nature by numbers reflection essay

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.

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed .

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) |

Related posts:

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

Leave a reply.

Name (required)

Email (required)

XHTML: You can use these tags: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>

Click here to cancel reply.

• 1,700 Free Online Courses
• 200 Online Certificate Programs
• 100+ Online Degree & Mini-Degree Programs
• 1,150 Free Movies
• 1,000 Free Audio Books
• 150+ Best Podcasts
• 800 Free eBooks
• 200 Free Textbooks
• 300 Free Language Lessons
• 150 Free Business Courses
• Free K-12 Education
• Get Our Daily Email

Free Courses

• Art & Art History
• Classics/Ancient World
• Computer Science
• Data Science
• Engineering
• Environment
• Political Science
• Writing & Journalism
• All 1700 Free 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
• 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

Sign up for Newsletter

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
• September 2024
• August 2024
• 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

©2006-2024 Open Culture, LLC. All rights reserved.

• Advertise with Us
• Copyright Policy
• Privacy Policy
• Terms of Use

• 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.

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:

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:

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.

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! :

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:

Recent Posts

• Winter Solstice 2020
• Comet NEOWISE
• Can we pull over?
• Fig Leibnizes
• Entries feed
• Comments feed
• WordPress.org

Math Worth Reading

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

• 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.

• 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.

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

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

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

Editor picks.

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

Aeon Video has a monthly newsletter!

Get curated editors’ picks, peeks behind the scenes, film recommendations and more.

The elegance of mathematics meets the breathtaking complexity of nature

When the Spanish filmmaker and graphic designer Cristóbal Vila looks at nature, he sees numbers, and the remarkable elegance of mathematics. Uniting music and animation with mathematics, Nature by Numbers is a sensory science film, an immersion in the world of the minute and microscopic, and an exciting introduction to some of the great geometric and scientific concepts.

Director: Cristóbal Vila

The rhythms of a star system inspire a pianist’s transfixing performance

Is it ethical to have a second child so that your first might live?

Watch as Japan’s surplus trees are transformed into forest-tinted crayons

Meaning and the good life

‘Everydayness is the enemy’ – excerpts from the existentialist novel ‘The Moviegoer’

A spectacular, close-up look at the starfish with a ‘hands-on’ approach to parenting

Technology and the self

A filmmaker finds a tactile beauty in the creation of her prosthetic leg

An Indigenous myth and a geological survey elicit two ways of knowing one place

Beetles take flight at 6,000 frames per second in this perspective-shifting short

War and peace

A war meteorologist’s riveting account of how the Allies averted a D-Day disaster

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).

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:

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:

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.

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.

• 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

What was lost when sxsw was canceled, ‘disease’ vs. ‘difference’: a question of eugenics, america's most widely misread literary work, atlantic experiments, what is a city, the truth about stalin’s prison camps, the last true hermit was alone for 27 years, more in this series, ‘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

Stay Connected

This site is made possible by member support . ❤️

Big thanks to Arcustech for hosting the site and offering amazing tech support.

When you buy through links on kottke.org, I may earn an affiliate commission. Thanks for supporting the site!

kottke.org. home of fine hypertext products since 1998.

🍔   💀   📸   😭   🕳️   🤠   🎬   🥔

Nature By Numbers

This lovely short film by Cristóbal Vila shows how the simple Fibonacci sequence manifests itself in natural forms like sunflowers, nautilus shells, and dragonfly wings.

See also Arthur Benjamin’s TED Talk on the Fibonacci numbers and the golden ratio and the Fibonacci Shelf . (via @stevenstrogatz )

• Arthur Benjamin
• Cristobal Vila
• Fibonacci sequence
• mathematics

Math and Multimedia

School math, multimedia, and technology tutorials.

Nature by Numbers: Watch and fall in love with math

This is a captivating video ‘inspired by numbers, geometry and nature’ and was created by Cristóbal Vila.   The video explains the connections between the Fibonacci sequence 1,1,2,3,5,8,13, … (can you see the pattern?), and nature (the golden rectangle, the nautilus, the sunflower, etc.).

For non-math people, you will appreciate this video if you know the  concepts behind it.

I came across with this video about two weeks ago, but I had no chance to post it until I was reminded by a post about it at the   MathFuture wiki. On the funny side, there were more than 10 thousand who liked the video in Youtube, but 122 disliked it (plus a few more recently). One user (GatorTomKK) had the following comment for those 122 (and possibly for future ‘ dislikers ‘) :

122 people don’t understand math in general.

I can’t help but grin after reading the comment, and I’m sure your doing the same.

Leave a Reply Cancel reply

Nature By Numbers

Comments ( 4 ) document.write('comments ( 4 )');.

Check out NeatoShop's large selection of T-shirts Science T-Shirts | Sci-Fi T-Shirts | Fantasy T-Shirts Funny T-Shirts | Horror T-Shirts | Video Game T-Shirts

Shirts available in sizes S to 10XL: 6XL T-Shirts | 7XL T-Shirts | 8XL T-Shirts 9XL T-Shirts | 10XL T-Shirts Big & Tall and Curvy Tees are our specialties! Customization and personalization available.

Specialty T-Shirts : Anime Big & Tall T-Shirts (3XL to 10XL)

Check out these new sites: Homes & Hues - Home design, decor and gadgets Infinite1UP - Video Games Laughosaurus - Humor & Amazing Videos Pictojam - Science & Tech Pop Culturista - Pop Culture Supa Fluffy - Cute Animals & Pets

Success! Your email has been sent!

This website uses cookies.

This website uses cookies to improve user experience. By using this website you consent to all cookies in accordance with our Privacy Policy.

Ask the publishers to restore access to 500,000+ books.

Internet Archive Audio

• This Just In
• Grateful Dead
• Old Time Radio
• 78 RPMs and Cylinder Recordings
• Audio Books & Poetry
• Computers, Technology and Science
• Music, Arts & Culture
• News & Public Affairs
• Spirituality & Religion
• Radio News Archive

• Flickr Commons
• Occupy Wall Street Flickr
• NASA Images
• Solar System Collection
• Ames Research Center

• All Software
• Old School Emulation
• MS-DOS Games
• Historical Software
• Classic PC Games
• Software Library
• Kodi Archive and Support File
• Vintage Software
• CD-ROM Software
• CD-ROM Software Library
• Software Sites
• Tucows Software Library
• Shareware CD-ROMs
• Software Capsules Compilation
• CD-ROM Images
• ZX Spectrum
• DOOM Level CD

• Smithsonian Libraries
• FEDLINK (US)
• Lincoln Collection
• American Libraries
• Canadian Libraries
• Universal Library
• Project Gutenberg
• Children's Library
• Biodiversity Heritage Library
• Books by Language
• Additional Collections

• Prelinger Archives
• Democracy Now!
• Occupy Wall Street
• TV NSA Clip Library
• Animation & Cartoons
• Arts & Music
• Computers & Technology
• Cultural & Academic Films
• Ephemeral Films
• Sports Videos
• Videogame Videos
• Youth Media

Search the history of over 866 billion web pages on the Internet.

Mobile Apps

• Wayback Machine (iOS)
• Wayback Machine (Android)

Browser Extensions

Archive-it subscription.

• Explore the Collections
• Build Collections

Save Page Now

Capture a web page as it appears now for use as a trusted citation in the future.

Please enter a valid web address

• Donate Donate icon An illustration of a heart shape

Nature's Numbers: The Unreal Reality Of Mathematics (Science Masters Series)

Bookreader item preview, share or embed this item, flag this item for.

• Graphic Violence
• Explicit Sexual Content
• Hate Speech
• Misinformation/Disinformation
• Marketing/Phishing/Advertising
• Misleading/Inaccurate/Missing Metadata

plus-circle Add Review comment Reviews

8 Favorites

DOWNLOAD OPTIONS

For users with print-disabilities

IN COLLECTIONS

Uploaded by Fast_Shadow777 on August 8, 2021

SIMILAR ITEMS (based on metadata)

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

• The Big Story
• Newsletters
• Steven Levy's Plaintext Column
• WIRED Classics from the Archive
• WIRED Insider
• WIRED Consulting

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.

A Summary and Analysis of Ralph Waldo Emerson’s ‘Nature’

By Dr Oliver Tearle (Loughborough University)

‘Nature’ is an 1836 essay by the American writer and thinker Ralph Waldo Emerson (1803-82). In this essay, Emerson explores the relationship between nature and humankind, arguing that if we approach nature with a poet’s eye, and a pure spirit, we will find the wonders of nature revealed to us.

You can read ‘Nature’ in full here . Below, we summarise Emerson’s argument and offer an analysis of its meaning and context.

Emerson begins his essay by defining nature, in philosophical terms, as anything that is not our individual souls. So our bodies, as well as all of the natural world, but also all of the world of art and technology, too, are ‘nature’ in this philosophical sense of the world. He urges his readers not to rely on tradition or history to help them to understand the world: instead, they should look to nature and the world around them.

In the first chapter, Emerson argues that nature is never ‘used up’ when the right mind examines it: it is a source of boundless curiosity. No man can own the landscape: it belongs, if it belongs to anyone at all, to ‘the poet’. Emerson argues that when a man returns to nature he can rediscover his lost youth, that wide-eyed innocence he had when he went among nature as a boy.

Emerson states that when he goes among nature, he becomes a ‘transparent eyeball’ because he sees nature but is himself nothing: he has been absorbed or subsumed into nature and, because God made nature, God himself. He feels a deep kinship and communion with all of nature. He acknowledges that our view of nature depends on our own mood, and that the natural world reflects the mood we are feeling at the time.

In the second chapter, Emerson focuses on ‘commodity’: the name he gives to all of the advantages which our senses owe to nature. Emerson draws a parallel with the ‘useful arts’ which have built houses and steamships and whole towns: these are the man-made equivalents of the natural world, in that both nature and the ‘arts’ are designed to provide benefit and use to mankind.

The third chapter then turns to ‘beauty’, and the beauty of nature comprises several aspects, which Emerson outlines. First, the beauty of nature is a restorative : seeing the sky when we emerge from a day’s work can restore us to ourselves and make us happy again. The human eye is the best ‘artist’ because it perceives and appreciates this beauty so keenly. Even the countryside in winter possesses its own beauty.

The second aspect of beauty Emerson considers is the spiritual element. Great actions in history are often accompanied by a beautiful backdrop provided by nature. The third aspect in which nature should be viewed is its value to the human intellect . Nature can help to inspire people to create and invent new things. Everything in nature is a representation of a universal harmony and perfection, something greater than itself.

In his fourth chapter, Emerson considers the relationship between nature and language. Our language is often a reflection of some natural state: for instance, the word right literally means ‘straight’, while wrong originally denoted something ‘twisted’. But we also turn to nature when we wish to use language to reflect a ‘spiritual fact’: for example, that a lamb symbolises innocence, or a fox represents cunning. Language represents nature, therefore, and nature in turn represents some spiritual truth.

Emerson argues that ‘the whole of nature is a metaphor of the human mind.’ Many great principles of the physical world are also ethical or moral axioms: for example, ‘the whole is greater than its part’.

In the fifth chapter, Emerson turns his attention to nature as a discipline . Its order can teach us spiritual and moral truths, but it also puts itself at the service of mankind, who can distinguish and separate (for instance, using water for drinking but wool for weaving, and so on). There is a unity in nature which means that every part of it corresponds to all of the other parts, much as an individual art – such as architecture – is related to the others, such as music or religion.

The sixth chapter is devoted to idealism . How can we sure nature does actually exist, and is not a mere product within ‘the apocalypse of the mind’, as Emerson puts it? He believes it doesn’t make any practical difference either way (but for his part, Emerson states that he believes God ‘never jests with us’, so nature almost certainly does have an external existence and reality).

Indeed, we can determine that we are separate from nature by changing out perspective in relation to it: for example, by bending down and looking between our legs, observing the landscape upside down rather than the way we usually view it. Emerson quotes from Shakespeare to illustrate how poets can draw upon nature to create symbols which reflect the emotions of the human soul. Religion and ethics, by contrast, degrade nature by viewing it as lesser than divine or moral truth.

Next, in the seventh chapter, Emerson considers nature and the spirit . Spirit, specifically the spirit of God, is present throughout nature. In his eighth and final chapter, ‘Prospects’, Emerson argues that we need to contemplate nature as a whole entity, arguing that ‘a dream may let us deeper into the secret of nature than a hundred concerted experiments’ which focus on more local details within nature.

Emerson concludes by arguing that in order to detect the unity and perfection within nature, we must first perfect our souls. ‘He cannot be a naturalist until he satisfies all the demands of the spirit’, Emerson urges. Wisdom means finding the miraculous within the common or everyday. He then urges the reader to build their own world, using their spirit as the foundation. Then the beauty of nature will reveal itself to us.

In a number of respects, Ralph Waldo Emerson puts forward a radically new attitude towards our relationship with nature. For example, although we may consider language to be man-made and artificial, Emerson demonstrates that the words and phrases we use to describe the world are drawn from our observation of nature. Nature and the human spirit are closely related, for Emerson, because they are both part of ‘the same spirit’: namely, God. Although we are separate from nature – or rather, our souls are separate from nature, as his prefatory remarks make clear – we can rediscover the common kinship between us and the world.

Emerson wrote ‘Nature’ in 1836, not long after Romanticism became an important literary, artistic, and philosophical movement in Europe and the United States. Like Wordsworth and the Romantics before him, Emerson argues that children have a better understanding of nature than adults, and when a man returns to nature he can rediscover his lost youth, that wide-eyed innocence he had when he went among nature as a boy.

And like Wordsworth, Emerson argued that to understand the world, we should go out there and engage with it ourselves, rather than relying on books and tradition to tell us what to think about it. In this connection, one could undertake a comparative analysis of Emerson’s ‘Nature’ and Wordsworth’s pair of poems ‘ Expostulation and Reply ’ and ‘ The Tables Turned ’, the former of which begins with a schoolteacher rebuking Wordsworth for sitting among nature rather than having his nose buried in a book:

‘Why, William, on that old gray stone, ‘Thus for the length of half a day, ‘Why, William, sit you thus alone, ‘And dream your time away?

‘Where are your books?—that light bequeathed ‘To beings else forlorn and blind! ‘Up! up! and drink the spirit breathed ‘From dead men to their kind.

Similarly, for Emerson, the poet and the dreamer can get closer to the true meaning of nature than scientists because they can grasp its unity by viewing it holistically, rather than focusing on analysing its rock formations or other more local details. All of this is in keeping with the philosophy of Transcendentalism , that nineteenth-century movement which argued for a kind of spiritual thinking instead of scientific thinking based narrowly on material things.

Emerson, along with Henry David Thoreau, was the most famous writer to belong to the Transcendentalist movement, and ‘Nature’ is fundamentally a Transcendentalist essay, arguing for an intuitive and ‘poetic’ engagement with nature in the round rather than a coldly scientific or empirical analysis of its component parts.

Discover more from Interesting Literature

Subscribe to get the latest posts sent to your email.

Type your email…

Subscribe now to keep reading and get access to the full archive.

Continue reading