in the sequence, we add the two previous terms. This means that the second term of the sequence will be \(0+1 = 1\); the third term will be \(1+1 = 2\); the fourth term will be \(1+2 = 3\); &c.

In mathematical notation, we can define the Fibonacci sequence in terms of the following equation: \(F_n = F_{n-1}+F_{n-2}\) where \(F_n\) is the \(n\)th term of the Fibonacci sequence and \(F_1=F_2=1\) (the first two terms are 1). It is conventional to define \(F_0 = 0\) 2 Chandra, Pravin and Weisstein, Eric W. “Fibonacci Number.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/FibonacciNumber.html .

The Golden Ratio

Skip ahead now to the 1:25 mark in the film, and watch until the 1:40 mark. The mathematics on display here go by quickly, but the content is definitely worth spending a little time on—the images presented represent one of the most pervasive mathematical concepts in art and nature.

We start by considering a particular rectangle. The rectangle that we are considering has a long side of length \(a\) units, and a short side of length \(b\) (shown in blue, below). If a square with sides of length \(a+b\) is appended to our rectangle, something interesting happens. The ratio \(a:b\) will be the same as the ratio \(a+b:a\).

This rectangle that we are considering is called the golden rectangle, and the ratio \(a:b\) is called the golden ratio. As this ratio appears to have been incorporated into many of the works of the ancient Greek sculptor Phidias, the 20th century mathematician Mark Barr suggested that this ratio be called \(\varphi\) (“phi”) in his honor 3 Weisstein, Eric W. “Golden Ratio.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GoldenRatio.html . Using this notation, we have \({a\over b}={a+b\over a}=\varphi\).

It is possible to get a numerical value for \(\varphi\). The details are presented below, but feel free to skip them if it isn’t your cup of tea.

If we simplify the middle fraction, we get \({a+b\over a} = 1+{b\over a}\). As \(\varphi={a\over b}\), this implies that \(1+{b\over a} = 1+{1\over\varphi}\). Putting this together with the original equation, we have \(1+{1\over\varphi} = \varphi\). Multiplying both sides by \(\varphi\), we get \(\varphi+1=\varphi^2\), and moving all of the terms to the left side, we end up with \(\varphi^2-\varphi-1 = 0\).

Applying the quadratic formula, we get \(\varphi = {1\pm\sqrt{1^2-4(1)(-1)}\over 2(1)} = {1\pm\sqrt{5}\over 2}\). Of the two possible solutions, only one is positive. As we are comparing the lengths of two sides of a rectangle a negative solution doesn’t make sense, so we can ignore that solution. The remaining solution, \(\varphi={1+\sqrt{5}\over 2} \approx 1.618\), is the golden ratio.

When we defined \(\varphi\approx 1.618\) above, we did it in terms of a rectangle that has the interesting property that, when a square is appended to one side, the new rectangle has the same proportions as the original rectangle. However, this ratio appears throughout nature and the arts. For instance, the golden ratio can be seen in the construction of the Parthenon, in the proportions of Da Vinci’s Vitruvian Man , and in the arrangement of leaves along the stems of plants.

The golden ratio even seems to have some relevance to how attractive people find each other. People seem to consider faces to be more attractive when they are proportioned according to the golden ratio. This includes the general shape of the head (which might fit into a golden rectangle), the proportions of the nose (the ratio of height to width may be the golden ratio), and the ratio of the distance between the pupils of the eyes and the edges of the eyes (also a golden ratio).

It is also interesting to note that the Fibonacci sequence and the golden ratio are, in fact, very closely related. Consider the ratio of two consecutive terms in the Fibonacci sequence. The first several such ratios are listed below.

\(F_2/F_1 = 1/1 = 1\) \(F_3/F_2 = 2/1 = 2\) \(F_4/F_3 = 3/2 = 1.5\) \(F_5/F_4 = 5/3 \approx 1.667\) \(F_6/F_5 = 8/5 = 1.6\) \(F_7/F_6 = 13/8 = 1.625\) \(F_8/F_7 = 21/13 \approx 1.615\) \(F_9/F_8 = 34/21 \approx 1.619\) \(F_{10}/F_9 = 55/34 \approx1.618\)

As we get deeper into the Fibonacci sequence, the ratio between successive terms gets closer and closer to \(\varphi\). In fact, while it is beyond the scope of this post, it is possible to prove that as \(n\) goes to infinity, the ratio \(F_{n+1}/F_n = \varphi\). This is a really interesting and startling result.

The Fibonacci and Golden Spirals

If we rewind the film a bit to the section that starts at the 0:25 mark, we can see yet another connection between the golden ratio and the Fibonacci sequence.

Consider a square which measures 1 unit on a side (a in the figure above). To that square, attach a second square which measures 1 unit on a side (b). Together, these squares form a rectangle that is 2 units by 1 unit. On the long side of this rectangle, attach a square which measures 2 units on a side (c). We now have a rectangle which measures 2 units on one side, and 3 units on the other. We can continue this process (d-g) to get the above figure.

Now, let’s play with that figure a little bit. In the first square that we drew, we are now going to make an arc that starts in one corner, and ends in the opposite corner. Continuing that arc, we get a semi-circle which goes through the first two squares that we drew. From there, we continue the arc, always going from the initial corner of a square to the opposite corner. The result is shown below.

Squares made from the Fibonacci sequence.

This figure is called the Fibonacci spiral. This spiral is closely related to another spiral, called the golden spiral. The golden spiral is created by plotting the equation \(r = e^{90\cdot\theta\cdot\ln(\varphi)}\) 4 Plotting points on a polar plane works much like plotting points on the Cartesian plane. On the Cartesian plane, we have an x -coordinate and a y -coordinate. The x -coordinate tells us how far to the left or right to go, and the y -coordinate tells us how far up or down to go. In this system, both coordinates give us a distance. In a polar system, we use the coordinates r and \(\theta\). r is a distance (it stands for radius), but \(\theta\) is an angle between 0 and 360 degrees. To get a handle on polar coordinates, think about trying to get from point A to point B using a map and compass. \(\theta\) tells you which direction to go—if \(\theta\) is 0 degrees, then you go due east; if \(\theta\) is 90 degrees, go due north; &c.—and r tells you how far to go. For more information, I suggest reading the Wikipedia article on polar coordinates . . This spiral grows by a factor of \(\varphi\) every quarter rotation. That is, if the distance from the center of the spiral to the edge of the spiral is 1 unit at some angle, then 90 degrees farther along the spiral, the distance from the center to the edge will be about 1.618 units.

At this time, please take a second to look at the two spirals above. They look very similar. The Fibonacci spiral is made up of quarter circles which grow in relation to the Fibonacci sequence, while the golden spiral grows at a constantly increasing rate. It seems odd that they should look so similar. Yet it turns out the the Fibonacci spiral is a very good approximation of the golden spiral 5 If you are mathematically inclined, the reason that this happens is that the ratio between Fibonacci numbers approaches the golden ratio. Hence as the Fibonacci squares get larger, the amount of growth from one square to the next approaches the golden ratio, meaning that growth over a quarter turn gets closer and closer to the golden ratio, thereby matching the golden spiral. . To hammer home the point, have a look at this:

The Fibonacci spiral and the golden spiral on the same plane.

Note that, while the two spirals are not exactly the same, they line up very closely. If we continued the figure farther out (by adding more squares), we would see the two spirals converge more and more.

Now for one quick connection to the natural world: in the film, a Fibonacci spiral morphs into the shell of a chambered nautilus. As it grows, the nautilus builds shell chambers around itself for protection. As the animal grows at a constantly increasing rate, the chambers get larger at a constantly increasing rate. It turns out that the creature’s rate of growth closely matches the golden ratio.

Ultimately, both the Fibonacci and golden spirals are easily described, and can be visualized with minimal expertise. However, because of constraints placed upon living organisms, the clinical mathematical curves are mirrored in the graceful, organic curves of those organisms. Mathematics is often used as a tool for describing the universe, but I can think of few examples that are more beautiful and arresting than the Fibonacci spiral and the chambered nautilus.

The Golden Angle

Imagine that you are a sunflower. Like all living organisms, you have evolved to reproduce your genes. In your case, this is best accomplished by producing as many offspring as you possibly can. An obvious question arises: how can you best produce offspring?

The answer is probably to produce as many seeds as possible, as cheaply as possible. More seeds mean more offspring, and cheaper seeds mean a better chance that you will have the resources to survive long enough to allow your seeds to mature.

You could produce a long string of seeds (legumes like peas and lima beans seem to be successful doing this), but there may be some problems. First, the string of seeds that you produce can only be as long as you are tall, otherwise they might rot on the ground before maturing. Also, the cost of producing those seeds goes up, as you have to get nutrients to all of them, many of which may be at the end of long strings. In the long run, this could hurt.

So, instead, you pack your seeds as tightly as you can at the head of your flowers. The next question is then: how do you most efficiently pack your seeds?

In order to address this question, we first need to figure out what constrains seed growth. There are some biological constraints which we must consider. New florets (which will become seeds when pollinated) form in the center of the sunflower head. As new florets are produced, older florets are pushed out from the center. Moreover, new florets form next to old florets, separated by a particular angle.

To model where the seeds end up as they mature, are going to get into some more notation. If you don’t like notation, just skip the box, and examine at the pictures after the box.

We can model the final locations of the seeds of a sunflower using the following equations:

\(r = c\sqrt{n}\) \(\theta = n\gamma\)

In these equations, \(r\) and \(\theta\) (the Greek letter “theta”) are the coordinates of a point in a polar plane; \(n\) is an index (this is basically the age of the seed); \(c\) is a constant scaling factor (this doesn’t actually have an effect on the overall configuration of the seeds, but is useful for fitting all of the plotted seeds into a smaller or larger area); and \(\gamma\) (the Greek letter “gamma”) is the angle between a new seed and its closest neighbor.

The value of \(r\) represents the fact that as new seeds are produced, older seeds are pushed out from the center. If the seeds were all in a straight line, then each new seed would push older seeds out by a constant unit. However, the seeds are filling a two dimensional space, so the farther out from the center a seed has been pushed, the less it has to move in order to accommodate new seeds. Because two dimensional spaces and one dimensional spaces are related by the square root, the distance that the seed has to move is related to the square root of its index. It turns out that the only thing affecting the distance of a seed from the center is the age of the seed.

The value of \(\gamma\) controls the angle between seeds. For instance, if \(\gamma = 90^{\circ}\), then each time a new seed is created, it will be offset from the previous seed by 90 degrees. This means that seed 1 will be at 0 degrees, seed 2 at 90 degrees, seed 3 at 180 degrees, and so on. Note that in this case all seeds with indices which are multiples of 4 will be at 0 degrees.

Ultimately, the configuration of seeds is dependent upon the angle between the seeds, and no other variables. Suppose that the angle between seeds is exactly 90 degrees. Then we end up with the following configuration of 605 seeds:

Configuration of seeds with gamma = 90 degrees.

There are a couple of problems here. First, note that near the end of each line, the seeds are practically on top of each other. If a real sunflower were configured in this manner, the seeds would probably kill each other. Each seed needs more space in order to grow and mature. Secondly, there is a ton of wasted space on the flower head. Wasted space means wasted resources. A more efficient configuration is definitely needed.

The next four images give other possible values of \(\gamma\). Starting in the upper left corner, and going clockwise, the values of \(\gamma\) are 45 degrees, 10 degrees, 138 degrees, and 14 degrees.

Configuration of seeds with gamma = {45, 10, 138, 14} degrees.

Looking at these configurations, it seems that \(\gamma = 138^{\circ}\) is the most efficient—it packs the seeds pretty closely together, and doesn’t waste as much space as the other configurations. But it still doesn’t look quite like a real sunflower (in fact, the \(\gamma = 14^{\circ}\) configuration looks closer to a real sunflower), and there is still quite a bit of wasted space. Can we find a more efficient configuration?

It turns out that we can, and that the ideal configuration is related to the golden ratio. The solution is demonstrated in the film starting around the 1:40 mark. Take a circle, and divide it into two arcs, such that the ratio of the length of the longer arc to the length of the shorter arc is the golden ratio. In the figure below, this implies that \(a/b = \phi \approx 1.618\). The smaller angle created by doing this is the golden angle (shown in blue), which measures about 137.5 degrees.

For reasons that are well beyond the scope of this post, this golden angle is the ideal angle for arranging sunflower seeds. Mathematically speaking, we cannot find a better angle for packing seeds onto a flower head. Using this angle, we get the following configuration 6 The images of hypothetical sunflower seed configurations were made with an Excel spreadsheet, which I have made available for download . The spreadsheet was made with Excel 2004, and should work happily with any relatively recent version of the program (say, 2003 and later). Have fun! :

Configuration of seeds with gamma = the golden angle.

Qualitatively, this looks quite a bit like a real sunflower head. Here is the truly humbling fact: the mathematics discussed in this section, which have been fairly well understood for a century or two, are the result of thousands of years of rational thought and reasoning. Nature came up with an almost identical solution, but did so without the benefit of rational and intelligent guidance, working only through the mechanisms of evolution. One of the pinnacles of human reason turns out to be a problem solved through an unthinking, uncaring, irrational process. We have not discovered or invented any mathematics, but are merely describing the complexity and splendor of the world that we inhabit. Even then, our descriptions are simplified and impoverished in comparison to the real thing. How can one feel anything but humility when presented with such majesty?

Search for:

Math Worth Reading

Good Math, Bad Math
Math Jokes 4 Mathy Folks
Spiked Math
The Math Less Traveled

Nature by numbers

Primary tabs.

View (active tab)

The animation begins by presenting a series of numbers. This is a very famous and recognized sequence since many centuries ago in the Western World thanks to Leonardo of Pisa , a thirteenth century Italian mathematician, also called Fibonacci . So it is known as Fibonacci Sequence , even although it had been described much earlier by Indian mathematicians.

This is an infinite sequence of natural numbers where the first value is 0 , the next is 1 and, from there, each amount is obtained by adding the previous two .

The values of this sequence have been appearing in numerous applications, but one of the most recognized is the Fibonacci Spiral , which has always been used as an approximation to the Golden Spiral (a type of logarithmic spiral ) because it is easier to represent with help of a simple drawing compass. This is the next thing to be shown on the animation, appearing just after the first values on the succession: the process of building one of these spirals.

We will create first a few squares that correspond to each value on the sequence: 1x1 - 1x1 - 2x2 - 3x3 - 5x5 - 8x8, etc. And they are arranged in the way how we see in the diagram at left. Then we draw a quarter circle arc (90°) within each little square and we can easily see how it builds step by step the Fibonacci Spiral , looking at right graphic. I have introduced a small optical correction in the animation in order to get the resulting curve more like a true Golden Spiral (more harmonious and balanced), as explained on this plate . It's something similar to what happens when we try to approach to an ellipse by drawing an oval using circular segments: the result is not the same as a true ellipse. And it shows.

IMPORTANT NOTE: while watching the animation conveys the idea that the Fibonacci spiral (or the Golden Spiral , it doesn't matter) is on the origin of the shape of a Nautilus , this isn't absolutely right. It's funny because if you perform this search at Google Images: “ spiral + nautilus ” you will see how many images suggest that this shell is really based on the construction system described above. But this isn't correct, as it's outlined on this other page .

The truth is that this is something I discovered when I had completely finished the screenplay for this project and I was too lazy to change. Therefore I must confess that I did a kind of cheat with this animation. Or you could explain in a more "genteel" way, saying that I have taken an artistic license ;-)

Once it has appeared the Nautilus we advance to the second part of the animation. It introduces the concept of Golden Ratio by constructing a Golden Rectangle . We start from a simple square to get that and use a classic method that requires only a ruler and drawing compass. See the complete process on the following series of illustrations:

This is very special rectangle known since ancient times. It fulfills this ratio, also known as the Golden Ratio or Divine Proportion : the ratio of the sum of the quantities ( a+b ) to the larger quantity ( a ) is equal to the ratio of the larger quantity ( a ) to the smaller one ( b ).

The result of this ratio (ie the division of a by b ) is an irrational number known as Phi —not to be confused with Pi — and an approximate value of 1.61803399… Formerly was not conceived as a true "unit" but as a simple relationship of proportionality between two segments. And we find in many works created by the mankind in art and architecture, from the Babylonian and Assyrian civilizations to our days, passing through ancient Greece or the Renaissance.

JUST A CURIOSITY: it isn't evident on the animation, but there is a deep connection between the Fibonacci Sequence and Golden Ratio . You have an example at right (we will see another one): if we divide each value in the Fibonacci Series by the previous, the result tends to Phi . The higher the value, the greater the approximation (consider that Phi, like any irrational number, has infinite decimals).

These two circular segments are accomplishing too with the same golden proportionality, but on this case the value of the angle formed by the smallest of them is another irrational number, we can simplify and round it as 137.5 º And this value is deeply present in nature. This is the next concept we see on the animation: how to configure the structure formed by the sunflower seeds . Look at the figures below:

We add a first red seed.
Turn 137.5º
Add a second green color seed and make the previous traveling to the center.
Turn other 137.5º
Add a third ocher seed and make the previous traveling to the center, to stay side by side with the first one.
Turn other 137.5º…

…and so on, seed after seed, we will obtain gradually a kind of distributions like the ones you have in the following figures.

This leads to the characteristic structure in which all seeds are arranged into a sunflower, which is as compact as possible. We have always said: nature is wise :-) ANOTHER CURIOSITY: Do you remember we had commented that there had a deep connection between the Fibonacci Sequence and Golden Ratio ? Well, next we have another meeting point between both concepts. Look at the following images of a sunflower:

By observing closely the seeds configuration you will see how appears a kind of spiral patterns . In the top left picture we have highlighted three of the spirals typologies that could be found on almost any sunflower. Well, if you look at one of the typologies, for example the one in green, and you go to the illustration above right you can check that there is a certain number of spirals like this, specifically 55 spirals . Coincidentally a number that is within the Fibonacci Sequence ;-)

And we have more examples in the two upper panels, cyan and orange, they are also arranged following values that are within the sequence: 34 and 21 spirals . In principle, all the sunflowers in the world show a number of spirals that are within the Fibonacci Sequence . You could go out to the countryside and look for a plantation to be sure :-) You can also use this image of a real sunflower or go to this website where this is explained, along with another curiosities. By the way, I recommend the rest of the Ron Knott site , a mathematician at the University of Surrey in England. His web is full of invaluable and educational information, all very well explained and with large doses of curious and funney elements .

Finally we reached the third segment of the animation in which we work with a concept that is a little less known than the others: the Voronoi Tessellations , also called Dirichlet Tessellation. I discovered this issue thanks to Hector Garcia's personal site , which I visit almost daily (and despite being a blog dedicated to Japanese culture and everything that is related to that country, also delights us from time to time with other interesting topics, like this one about Delaunay and Voronoi ). These geometric formations are based on a distribution pattern that is easily recognizable in many natural structures, like the wings of some insects or these small capillary ramifications in some plant's leaves. It is also widely used to optimize the distribution systems based on areas of influence, at the time to decide, for example, where to install phone antennas, or where to build the different delegations for a pizza chain. Let me show you a very intuitive way to understand how it forms a Voronoi Tiling:

Imagine we have two points: one red and another blue (top left). Start by drawing a segment joining these dots and then a second orthogonal line who is right in the middle. We have just found the bisector of the segment joining these two points. Above right we added a third green point, generating two new bisectors that intersect with the first. If we continue adding points to generate succesive bisectors, with their intersections, will lead to a series of polygons —Voronoi Tiles— around a set of "control points". Thus, the perimeter of each one of these tiles is equidistant to neighboring points and defines their area of influence. All these segments that interconnect the points form a triangular structure called Delaunay Triangulation . In the illustration below you can see the process as we continue adding points:

We can find interactive sites on the internet (like this ) to draw points, move them, and check how the structure becomes updated in real time. In fact, if we have a series of random dots scattered in the plane, the best way of finding the correct Voronoi Telesación for this set is using the Delaunay triangulation. And in fact, this is precisely the idea shown on the animation: first the Delaunay Triangulation and then, subsequently, the Voronoi Tessellation . But to draw a correct Delaunay Triangulation is necessary to meet the so-called “Delaunay Condition”. This means that: a network of triangles could be considered Delaunay Triangulation if all circumcircles of all triangles of the network are “empty”. Notice that actually, given a certain number of points in the plane there is no single way to draw triangles, there are many. But only one possible triangulation meets this condition. It is very simple: we draw a triangle using 3 points only if the circumcircle created using these 3 points is "empty" (not enclosing any other dot). You see that in the graph below, extracted from Wikipedia:

prossimo articolo

How far has scientific culture come in italy in the last twenty years.

It will be presented on March 18 the 20th edition of the Science Technology and Society Yearbook by Observa, which gathers twenty years of data to provide an overview of the most significant dynamics and trends in the relationships between science, technology, and society. Here is our review of the report.

Often when the Italian speaker discusses any topic, they express their opinions. The Anglo-Saxon speaker, on the other hand, often starts by presenting data, and then, if really necessary, offers their opinion.

Skip to main content
Keyboard shortcuts for audio player

13.7 Cosmos & Culture

13.7: cosmos and culture, nature by numbers.

Something beautiful for a sunday morning.

This is extraordinary animation from Etérea studios which is the work of a single person: Cristóbal Vila.

Note the appearence of the golden mean or golden ratio which is an irrational mathematical constant , approximately 1.6180339887. It appears everywhere in nature and in art and appears somehow hardwired into our appreciation of beauty.

Wondrous indeed.

The best free cultural &

educational media on the web

Online Courses
Certificates
Degrees & Mini-Degrees
Audio Books

Nature by Numbers: Short Film Captures the Geometrical & Mathematical Formulas That Reveal Themselves in Nature

in Film , Math | March 21st, 2010 5 Comments

For centuries, artists and architects have used some well-known geometrical and mathematical formulas to guide their work: The Fibonacci Series and Spiral, The Golden and Angle Ratios, The Delauney Triangulation and Voronoi Tessellations, etc. These formulas have a reality beyond the minds of mathematicians. They present themselves in nature, and that’s what a Spanish filmmaker, Cristóbal Vila, wanted to capture with this short film, Nature by Numbers . You can learn more about the movie at the filmmaker’s web site , and also find his latest film here: Inspirations: A Short Film Celebrating the Mathematical Art of M.C. Escher .

If you would like to sign up for Open Culture’s free email newsletter, please find it here . Or follow our posts on Threads , Facebook , BlueSky or Mastodon .

If you would like to support the mission of Open Culture, consider making a donation to our site . It’s hard to rely 100% on ads, and your contributions will help us continue providing the best free cultural and educational materials to learners everywhere. You can contribute through PayPal , Patreon , and Venmo (@openculture). Thanks!

by OC | Permalink | Comments (5) |

Comments (5), 5 comments so far.

This is completely and utterly awsome!!!!

Absolutely wonder-filled! As a newbie photographer I’ve become more and more aware of the geometries like those in the sunflower center. Thanks for sharing this.

wonderfully made…

This is such a poignant and artfully concise illustration of a concept so complex we may never fully understand it. Well done!!!

Beautiful work, attentive to details that take us long to realize. The presence of mathematics and geometry in nature is something striking, I conclude that we know nothing. Although I was already aware of it, the beauty of this filmmaker’s demonstration increased my too simple notion. Thank you!

Add a comment

Newsletter Sign-up

Free courses.

Art & Art History
Classics/Ancient World
Computer Science
Data Science
Engineering
Environment
Political Science
Writing & Journalism
All 1500 Free Courses
1000+ MOOCs & Certificate Courses

Receive our Daily Email

Free updates, get our daily email.

Get the best cultural and educational resources on the web curated for you in a daily email. We never spam. Unsubscribe at any time.

FOLLOW ON SOCIAL MEDIA

Free Movies

1150 Free Movies Online
Free Film Noir
Silent Films
Documentaries
Martial Arts/Kung Fu
Free Hitchcock Films
Free Charlie Chaplin
Free John Wayne Movies
Free Tarkovsky Films
Free Dziga Vertov
Free Oscar Winners
Free Language Lessons
All Languages

Free eBooks

700 Free eBooks
Free Philosophy eBooks
The Harvard Classics
Philip K. Dick Stories
Neil Gaiman Stories
David Foster Wallace Stories & Essays
Hemingway Stories
Great Gatsby & Other Fitzgerald Novels
HP Lovecraft
Edgar Allan Poe
Free Alice Munro Stories
Jennifer Egan Stories
George Saunders Stories
Hunter S. Thompson Essays
Joan Didion Essays
Gabriel Garcia Marquez Stories
David Sedaris Stories
Stephen King
Golden Age Comics
Free Books by UC Press
Life Changing Books

Free Audio Books

700 Free Audio Books
Free Audio Books: Fiction
Free Audio Books: Poetry
Free Audio Books: Non-Fiction

Free Textbooks

Free Physics Textbooks
Free Computer Science Textbooks
Free Math Textbooks

K-12 Resources

Free Video Lessons
Web Resources by Subject
Quality YouTube Channels
Teacher Resources
All Free Kids Resources

Free Art & Images

All Art Images & Books
The Rijksmuseum
Smithsonian
The Guggenheim
The National Gallery
The Whitney
LA County Museum
Stanford University
British Library
Google Art Project
French Revolution
Getty Images
Guggenheim Art Books
Met Art Books
Getty Art Books
New York Public Library Maps
Museum of New Zealand
Smarthistory
Coloring Books
All Bach Organ Works
All of Bach
80,000 Classical Music Scores
Free Classical Music
Live Classical Music
9,000 Grateful Dead Concerts
Alan Lomax Blues & Folk Archive

Writing Tips

William Zinsser
Kurt Vonnegut
Toni Morrison
Margaret Atwood
David Ogilvy
Billy Wilder
All posts by date

Personal Finance

Open Personal Finance
Amazon Kindle
Architecture
Artificial Intelligence
Beat & Tweets
Comics/Cartoons
Current Affairs
English Language
Entrepreneurship
Food & Drink
Graduation Speech
How to Learn for Free
Internet Archive
Language Lessons
Most Popular
Neuroscience
Photography
Pretty Much Pop
Productivity
UC Berkeley
Uncategorized
Video - Arts & Culture
Video - Politics/Society
Video - Science
Video Games

Great Lectures

Michel Foucault
Sun Ra at UC Berkeley
Richard Feynman
Joseph Campbell
Jorge Luis Borges
Leonard Bernstein
Richard Dawkins
Buckminster Fuller
Walter Kaufmann on Existentialism
Jacques Lacan
Roland Barthes
Nobel Lectures by Writers
Bertrand Russell
Oxford Philosophy Lectures

Open Culture scours the web for the best educational media. We find the free courses and audio books you need, the language lessons & educational videos you want, and plenty of enlightenment in between.

Great Recordings

T.S. Eliot Reads Waste Land
Sylvia Plath - Ariel
Joyce Reads Ulysses
Joyce - Finnegans Wake
Patti Smith Reads Virginia Woolf
Albert Einstein
Charles Bukowski
Bill Murray
Fitzgerald Reads Shakespeare
William Faulkner
Flannery O'Connor
Tolkien - The Hobbit
Allen Ginsberg - Howl
Dylan Thomas
Anne Sexton
John Cheever
David Foster Wallace

Book Lists By

Neil deGrasse Tyson
Ernest Hemingway
F. Scott Fitzgerald
Allen Ginsberg
Patti Smith
Henry Miller
Christopher Hitchens
Joseph Brodsky
Donald Barthelme
David Bowie
Samuel Beckett
Art Garfunkel
Marilyn Monroe
Picks by Female Creatives
Zadie Smith & Gary Shteyngart
Lynda Barry

Favorite Movies

Kurosawa's 100
David Lynch
Werner Herzog
Woody Allen
Wes Anderson
Luis Buñuel
Roger Ebert
Susan Sontag
Scorsese Foreign Films
Philosophy Films
February 2024
January 2024
December 2023
November 2023
October 2023
September 2023
August 2023
February 2023
January 2023
December 2022
November 2022
October 2022
September 2022
August 2022
February 2022
January 2022
December 2021
November 2021
October 2021
September 2021
August 2021
February 2021
January 2021
December 2020
November 2020
October 2020
September 2020
August 2020
February 2020
January 2020
December 2019
November 2019
October 2019
September 2019
August 2019
February 2019
January 2019
December 2018
November 2018
October 2018
September 2018
August 2018
February 2018
January 2018
December 2017
November 2017
October 2017
September 2017
August 2017
February 2017
January 2017
December 2016
November 2016
October 2016
September 2016
August 2016
February 2016
January 2016
December 2015
November 2015
October 2015
September 2015
August 2015
February 2015
January 2015
December 2014
November 2014
October 2014
September 2014
August 2014
February 2014
January 2014
December 2013
November 2013
October 2013
September 2013
August 2013
February 2013
January 2013
December 2012
November 2012
October 2012
September 2012
August 2012
February 2012
January 2012
December 2011
November 2011
October 2011
September 2011
August 2011
February 2011
January 2011
December 2010
November 2010
October 2010
September 2010
August 2010
February 2010
January 2010
December 2009
November 2009
October 2009
September 2009
August 2009
February 2009
January 2009
December 2008
November 2008
October 2008
September 2008
August 2008
February 2008
January 2008
December 2007
November 2007
October 2007
September 2007
August 2007
February 2007
January 2007
December 2006
November 2006
October 2006
September 2006

Advertise with Us
Copyright Policy
Privacy Policy
Terms of Use

Nature's numbers

Human beings have been aware of patterns in nature for as long as they have been human. Without such an awareness, human culture cannot function. Agriculture is impossible without an understanding of the seasons; even before agriculture, hunting depended upon animal life cycles and patterns as simple as the trajectory of a thrown rock. Humans have discovered that one of the most effective tools for grasping nature's patterns is mathematics.

Some physicists think that the universe is made of mathematics. Even if this is an exaggeration, there is no doubt that the universe often behaves as if mathematics is a central component of it. It is a message that goes back well before Plato, with his celebrated image of the real world as the shadow on a wall, cast by an ideal mathematical world that we glimpse only imperfectly. It was Pythagorean philosophy which maintained that all is number. Later thinkers added geometry as well; Johannes Kepler discovered that the orbit of Mars is an ellipse, a curve that featured prominently in the geometry of Euclid. It was this geometric pattern which led Isaac Newton to formulate the law of gravity. In so doing, he introduced new mathematical tools such as calculus and dynamics, and his "System of the World" was so successful that it became a model for succeeding generations.

But models change, and today we are starting to glimpse a new vision of mathematics in nature, one that is more flexible and adaptable than the numerology of Pythagoras or the rigid geometry of Euclid. This vision is providing a mathematical understanding of nature's processes as well as its structures.

hidden within the Newtonian paradise was a philosophical time bomb; it has just exploded. For centuries science made a plausible assumption: that the key to understanding nature is to find the simple laws which underpin it. Once the laws are known, all else follows. Not so. Thanks to various scientific mavericks, we now know that simple laws can give rise to behaviour of endless complexity-and conversely, that highly complex systems often exhibit well defined, but apparently acausal, patterns.

One area that stands to benefit from this new knowledge is biology. There are many biological phenomena that we do not understand in depth. One is the evolution of organised structure in the living world; another, our main focus here, is the occurrence of mathematical form in an organism.

The maverick who led the way in this field was Sir D'Arcy Wentworth Thompson: a Scot born in Edinburgh in 1860, both zoologist and classical scholar. At a time when most zoological studies were purely descriptive, D'Arcy Thompson became convinced that many aspects of biological form hinged on universal mathematical principles. In 1917 he assembled the evidence in his book On Growth and Form.

It led its readers from basic principles to generalities-demonstrating, for instance, that the body-plans of two different species of fish can be matched one-to-one by a simple co-ordinate transformation. It related the biology of dividing cells to the mathematics of surfaces of least area and the physics of soap bubbles. It assembled insights into the geometry and numerology of plants that spanned several centuries, centred upon the strange predilection of the plant kingdom for Fibonacci numbers-the sequence 1, 2, 3, 5, 8, 13, 21, 34, 55...-in which each number is the sum of the preceding two. Most flowers have a Fibonacci number of petals.

Despite Thompson's dazzling collection of biomathematical relationships, the mathematics of living form has proved less tractable than Sir D'Arcy expected. One reason is that genetic influences play a far greater role in determining biological form than he realised. A living creature does not follow some free-flowing pattern of growth; instead, chemical instructions written in its genes organise and constrain its growth in a manner that ultimately derives from the evolutionary imperative of survival. The simplest way to sum up what we now know is to say that Thompson was almost always wrong in important details, but almost always right in spirit.

One of the problems with being ahead of your time is that you have to work with inadequate tools. But a series of recent discoveries have given us a much more organic understanding of how mathematical rules create patterns. Indeed, our very conception of what a pattern is has begun to change. Chaos theory has taught us that simple deterministic rules can create intricate and apparently random effects, which means that we must distinguish between the patterns of structure that are made explicit by the rules, and the more subtle patterns of behaviour that the rules implicitly define. Complexity theory has shown that simple large scale patterns can "emerge" from the interactions between individual units, so that the capabilities of the whole may transcend those of its parts. Global principles such as symmetry and topology have provided ways to characterise and organise not only static patterns, but the dynamics that give birth to them, and how they grow or evolve.

Some of these discoveries originated thanks to another maverick who embellished Thompson's insights-a homosexual genius who, in an age of moral intolerance, took his own life. He was Alan Mathison Turing, remembered today as a father of the electronic computer and artificial intelligence. But Turing was also a pure mathematician, making important discoveries in number theory and abstract algebra; a logician, who found a simple route to Kurt Gödel's epic discoveries about the limitations of axiomatic systems as a base for mathematics; as well as a biomathematician. He sought a mathematical theory for the patterns which appear on animals-stripes, spots, dapples, patches. Unlike Thompson, his ideas went far beyond description: he proposed a general mechanism for the formation of biological pattern, inspiring a new branch of mathematics known as reaction-diffusion equations.

According to Turing, the development of mathematical patterns in animal markings arises through the combination of two processes. One involves some chemical "trigger" for pattern formation, a morphogen, which diffuses across the embryonic creature's surface, or even through its body. The other is a local chemical reaction, taking place inside each cell, which interacts with the diffusing morphogen. Mathematicians call the resulting system a reaction-diffusion equation.

One fascinating example is a celebrated laboratory experiment known as the Belousov-Zhabotinskii reaction (mercifully shortened to BZ), after the Russian who discovered it. Until then chemists expected all reactions to happen and then stop, but Belousov's reaction cycled through the same sequence of changes over and over again; in the process forming intricate spatial patterns. A dish containing a thin layer of evenly mixed red BZ liquid, left to its own devices, will develop a blue spot which grows into an ever-expanding series of concentric red and blue rings like an archery target. If the rings are broken they curl up into spirals.

Where do the patterns come from? The answer is that the patterns arise when a uniform distribution of morphogen becomes unstable. In a stable distribution, tiny irregularities are automatically smoothed out, but this does not always happen. In suitable circumstances, a tiny random fluctuation-triggered perhaps by the thermal vibration of atoms-may grow, and the bigger it becomes the faster it grows, until eventually other influences limit its size.

Like Thompson, Turing suffered the usual fate of brilliant mavericks: although right in spirit, he was wrong in detail. His choice of equations fails to predict such things as the dependence of pattern on temperature. The large iridescent spots on the tail of a peacock seem, at first sight, to be prime candidates for Turing's theory, but the spots extend across the many filaments of a single feather as if there were no joins, whereas a diffusing morphogen would get trapped at the boundaries. Moreover, the patterns are laid down long before the mature shape develops, but it is on the adult that we see the perfect Turingesque spots. So biologists lost confidence in Turing's equations, and forgot the greater insight, which is about mathematical structure, not specific equations.

Even now, Turing's theory undergoes periodic revivals. Last year two Japanese biomathematicians found that stripes on a particular species of tropical fish obey Turing's equations accurately. Their clinching argument was that over about three months the stripes on the fish move-and the changes are precisely as Turing's equations predict. But Turing's reaction-diffusion equations are only one example of a million possibilities with similar pattern-forming features. Over the last decade variants on Turing's scheme have been devised which come much closer to capturing biological realities: among these are the "mechano-chemical" equations, which perceive growth and pattern as an interaction between chemistry and geometry. The concentrations of various chemicals, at some position in or on an organism, determine not just its surface patterns, but also its growth rate, and hence the way its shape changes. On the other hand, changes in shape can stimulate or suppress the production of those very chemicals, so that shape and chemistry form an endless feedback loop.

while mathematically-minded scientists were developing general theories of pattern formation and applying them to everything they could lay their hands on, biology was marching to a different drumbeat. Two more mavericks, Francis Crick and James Watson, had discovered what Crick modestly called "the secret of life." With a mixture of calculation and guesswork, published results and gossip, and a spate of molecular model-building, Crick and Watson stumbled upon the double helix structure of DNA. Following this breakthrough, biologists were led to the genetic code, whereby triplets of DNA "bases" define the structure of the complex proteins from which living creatures are made. The genetic code underpins nearly all of our current understanding of how living organisms replicate, develop, mutate, and evolve; and it has led to the image of life as a kind of computer algorithm, programmed in DNA and implemented in protein. At first sight, such an image leaves little room for mathematical equations such as Turing's: programmes are flexible-you can write anything you want-but equations are constrained, they can do only what the mathematics permits. So today's biology has mostly turned its back on the mathematicians, seduced by the molecular biologists.

But the latest generation of science mavericks, still in hot pursuit of the principles underlying biological form and pattern, is attempting a kind of mathematico-biological fusion which, like the jazz-rock fusion of some 1980s pop music, tries to get the best of both worlds. Successes include an explanation of those puzzling Fibonacci patterns in flowers. Stephane Douady and Yves Couder, two French mathematicians, have traced the numerology to the dynamics of cell movement: seeds, sepals, and petals begin their existence as "primordia," little clumps of cells that emanate from the tip of the plant and drift outwards from it as the plant continues to grow. Each new primordium is crowded by the previous ones, so gets pushed into the biggest available gap. The result is that successive primordia appear along a tightly wound spiral, separated at intervals by the "golden angle," which is 137.5 degrees; the Fibonacci numbers show up as fractional approximations to this golden angle. So genetics can choose between Fibonacci numbers, or even (within limits defined by the same dynamical theory) choose a few other numbers instead, but it cannot escape the mathematical constraints of plant cell dynamics.

This kind of synthesis is wonderful when it succeeds, but it is a dangerous game because anybody can stick two different things together in a superficial way. The point is to create something that transcends either component-a kind of synergy that Jack Cohen and I, in The Collapse of Chaos, call "complicity." The American biomathematician Stuart Kauffman believes he can already discern the general framework needed. His inspiration is drawn from different areas of late 20th century science: chaos theory, artificial intelligence, computer simulation, non-equilibrium thermodynamics, fuzzy logic, cellular automata. It doesn't greatly matter what these names mean, but it is important to understand that the sources of inspiration are diverse and that each, individually, is seen by the orthodox as distinctly flaky. This makes the task of today's mavericks even harder, because a strange mixture of knowledge is required to understand what they are trying to say.

The concept that sums up this flaky proto-framework is "complexity theory," which is associated with the Santa Fe Institute set up in 1984 by the Nobel prize-winning physicist Murray Gell-Mann. Complexity theory is strong on computer simulations and short on rigorous mathematics. It has an unfortunate-but, as we shall see, inevitable-flavour of mysticism. Wildly interdisciplinary, it hops from history to economics to quantum mechanics and back. It stresses the failures of much current orthodoxy, instead of basking in its own successes. It attracts scientific journalists who can always find a good story; but it repels scientific conservatives who see the public face of complexity theory and don't like it, and who are not going to waste time trying to grasp the weird concepts which allegedly lie behind it.

The underlying theme of complexity theory is that systems composed of large numbers of simple agents, interacting with each other according to simple rules, often develop curious large scale patterns. Instead of trying to solve the mathematical equations for such systems (currently impossible), or to describe them verbally (too imprecise), complexity theorists simulate them on computers and analyse the results with strange new techniques.

What they find is stunning. For example, assemblies of simple mathematical "cells" (far simpler than any biological cell) equipped with simple rules for growth, division, relocation, and chemical constitution, spontaneously assemble themselves into complex structures, patterns, shapes. You don't have to try hard to get them to do this: almost any reasonable set of rules will do. Similarly, in complexity models of evolution, the typical phenomena are those which have puzzled evolutionary theorists for decades: sudden bursts of diversity, long periods of stagnation, the spontaneous increase of order, rudimentary forms of "social" and "sexual" behaviour. In complexity models of economics the typical events are stockmarket crashes, which conventional economic theory could never explain. Complexity models have a habit of corresponding to the real world. (Adjusting more conventional models to mimic such behaviour usually requires a great deal of effort and special pleading.)

What do these simulations teach us? Literal- minded biologists look at the simple rules programmed into the complexity theorist's model of cells, and observe that a real cell is far more complex. So they argue that the model is unrealistic, hence no use. But that's only true if you think the game is to reproduce nature exactly. Complexity theorists are playing a different game: gaining insight. Simple rule-based models often produce the same kind of structures and self-organisation which real biology does. There seems to be a kind of "universality" to the spontaneous patterns of rule-based complex systems. The hope is that this can be understood, so that the simplified models can be used to explain how real cells perform some-not all, but some important ones- of their remarkable functions.

Unfortunately, it is technically difficult to pin down what this universality consists of, when it is valid, and why it occurs. The current way to get round this is to examine dozens of different sets of rules and see whether the same kinds of patterns occur-a sort of Monte Carlo statistical sampling. What most scientists, orthodox or not, would prefer is a coherent logical deduction along mathematical lines. But this may be difficult or even impossible, because the patterns observed in complex systems are usually emergent: that is, they are not obviously built into the component parts or how these interact; they seem to transcend the possibilities available to their components.

This is where complexity theory cannot avoid becoming mystical-or, at least, giving that appearance. Philosophers have struggled for a long time with emergence, and they have reached very few significant conclusions. Emergence has a mystic feel- for example, human consciousness may well be an emergent property of nerve cells, but saying so doesn't bring us any closer to understanding consciousness scientifically. Some philosophers have become very confused about emergence and causality: they feel that because mind is emergent, and (apparently) possesses free will, a conscious mind can choose to interfere with the operation of its own nerve cells and thereby subvert the laws of physics.

The solution lies in a better understanding of emergence. The word "transcend" is too strong. Emergent phenomena do not disobey rules of causality. What makes them appear to lack a cause is that the chain of cause-and-effect leading from the behaviour of the components to that of the whole is so intricate that human beings cannot grasp it. Thus, even though a mind with free will cannot discern the complex causal chain from which its freedom emerges, it cannot choose to disrupt that chain without affecting itself. Emergence is like a mathematical theorem whose statement is simple but whose proof is so long that nobody can follow it.

Complexity theory seems mystical because it is a serious attempt to develop a coherent mathematical theory of emergence. And many of the problems that worry scientists centre upon emergent phenomena. A stock market crash, for example, is the result of a very complex series of trading decisions made by thousands of individuals. It need not have a single obvious cause; it is just one of the universal emergent properties of that sort of system. This does not answer the question "why do stock markets crash?" and it certainly does not help us predict crashes. But the first step in doing this must be to place the problem in the right context-and that context is emergent behaviour in complex systems, not the over-simplifications of conventional mathematical economics.

There are drawbacks too. Although complexity theory is good at reproducing the qualitative behaviour of the real world, it is not yet developed enough on the quantitative side. To critics this shows that the new ideas are all form and no substance, but to enthusiasts it merely underlines the need for further development. Cohen and I call this the "Rolls Royce problem." Someone comes along to your car factory with (they claim) a revolutionary new idea, and you say "I'll believe you if you can improve on a Rolls Royce." But years of effort by thousands of people went into the creation of the Rolls Royce. However good the new idea is, it won't be able to compete unless it is given sufficient opportunity to develop.

Should it be given that chance? Does anything important hinge on the tendency of the natural world to produce mathematical patterns? Why do we want to know how such patterns form, anyway? It would be hard to suggest that the occurrence of, say, Fibonacci numbers in flower petals has any great economic or technological significance. Our new-found understanding of the processes that create this particular pattern is a contribution not to technology, but to what used to be called "natural philosophy." This is often characterised as knowledge for its own sake, but knowledge doesn't have a sake-we do. The natural philosophers sought knowledge because they saw it as the key to understanding, and understanding as the key to power.

More than 250 years ago a group of mathematicians became interested in the vibrations of a violin string. The long term result was the discovery of radio waves, leading to radar, television, and satellite navigation systems. The connection is that electromagnetic fields can form waves, just as the violin string does-on a mathematical level there is no significant difference. The mathematicians were not looking for wireless telegraphy when they studied the vibrations of the violin. They weren't even trying to build a better violin. They just thought that waves were interesting, and wanted to understand them. Had they not done so, physicists would not have noticed that the laws of electromagnetism imply the existence of radio waves, and engineers would not have tried to build the requisite equipment to turn them into a communications medium.

There is an even better reason for studying the patterns of living creatures than just the interconnectedness of the whole. The distinction between BZ style spirals and target patterns is crucial to our own continued existence. The pacemaker wave of electrical activity that spreads over the muscle tissue of your heart, causing it to contract as a whole and thereby function as an effective pump, is a target pattern written in electricity rather than chemicals. If that pattern is disrupted, then-just like the chemical BZ pattern-it turns into masses of swirling spirals. Your heart fibrillates, its muscle fibres no longer contracting in synchrony, and you rapidly die.

More generally, the new ideas tell us that biological form and behaviour are determined by mathematics as well as by recipes encoded in DNA. A recipe alone is not enough: it has to be cooked, and as every cook knows, a perfectly reasonable recipe can sometimes turn into a disaster when it is implemented using real ingredients. You will not avoid such a disaster by studying the recipe book; instead, you must study the dynamics of cookery.

The financial structure of today's science makes it poor at supporting fascinating but flaky new ideas. A committee which must justify its actions to political or industrial masters (who may view any failure as wastage) will prefer to fund yet another piece of mainstream work rather than to gamble on novelty. This is why we need science mavericks like those who are uncovering nature's numbers.

Design Thinking
Data Visualization
Data Journalism
Film and Video
Gaming and Game Design
Infographics
3D Graphics
Services and Products

Cristobal Vila’s Nature by Numbers

See it below, and then be sure to read his writeup about the various formula’s within the film.

Nature by Numbers Movie .

Randall Hand

Randall Hand is a computer graphics programmer and news junky that's been working in the field for the last 15 years. He's responsible for visualizations generated on some of the most powerful supercomputers in the world, ytnef, mullion support in ParaView, and VizWorld.com.

Dynamic Drawing from Autodesk offers animation with no timeline

MAXON Announces Immediate Availability of Cinema 4D Release 18

Updated features to the new version of MOHO (formerly Anime Studio Pro)

Anime Studio for 2D animation gets updated and renamed Moho 12

Harmony 14 from Toon Boom released with enhanced 3D workflow

Google spotlight stories come to youtube with interactive 360-degree storytelling.

(L-r) The Iron Giant (voiced by VIN DIESEL) and Hogarth (voiced by ELI MARIENTHAL) in Warner Bros.’ animated feature, “The Iron Giant.” Courtesy of Warner Bros. Pictures

‘The Iron Giant: Signature Edition’ Back in Theaters Remastered With New Scenes

Using Bias through Visuals in Political Advertising Matters Today

esports gamer: https://www.flickr.com/photos/denisdervisevic/

Gaming as a Service? And a Gaming ETF (HERO) emerges

selected New Apple iOS 13.2 emojis showing gender neutral characters, diversity and differently abled people

Gender Neutral Emojis introduced in Apple iOS 13.2 to mixed reactions

Multigenerational Workforce Vital But Needs Attention, Support

24 Cognitive Biases that block Critical Thinking (Infographic)

Interview with Ge Wang by Dean Meyers on morality, ehthics and what is sublime in design

Artful Design asks us to design with ethics and create the sublime

Share on Facebook
The Atlantic Selects

Nature by Numbers, Visualized

Jun 25, 2014 | 831 videos video by cristóbal vila.

We want to hear what you think about this article. Submit a letter to the editor or write to [email protected].

Author: Chris Heller

About this series.

A showcase of cinematic short documentary films, curated by The Atlantic.

Most Watched

The loudest underwater sound ever recorded has no scientific explanation, more in this series, what was lost when sxsw was canceled, the last true hermit was alone for 27 years, ‘my boyfriend died of covid-19’.

Atlantic Documentaries

The Iguana King

The Idea File

The Great Giraffe Illusion

A substance like nothing else on earth, a coronavirus prayer, ‘don’t let us die in here’.

The Atlantic Argument
Home School
Visual Histories
Atlantic Interviews
You Are Here
School Myths

To revisit this article, visit My Profile, then View saved stories .

Backchannel
Newsletters
WIRED Insider
WIRED Consulting

Brian Romans

Nature by Numbers

Need a little break from your hectic day? Or, maybe you are still on vacation and very relaxed? Either way, check out this short video from artist and 3D illustrator Cristóbal Vila exploring the relationship of numbers, geometry, and nature. If you're into the details, you should check out the explanation the mathematics shown in the video.

I hope everyone is having a nice holiday break – I'll get back to regular blogging in a few days.

This Woman Will Decide Which Babies Are Born

Karen Williams

These Women Came to Antarctica for Science. Then the Predators Emerged

David Kushner

Mexico City’s Metro System Is Sinking Fast. Yours Could Be Next

Reece Rogers

Frankie Adkins

He Got a Pig Kidney Transplant. Now Doctors Need to Keep It Working

Emily Mullin

IMAGES

Nature's by Numbers (Reflection Paper)
Nature Essay in English 10 Lines || Short Essay on Nature
REFLECTION-MATH.docx
NATURE BY NUMBERS.docx
Nature by Numbers (Reflection paper)
REFLECTION PAPER ON NATURE BY NUMBERS.docx

VIDEO

IH 51: On the Personal Reflection Essay
WILD Mag Jr
Evidence 1. Personal Reflection Essay
Evidence 1 A00838865
Task 7.1. Signed Reflection Essay DEDASTU
Kiva Reflection Essay

COMMENTS

Nature's by Numbers (Reflection Paper)
The Fibonacci sequence, for example is a mathematical property that pervades nature. Every living thing, whether a single cell, a grain of wheat, a hive of bees, and even the entire human race, grows according to the Fibonacci numbers. The counting method used by nature is called the Fibonacci numbers.
REFLECTION PAPER ON NATURE BY NUMBERS.docx
REFLECTION PAPER ON NATURE BY NUMBERS Cristobal Vila presents a series of animations illustrating various mathematical principles, beginning with a breathtaking animation of the Fibonacci Sequence before moving on to the Golden Ratio, the Angle Ratio, the Delaunay Triangulation, and Voronoi Tessellations. The words are scary sounding, but the truth is math very simple and easy to learn.
Nature by Numbers
Tue 19 Oct 2010 04.19 EDT. In this beautiful video, "Nature by Numbers," filmmaker Cristobal Vila presents a series of animations illustrating various mathematic principles, beginning with a ...
Essay about nature of numbers.docx
Essay about nature of numbers I've learn from the movie inspired by nature of number, geometry and natures are all of the things in this world are inspired by number, measurement, and computation. In short all of the things that we see are examples of mathematics. The film composed of beautiful image that can clearly represent the mathematics in the nature world, and it also represent a work ...
Nature by numbers: The hidden beauty of mathematics
Fractal and wallpaper symmetry are the two types I wanted to discuss. However, this article would be incomplete without a nod to the spirals that are too often seen in nature. Some of these spirals arise due to the golden ratio of 1.618[…] which is the most irrational number we can get. Put simply, it is the furthest away we can be from a ...
Nature by Numbers
A short movie inspired on numbers, geometry and nature. Project completed in March, 2010. Artists and architects have used since ancient times many geometrical and mathematical properties: we could take some examples simply by observing the refined use of the proportions by architects from Ancient Egypt, Greece and Rome or other Renaissance artists like Michelangelo, Da Vinci or Raphael.
A Reaction Paper on Nature's Numbers: The Unreal Reality ...
Essay on A Reaction Paper on Nature's Numbers: The Unreal Reality of Mathematics Ian Stewart's Nature's Numbers: The Unreal Reality of Mathematics (New York: Basic Books, 1995) is a book that lets us see nature from a mathematician's ... (translation, rotation, reflection) and shapes are so widespread in the nature that it is difficult not to ...
Nature By Numbers
Nature Meditation: Walk or sit silently in the forest. Focus on any one sense at a time for a short period of 2-3 minutes. Make a note of all the unique things you can observe and sense. Repeat the exercise with another of your senses. Notice how each sense reveals something new about the nature around us.
Nature by Numbers
To that square, attach a second square which measures 1 unit on a side (b). Together, these squares form a rectangle that is 2 units by 1 unit. On the long side of this rectangle, attach a square which measures 2 units on a side (c). We now have a rectangle which measures 2 units on one side, and 3 units on the other.
Nature by numbers
It fulfills this ratio, also known as the Golden Ratio or Divine Proportion: the ratio of the sum of the quantities ( a+b) to the larger quantity ( a) is equal to the ratio of the larger quantity ( a) to the smaller one ( b ). The result of this ratio (ie the division of a by b) is an irrational number known as Phi —not to be confused with Pi ...
The Hidden Mathematics of Nature : 13.7: Cosmos And Culture : NPR
The Hidden Mathematics of Nature : 13.7: Cosmos And Culture A wonderful vision of natures mathematics. 13.7 Cosmos & Culture ... Nature by Numbers. July 25, 2010 6:54 AM ET. By . Adam Frank
Nature by Numbers: Short Film Captures the Geometrical & Mathematical
They present themselves in nature, and that's what a Spanish filmmaker, Cristóbal Vila, wanted to capture with this short film, Nature by Numbers. You can learn more about the movie at the filmmaker's web site , and also find his latest film here: Inspirations: A Short Film Celebrating the Mathematical ...
Nature by Numbers Reaction Paper
Partial preview of the text. Download Nature by Numbers Reaction Paper and more Mathematics High school final essays in PDF only on Docsity! In this world, millions of beautiful unique things or phenomena exist that human are yet to find answers to. And people are still undoubtedly distant from knowing all the truth about the world we live in.
Reflection Paper Nature by Numbers
Reflection Paper Nature by Numbers - Free download as Word Doc (.doc / .docx), PDF File (.pdf), Text File (.txt) or read online for free. Nature by Numbers
Nature's numbers
Nature's numbers. In the spirit of Plato, a new generation of science mavericks is seeking to find a mathematical understanding of nature, from the number of petals in a flower to the shape of spots on a peacock. Ian Stewart explains why we should take their flaky theory seriously. February 20, 1996. Human beings have been aware of patterns in ...
Cristobal Vila's Nature by Numbers
6.3 K. As visualization scientists and visual effects artists, we spend much of our time trying to use math to recreate physical nature and effects, we don't spend much time trying to derive math from nature. A new short film from Cristobal Vila and Eterea studios aims do to just that by showing us how the fibonnaci sequence, the golden ratio ...
Nature by Numbers, Visualized
Nature by Numbers, Visualized Jun 25, 2014 | 831 videos Video by Cristóbal Vila
Reflection-Paper-on-Nature-of-Numbers.docx
Reflection Paper on Nature of Numbers As I watched the film which is Nature by Numbers by Cristobal Vila, I realized that mathematics is truly everywhere and even the things that is nature or the nature itself is also involves numbers or mathematics. In the first part of the video, it shows a series of numbers or rather a number sequence that discovered by Leonardo Pisano who is also known as ...
PAULINO
Nature by numbers by Cristobal Villa, is a must watch video. The important points from the film are everything we see, touch, smell and hear have a meaning. I'm amazed at what I've watched, one of the important examples in the film are how the patterns and the sequence is being formed from that object or living things.
Reflection Paper On Nature of Numbers
Reflection_Paper_on_Nature_of_Numbers.docx - Free download as PDF File (.pdf), Text File (.txt) or read online for free.
Reflection paper on nature by numbers.docx
View Reflection paper on nature by numbers.docx from MATH 1 at De Lasalle University Dasmariñas. Reflection Paper on 'Nature by Numbers' by Cristobal Vila First of all, nature is beautiful, and we ... View Essay - MAT 135 7-1.docx from MAT 135 at Southern New Hampshire University. For th... essay. GoldenRatio_MA1015.docx. Ultimate Medical ...
Nature by Numbers
Either way, check out this short video from artist and 3D illustrator Cristóbal Vila exploring the relationship of numbers, geometry, and nature. If you're into the details, you should check out ...

Nature by Numbers

Comments (…)

Nature by numbers: The hidden beauty of mathematics

Leave a Reply Cancel reply

Nature by Numbers

A short movie inspired on numbers, geometry and nature

The concepts behind Nature by Numbers

Making of Nature by Numbers

Nautilus shell: 3D model process

Building the wing structure

Sunflower 3D model

Dragonfly 3D model

Procedural animation

Forest Walks, Nature Art, Mindfulness Activities, Healing with Nature

NATURE BY NUMBERS FILM

NUMBERS IN NATURE & MEDITATION

MEDITATIONS ON NATURE | FREE DOWNLOAD

Spread A Little Calm. Our World Needs It.

4 Comments on “ Nature By Numbers | Nature Meditations on Life and Self ”

Share your story Cancel reply

150+ Ways To Find Your Calm

Join Our Monthly Mailer

Nature by Numbers

The Fibonacci Sequence

The Golden Ratio

The Fibonacci and Golden Spirals

The Golden Angle

Recent Posts

Math Worth Reading

Nature by numbers

prossimo articolo

13.7 Cosmos & Culture

Nature by Numbers: Short Film Captures the Geometrical & Mathematical Formulas That Reveal Themselves in Nature

Related posts:

Add a comment

Newsletter Sign-up

Receive our Daily Email

Free Movies

Free eBooks

Free Audio Books

Free Textbooks

K-12 Resources

Free Art & Images

Writing Tips

Personal Finance

Great Lectures

Great Recordings

Book Lists By

Favorite Movies

Nature's numbers

Cristobal Vila’s Nature by Numbers

Randall Hand

You might also like

Dynamic Drawing from Autodesk offers animation with no timeline

MAXON Announces Immediate Availability of Cinema 4D Release 18

Anime Studio for 2D animation gets updated and renamed Moho 12

Harmony 14 from Toon Boom released with enhanced 3D workflow

‘The Iron Giant: Signature Edition’ Back in Theaters Remastered With New Scenes

Using Bias through Visuals in Political Advertising Matters Today

Gaming as a Service? And a Gaming ETF (HERO) emerges

Gender Neutral Emojis introduced in Apple iOS 13.2 to mixed reactions

Multigenerational Workforce Vital But Needs Attention, Support

24 Cognitive Biases that block Critical Thinking (Infographic)

Artful Design asks us to design with ethics and create the sublime

Nature by Numbers, Visualized

Author: Chris Heller

Most Watched

The Iguana King

The Great Giraffe Illusion

Nature by Numbers

IMAGES

VIDEO

COMMENTS