mathematics in nature 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 .

share this!

May 20, 2020

The mystique of mathematics: 5 beautiful math phenomena

by Sherry Landow, University of New South Wales

Mathematics is visible everywhere in nature, even where we are not expecting it. It can help explain the way galaxies spiral, a seashell curves, patterns replicate, and rivers bend.

Even subjective emotions, like what we find beautiful, can have mathematic explanations.

"Maths is not only seen as beautiful—beauty is also mathematical," says Dr. Thomas Britz, a lecturer in UNSW Science's School of Mathematics & Statistics. "The two are intertwined."

Dr. Britz works in combinatorics, a field focused on complex counting and puzzle solving. While combinatorics sits within pure mathematics, Dr. Britz has always been drawn to the philosophical questions about mathematics.

He also finds beauty in the mathematical process.

"From a personal point of view, maths is just really fun to do. I've loved it ever since I was a little kid.

"Sometimes, the beauty and enjoyment of maths is in the concepts, or in the results, or in the explanations. Other times, it's the thought processes that make your mind turn in nice ways, the emotions that you get, or just working in the flow—like getting lost in a good book."

Here, Dr. Britz shares some of his favorite connections between maths and beauty.

1. Symmetry—but with a touch of surprise

In 2018, Dr. Britz gave a TEDx talk on the Mathematics of Emotion, where he used recent studies on maths and emotions to touch on how maths might help explain emotions, like beauty.

"Our brains reward us when we recognize patterns, whether this is seeing symmetry, organising parts of a whole, or puzzle-solving," he says.

"When we spot something deviating from a pattern—when there's a touch of the unexpected—our brains reward us once again. We feel delight and excitement."

For example, humans perceive symmetrical faces as beautiful. However, a feature that breaks up the symmetry in a small, interesting or surprising way—such as a beauty spot—adds to the beauty.

"This same idea can be seen in music," says Dr. Britz. "Patterned and ordered sounds with a touch of the unexpected can have added personality, charm and depth."

Many mathematical concepts exhibit a similar harmony between pattern and surprise, elegance and chaos, truth and mystery.

"The interwovenness of maths and beauty is itself beautiful to me," says Dr. Britz.

2. Fractals: infinite and ghostly

Fractals are self-referential patterns that repeat themselves, to some degree, on smaller scales. The closer you look, the more repetitions you will see—like the fronds and leaves of a fern.

"These repeating patterns are everywhere in nature," says Dr. Britz. "In snowflakes, river networks, flowers, trees, lightning strikes—even in our blood vessels."

Fractals in nature can often only replicate by several layers, but theoretic fractals can be infinite. Many computer-generated simulations have been created as models of infinite fractals.

"You can keep focusing on a fractal, but you'll never get to the end of it," says Dr. Britz.

"Fractals are infinitely deep. They are also infinitely ghostly.

"You might have a whole page full of fractals, but the total area that you've drawn is still zero, because it's just a bunch of infinite lines."

3. Pi: an unknowable truth

Pi (or 'π') is a number often first learned in high school geometry. In simplest terms, it is a number slightly more than 3.

Pi is mostly used when dealing with circles, such as calculating the circumference of a circle using only its diameter. The rule is that, for any circle, the distance around the edge is roughly 3.14 times the distance across the center of the circle.

But Pi is a lot more than this.

"When you look into other aspects of nature, you will suddenly find Pi everywhere," says Dr. Britz. "Not only is it linked to every circle, but Pi sometimes pops up in formulas that have nothing to do with circles, like in probability and calculus."

Despite being the most famous number (International Pi Day is held annually on 14 March, 3.14 in American dating), there is a lot of mystery around it.

"We know a lot about Pi, but we really don't know anything about Pi," says Dr. Britz.

"There's a beauty about it—a beautiful dichotomy or tension."

Pi is infinite and, by definition, unknowable. No pattern has yet been identified in its decimal points. It's understood that any combination of numbers, like your phone number or birthday, will appear in Pi somewhere (you can search this via an online lookup tool of the first 200 million digits).

We currently know 50 trillion digits of Pi, a record broken earlier this year. But, as we cannot calculate the exact value of Pi, we can never completely calculate the circumference or area of a circle—although we can get close.

"What's going on here?" says Dr. Britz. "What is it about this strange number that somehow ties all the circles of the world together?

"There's some underlying truth to Pi, but we don't understand it. This mystique makes it all the more beautiful."

4. A golden and ancient ratio

The Golden Ratio (or 'ϕ') is perhaps the most popular mathematical theorem for beauty. It's considered the most aesthetically pleasing way to proportion an object.

The ratio can be shortened, roughly, to 1.618. When presented geometrically, the ratio creates the Golden Rectangle or the Golden Spiral.

"Throughout history, the ratio was treated as a benchmark for the ideal form, whether in architecture, artwork, or the human body," says Dr. Britz. "It was called the "Divine Proportion."

"Many famous artworks, including those by Leonardo da Vinci, were based on this ratio."

The Golden Spiral is frequently used today, especially in art, design and photography. The center of the spiral can help artists frame image focal points in aesthetically pleasing ways.

5. A paradox closer to magic

The unknowable nature of maths can make it seem closer to magic.

A famous geometrical theorem called the Banach-Tarski paradox says that if you have a ball in 3-D space and split it into a few specific pieces, there is a way to reassemble the parts so that you create two balls.

"This is already interesting, but it gets even weirder," says Dr. Britz.

"When the two new balls are created, they will both be the same size as the first ball."

Mathematically speaking, this theorem works—it is possible to reassemble the pieces in a way that doubles the balls.

"You can't do this in real life," says Dr. Britz. "But you can do it mathematically.

"That's sort of magic. That is magic."

Fractals, the Banach-Tarski paradox and Pi are just the surface of the mathematical concepts he finds beauty in.

"To experience many beautiful parts of maths, you need a lot of background knowledge," says Dr. Britz. "You need a lot of basic—and often very boring—training. It's a bit like doing a million push ups before playing a sport.

"But it is worth it. I hope that more people get to the fun bit of maths. There is so much more beauty to uncover."

Provided by University of New South Wales

Explore further

Feedback to editors

New in-vitro technique provides glimpse into the chloroplast workshop

1 minute ago

Researchers develop proton barrier films using pore-free graphene oxide

14 minutes ago

Wave scattering simulation unlocks potential metamaterials

15 minutes ago

'Fussy' molecules prefer one direction over the other, chiral copolymer study finds

Study accelerates AI-based particle size probe for medication manufacturing

Parasitoid wasp that lays its eggs inside of adult fruit fly discovered

16 minutes ago

Novel coupled nanopore platform offers greater precision for detecting molecules

Two-way mathematical 'dictionary' could connect quantum physics with number theory

California has dramatically improved its air quality, but racial disparities persist

23 minutes ago

Unbiased look at plants reveals how they achieve transcriptional regulation

48 minutes ago

Relevant PhysicsForums posts

What is a distinct feature of an ambiguous result.

Sep 10, 2024

Difficult to understand the solution provided in the video (travelling salesman problem)

Sep 6, 2024

Questions regarding Kurepa's Conjecture

Can higher degree nested radicals be simplified.

Sep 4, 2024

Raising to the power of 0 or 1

Calculate new height of truncated cone.

Sep 3, 2024

More from General Math

Related Stories

People can see beauty in complex mathematics, study shows

Sep 5, 2019

Mathematics is beautiful (no, really)

Feb 20, 2017

Chemists develop foolproof new test to track the fats we eat

May 15, 2020

Mathematical beauty activates same brain region as great art or music

Feb 12, 2014

Researchers discover a rare case of internal differences between the sexes of one fish genus

Jul 21, 2014

Study shows we like our math like we like our art: Beautiful

Aug 9, 2019

Recommended for you

Mathematical modeling explores the statistical mysteries of successfully scheduling a meeting

Sep 9, 2024

Virtual learning linked to rise in chronic absenteeism, study finds

Sep 5, 2024

Mathematicians model a puzzling breakdown in cooperative behavior

Mathematicians debunk GPS assumptions to offer improvements

Aug 28, 2024

Framework for solving parabolic partial differential equations could guide computer graphics and geometry processing

AI tools like ChatGPT popular among students who struggle with concentration and attention

Let us know if there is a problem with our content.

Use this form if you have come across a typo, inaccuracy or would like to send an edit request for the content on this page. For general inquiries, please use our contact form . For general feedback, use the public comments section below (please adhere to guidelines ).

Please select the most appropriate category to facilitate processing of your request

Thank you for taking time to provide your feedback to the editors.

Your feedback is important to us. However, we do not guarantee individual replies due to the high volume of messages.

E-mail the story

Your email address is used only to let the recipient know who sent the email. Neither your address nor the recipient's address will be used for any other purpose. The information you enter will appear in your e-mail message and is not retained by Phys.org in any form.

Newsletter sign up

Get weekly and/or daily updates delivered to your inbox. You can unsubscribe at any time and we'll never share your details to third parties.

More information Privacy policy

Donate and enjoy an ad-free experience

We keep our content available to everyone. Consider supporting Science X's mission by getting a premium account.

E-mail newsletter

Home — Essay Samples — Science — Mathematics in Everyday Life — The Relation of Mathematics and Nature

The Relation of Mathematics and Nature

• Categories: Mathematics in Everyday Life

About this sample

Words: 724 |

Published: Aug 31, 2023

Words: 724 | Pages: 2 | 4 min read

Cite this Essay

To export a reference to this article please select a referencing style below:

Let us write you an essay from scratch

• 450+ experts on 30 subjects ready to help
• Custom essay delivered in as few as 3 hours

Get high-quality help

Dr Jacklynne

Verified writer

• Expert in: Science

+ 120 experts online

By clicking “Check Writers’ Offers”, you agree to our terms of service and privacy policy . We’ll occasionally send you promo and account related email

No need to pay just yet!

Related Essays

3 pages / 1172 words

2 pages / 1086 words

1 pages / 582 words

9 pages / 4061 words

Remember! This is just a sample.

You can get your custom paper by one of our expert writers.

121 writers online

Still can’t find what you need?

Browse our vast selection of original essay samples, each expertly formatted and styled

Related Essays on Mathematics in Everyday Life

Mathematics has thousands of branches, and each branch means something different to every person. Some may know it as a useful tool that is a key to getting civilizations rolling. Others may just see it as bothersome and a tough [...]

Statistics are an integral part of daily life, influencing decisions and shaping our understanding of the world. Statistics are a branch of mathematics that involves the collection, analysis, interpretation, presentation, and [...]

The importance of polynomials in our daily life cannot be understated. From solving practical problems to shaping the foundations of various disciplines, polynomials play a crucial role in mathematics and its applications. In [...]

Mathematics is often described as the language of the universe. It is an essential field of study that underpins many aspects of daily life and numerous other academic disciplines. The importance of math extends beyond simple [...]

Statistics is a topic that I have never really thought of until now. I didn't know how it worked or how important and interesting it really is. It can open a whole new world of understanding the way we live as humans and why we [...]

Algebra bears major importance throughout youth and adulthood by providing mental assistance in solving daily tasks quicker, a backbone to other school related subjects, aid in understanding the mathematical work done by others, [...]

Related Topics

By clicking “Send”, you agree to our Terms of service and Privacy statement . We will occasionally send you account related emails.

Where do you want us to send this sample?

By clicking “Continue”, you agree to our terms of service and privacy policy.

Be careful. This essay is not unique

This essay was donated by a student and is likely to have been used and submitted before

Download this Sample

Free samples may contain mistakes and not unique parts

Sorry, we could not paraphrase this essay. Our professional writers can rewrite it and get you a unique paper.

Please check your inbox.

We can write you a custom essay that will follow your exact instructions and meet the deadlines. Let's fix your grades together!

Get Your Personalized Essay in 3 Hours or Less!

We use cookies to personalyze your web-site experience. By continuing we’ll assume you board with our cookie policy .

• Instructions Followed To The Letter
• Deadlines Met At Every Stage
• Unique And Plagiarism Free

What if Math Is a Fundamental Part of Nature, Not Something Humans Came Up With?

Nature is an unstoppable force, and a beautiful one at that. Everywhere you look, the natural world is laced with stunning patterns that can be described with mathematics. From bees to blood vessels, ferns to fangs, math can explain how such beauty emerges.

Math is often described this way, as a language or a tool that humans created to describe the world around them, with precision.

But there's another school of thought which suggests math is actually what the world is made of; that nature follows the same simple rules, time and time again, because mathematics underpins the fundamental laws of the physical world.

This would mean math existed in nature long before humans invented it, according to philosopher Sam Baron of the Australian Catholic University.

"If mathematics explains so many things we see around us, then it is unlikely that mathematics is something we've created," Baron writes .

Instead, if we think of math as an essential component of nature that gives structure to the physical world, as Baron and others suggest, it might prompt us to reconsider our place in it rather than reveling in our own creativity.

A world made of math

This thinking dates back to Greek philosopher Pythagoras (around 575-475 BCE), who was the first to identify mathematics as one of two languages that can explain the architecture of nature; the other being music. He thought all things were made of numbers; that the Universe was 'made' of mathematics, as Baron puts it .

More than two millennia later, scientists are still going to great lengths to uncover where and how mathematical patterns emerge in nature, to answer some big questions – like why cauliflowers look oddly perfect .

"We spent many hours frantically dismantling [cauliflower] florets, counting them, measuring angles between them," writes University of Nottingham mathematician Etienne Farcot, who studied cauliflower growth in an effort to understand these "mysterious cabbages."

Fractals are exquisite, self-repeating patterns which, besides some cauliflowers, are also found in fern fronds, branching blood vessels, and the rings of Saturn . Fractals are geometrical shapes made up of smaller and smaller copies of themselves, creating a mesmerizing 'self-similarity' that is infinitely deep.

Although only mathematical or computer-generated fractals are truly perfect fractals, nature comes pretty close.

"These repeating patterns are everywhere in nature," says mathematician Thomas Britz of the University of New South Wales in Sydney, Australia. "In snowflakes, river networks, flowers, trees, lightning strikes – even in our blood vessels."

Part of the charm of fractals is that they help to explain how complexity is born out of simplicity. As Benoît Mandelbröt, the Polish-born mathematician who coined the term fractal, said in 2010: "Bottomless wonders spring from simple rules which are repeated without end."

Branching river systems also carve near-perfect fractal patterns in the landscape.

So persistent are these patterns that in one instance, archaeologists looked for missing fractals to deduce ancient Egyptians might have modified river channels when building pyramids nearby.

Insects appear to follow mathematical principles, too.

Whether they know it or not, bees build hexagonal honeycomb in a way that produces the most storage space using the least materials – a theory known as the 'honeycomb conjecture' which was finally demonstrated by American mathematician Thomas Hales in 1999.

Some species of cicadas also have a life cycle geared towards prime numbers. Swarms of two North American species emerge from their subterranean burrows every 13 or 17 years , a trick which scientists think helps cicadas avoid predators with more regular rhythms.

Let's not forget nature's 'favorite' numbers, Fibonacci numbers , where each number in the sequence is the sum of the previous two. Fibonacci numbers are seen in sunflower seeds , pine cones and pineapples.

Spiral galaxies and nautilus shells also mimic so-called golden spirals by growing in a logarithmic ratio with every quarter turn.

But even though mathematical patterns are everywhere to be seen in nature, recent discoveries suggest the connection between math and nature runs deeper still, in ways we're only just beginning to appreciate.

Earlier this year, researchers discovered what they described as a previously unknown law of nature : a growth pattern which describes how pointed shapes form again and again in nature – from shark teeth and spider fangs to bird beaks and dinosaur horns.

"The diversity of animals, and even plants, that follow this rule is staggering," evolutionary biologist Alistair Evans from Monash University in Australia said at the time they discovered the mathematical formula, dubbed the 'power cascade'.

"We found it almost everywhere we looked across the kingdoms of life – in living animals, and those extinct for millions of years."

Back in 2015, scientists were also delighted to find a classic formula for Pi – the ever-constant ratio between a circle's circumference and its diameter – lurking in hydrogen atoms .

In a roundabout way, that discovery leads us back to the idea that mathematics provides a structural framework for the physical world. It's an interesting idea to entertain – so long as your head doesn't explode.

24/7 writing help on your phone

To install StudyMoose App tap and then “Add to Home Screen”

Mathematics In Nature - Free Essay Examples and Topic Ideas

Mathematics and nature have a close relationship. Many patterns and designs found in nature follow mathematical principles, such as the golden ratio and Fibonacci sequence. Mathematics can also be used to describe and understand natural phenomena, such as the movement of planets, the growth of populations, and the behavior of waves. The study of mathematics in nature is important for both mathematicians and scientists, as it can lead to new discoveries and a greater appreciation of the beauty and complexity of the natural world.

• 📘 Free essay examples for your ideas about Mathematics In Nature
• 🏆 Best Essay Topics on Mathematics In Nature
• ⚡ Simple & Mathematics In Nature Easy Topics
• 🎓 Good Research Topics about Mathematics In Nature

Essay examples

Essay topic.

Save to my list

Remove from my list

• Mathematics in Nature and Society
• The Nature of Mathematics
• The Perspectives on Discovering in Mathematics
• Mathematics & Natural Sciences with absolute certainty (TOK)
• Mathematics
• Literacy in Mathematics
• The Beauty of Mathematics Is Cold and Austere
• Babylonian Mathematics
• The Contributions of Greeks and Arabs to Mathematics
• Importance of Learning Mathematics
• Mathematics: Discipline Awareness Paper
• Poor Performance in Mathematics
• Learning Mathematics or Math as they call is not easy DP
• The Mozart Effect and Mathematics
• A Literature Review of Goals and Self-Efficacy Beliefs in Mathematics
• Objective Function vs Constraints in Mathematics

FAQ about Mathematics In Nature

👋 Hi! I’m your smart assistant Amy!

Don’t know where to start? Type your requirements and I’ll connect you to an academic expert within 3 minutes.

• Architecture and Design
• Asian and Pacific Studies
• Business and Economics
• Classical and Ancient Near Eastern Studies
• Computer Sciences
• Cultural Studies
• Engineering
• General Interest
• Geosciences
• Industrial Chemistry
• Islamic and Middle Eastern Studies
• Jewish Studies
• Library and Information Science, Book Studies
• Life Sciences
• Linguistics and Semiotics
• Literary Studies
• Materials Sciences
• Mathematics
• Social Sciences
• Sports and Recreation
• Theology and Religion
• Publish your article
• The role of authors
• Promoting your article
• Abstracting & indexing
• Publishing Ethics
• Why publish with De Gruyter
• How to publish with De Gruyter
• Our book series
• Our subject areas
• Your digital product at De Gruyter
• Contribute to our reference works
• Product information
• Tools & resources
• Product Information
• Promotional Materials
• Orders and Inquiries
• FAQ for Library Suppliers and Book Sellers
• Repository Policy
• Free access policy
• Open Access agreements
• Database portals
• For Authors
• Customer service
• People + Culture
• Journal Management
• How to join us
• Working at De Gruyter
• Mission & Vision
• De Gruyter Foundation
• De Gruyter Ebound
• Our Responsibility
• Partner publishers

Your purchase has been completed. Your documents are now available to view.

Mathematics in Nature

Modeling patterns in the natural world.

• John A. Adam
• X / Twitter

Please login or register with De Gruyter to order this product.

• Language: English
• Publisher: Princeton University Press
• Copyright year: 2003
• Edition: Course Book
• Audience: Professional and scholarly;College/higher education;
• Main content: 392
• Other: 24 color illus. 84 line illus. 9 tables.
• Keywords: Quantity ; Theorem ; Wavelength ; Surface area ; Mathematics ; Calculation ; Convection ; Approximation ; Geometry ; Amplitude ; Refraction ; Equation ; Proportionality (mathematics) ; Instability ; Diameter ; Gravity wave ; Dimensional analysis ; Square root ; Reynolds number ; Lee wave ; Temperature ; Drop (liquid) ; Line segment ; Surface tension ; Curvature ; Differential equation ; Partial differential equation ; Fermi problem ; Length scale ; Density ; Earth ; Angular diameter ; Differentiable function ; Gravity ; Mathematical model ; Dimension ; Buckling ; Natural number ; Dune ; Wavenumber ; Refractive index ; Gravitational acceleration ; Kinetic energy ; Dispersion relation ; Mathematician ; Variable (mathematics) ; Boundary value problem ; Soap bubble ; Fibonacci number ; Circumference ; Initial condition ; Scattering ; Solitary wave ; Prediction ; Three-dimensional space (mathematics) ; Viscosity ; Trigonometry ; Diagram (category theory) ; Molecule ; Wave equation ; Wing loading ; Summation ; Wave ; Dimensionless quantity ; Theory ; Air mass ; Golden angle ; Vibration ; Pattern formation ; Natural frequency ; Estimation ; Cloud ; Without loss of generality ; Iridescence ; Ray (optics) ; Diffusion equation ; Iteration ; Kootenay Lake ; Notation ; Wind shear ; Arc length ; Eye (cyclone) ; Aerosol ; Ordinary differential equation ; Applied mathematics ; Surface wave ; Rayleigh scattering ; Oscillation ; Mathematical analysis ; Sine wave ; Mathematical physics ; Probability ; Temperature gradient ; Snell's law ; Wave power ; Radius of curvature ; Vertical direction ; Differential calculus ; Golden rectangle ; Two-dimensional space
• Published: October 2, 2011
• ISBN: 9781400841011

Mathematics in Nature

Before you purchase audiobooks and ebooks.

Please note that audiobooks and ebooks purchased from this site must be accessed on the Princeton University Press app. After you make your purchase, you will receive an email with instructions on how to download the app. Learn more about audio and ebooks .

Support your local independent bookstore.

• United States
• United Kingdom

Mathematics

Mathematics in Nature: Modeling Patterns in the Natural World

• Download Cover

From rainbows, river meanders, and shadows to spider webs, honeycombs, and the markings on animal coats, the visible world is full of patterns that can be described mathematically. Examining such readily observable phenomena, this book introduces readers to the beauty of nature as revealed by mathematics and the beauty of mathematics as revealed in nature. Generously illustrated, written in an informal style, and replete with examples from everyday life, Mathematics in Nature is an excellent and undaunting introduction to the ideas and methods of mathematical modeling. It illustrates how mathematics can be used to formulate and solve puzzles observed in nature and to interpret the solutions. In the process, it teaches such topics as the art of estimation and the effects of scale, particularly what happens as things get bigger. Readers will develop an understanding of the symbiosis that exists between basic scientific principles and their mathematical expressions as well as a deeper appreciation for such natural phenomena as cloud formations, halos and glories, tree heights and leaf patterns, butterfly and moth wings, and even puddles and mud cracks. Developed out of a university course, this book makes an ideal supplemental text for courses in applied mathematics and mathematical modeling. It will also appeal to mathematics educators and enthusiasts at all levels, and is designed so that it can be dipped into at leisure.

Awards and Recognition

• John Adam, Winner of the 2007 recipient of the Virginia Outstanding Faculty Award, State Council of Higher Education for Virginia
• Winner of the 2003 for Professional/Scholarly Award in Mathematics and Statistics, Association of American Publishers
• One of Choice's Outstanding Academic Titles for 2004

" Mathematics in Nature is an excellent resource for bringing a greater variety of patterns into the mathematical study of nature, as well as for teaching students to think about describing natural phenomena mathematically. . . . [T]he breadth of patterns studied is phenomenal."—Will Wilson, American Scientist

"John Adam has combined his interest in the great outdoors and applied mathematics to compile one surprising example after another of how mathematics can be used to explain natural phenomena. And what examples! . . . [He] has done a great deal of reading and exposition, indulging his passions to create this compilation of mathematical models of natural phenomena, and the sheer number of examples he manages to cram into this book is testament to his efforts. There are other texts on the market which explore the connection between mathematics and nature . . . but none this wide-ranging."—Steven Morics, MAA Online

"Adam has laced his mathematical models with popular descriptions of the phenomena selected. . . . Mathematics in Nature can accordingly be read for pleasure and instruction by the select laity who are not afraid of reading between the lines of equations."—Philip J. Davis, SIAM News

"John Adam's quest is a very simple one: that is, to invite one to look around and observe the wonders of nature, both natural and biological; to ponder them; and to try to explain them at various levels with, for the most part, quite elementary mathematical concepts and techniques."—Brian D. Sleeman, Notices of the American Mathematical Association

"Reading this book progressively creates a course in mathematical modeling built around familiar, tangible, human-scale examples, with a trajectory that takes readers from dimensional estimates through geometrical modeling, linear and nonlinear dynamics, to pattern formation."— Choice

"John Adam's Mathematics in Nature illustrates how, in a friendly and lucid manner, mathematicians think about nature. Adam lets us see how mathematics is not only an ally, but is perhaps the very language that nature uses to express the beautiful. . . . This is a book that will challenge while it intrigues and excites."—Stanley David Gedzelman, Weatherwise

"Although Mathematics in Nature has not been written as a textbook, availability of such a manual shall help instructors who choose this delightful book for teaching a course in applied mathematics or mathematical modeling."—Yuri V. Rogovchenko, Zentralblatt Math

"Spanning a range of mathematical levels, this book can be used as an undergraduate textbook, a source of high school math enrichment, or can be read for pleasure by folks with an appreciation of nature but without advanced mathematical background."— Southeastern Naturalist

"Have you wondered how rainbows or sand dunes form? Does it puzzle you why drying mud forms polygonally shaped cracks? Can you explain the patterns on a butterfly's wings or how birds fly? In this delightful book, John Adam invites us to question and to share his enthusiasm for developing mathematical models to explore a wide range of everyday natural phenomena. Mathematics in Nature can be used as a text on mathematical modeling or as a book to dip into and encourage us to observe and wonder at the beauty of nature. It has the potential of becoming a classic."—Brian Sleeman, University of Leeds

"This is a book that I will want to keep close to hand so that I will not be stumped by all those seemingly simple yet subtle questions about nature: Why can fleas jump so high? Why is visibility better in rain than in fog? Why does a river meander? How high can trees grow? But it is much more than a compendium of useful facts and explanations. It is the clearest guide I have seen to the art of conceptualizing, simplifying, and modeling natural phenomena—no less than an exegesis on how good quantitative science is done."—Phillip Ball, Consultant Editor, Nature

" Mathematics in Nature leads the calculus-literate reader on a vigorous tour of nature's visible patterns—from the radiator-sailed dinosaur Dimetrodon to fracturing of dried mud and ceramic glazes, from the dispersion of rainbows and iridescence of beetles to the pearling of spider silk. Eschewing phenomena that are too small to see or too large to grasp, Adam shows how elementary college mathematics, rigorously applied, can give precise expression to everyday natural phenomena. His extraordinary range of examples and meticulous explanations document mathematics' wonderful capacity to describe and explain nature's patterns."—Lynn Arthur Steen, St. Olaf College

"This work is outstanding! The color photographs are beautiful. The writing style is splendid."—Robert B. Banks, author of Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics

"This is a unique, even great book. It is in the spirit of a number of books on topics like symmetry and chaos that look at mathematics in the context of visually striking natural and other phenomena but is more broadly based. The author leads with the phenomena and follows with the math, making the book accessible to a wider audience while still appealing to math students and faculty."—Frank Wattenberg

"This is one of the best contemporary texts on the subject, appealing to a very broad audience that will definitely love this excellent book."—Yuri V. Rogovchenko, Zentralblatt Math (European Mathematical Society)

John A. Adam’s website

Stay connected for new books and special offers. Subscribe to receive a welcome discount for your next order. 

Buy a gift card for the book lover in your life and let them choose their next great read.

• ebook & Audiobook Cart

THE HIDDEN BEAUTY OF MATHEMATICS

By Museu do Amanhã

Mathematics is everywhere. It is in the objects we create, in the works of art we admire. Although we may not notice it, mathematics is also present in the nature that surrounds us, in its landscapes and species of plants and animals, including the human species. Our attraction to other humans and even our mobility depend on it. But how does this happen?

Modern architecture by Unknown Author Museu do Amanhã

From the structure of buildings to the discovery of new planets, from trade to fashion and new technologies, mathematics has always served as an important tool in the advancement of science and technology, in fields as diverse as Engineering, Biology, Philosophy and Arts. And it is also present in nature, concealing- and revealing- its charms in various forms, intriguing researchers and inspiring poets. One of the ideas that best embodies mathematics in all its elegance is the concept of symmetry .

Ceiling of Lotfollah Mosque in Isfahan, Iran by Phillip Maiwald, Wikimedia Commons Museu do Amanhã

The roof of the Lotfollah mosque in Isfahan (Iran) is a great example of symmetry endowed with beauty in architecture.

Inside the mosque there are several rooms with diverse symmetrical motifs.

The Three Graces by Unknown Author Museu do Amanhã

Harmony and beauty

An object is symmetrical when there is "harmony in the proportions" of its parts in relation to the whole: height, width and length are balanced. Strictly associated with harmony and beauty, symmetry is also a decisive concept in theories about nature. Ancient Greece was apparently the first place where this idea had room to develop.

Hexagons by Unknown Author Museu do Amanhã

The Stanford Dictionary of Philosophy reminds us that in Timaeus, the work of Greek philosopher Plato (429-347 BCE or Before the Common Era), for example, regular geometric forms take center stage in the doctrine of natural elements because of the proportions they contain and the beauty of their forms. The four elements - Fire, Water, Earth and Air - could be represented by regular geometric shapes (with polyhedrons of four, twenty, six and eight equal sides, respectively). The Universe could also be represented by a 12-sided polyhedron - or a symmetrical dodecahedron. When particles with these different forms are combined, they give rise to all the natural elements we know. Although the word or concept of "symmetry" did not exist in Greek vocabulary in the days of Plato, the concept was already developing. The Greek noun "summetria", which literally means "of the same size", was already being used to refer to "proportion”.

Espiral áurea by Dicklyon/Wikimedia Commons Museu do Amanhã

The golden ratio

Some say that the size and proportion of perfect solids described by Plato are related to each other - the sides of particles of fire, water, and air could be combined together because they are proportional. They were described as having a "golden proportion" among themselves - or a type of symmetry that marks the growth rate in the development of several species. The leaves of a tree, for example, multiply more or less at this speed after they sprout: 2, 3, 5, 8, 13, 21, 34, 55 ... and so on. The last number is always the sum of the two preceding numbers - and when we divide each number by its predecessor, the result will be very close to 1.6180, or what mathematicians like the Italian Leonardo Fibonacci (1175 - 1250) consider as "the golden proportion”. When applied to a succession of proportional squares within a rectangle, this sequence of numbers - always with the golden ratio between them - generates a "golden rectangle”. If we draw a line, formed by quarters of a circle, following the progression of the figures formed in it, we have the "golden spiral", as you can see in the drawing on the side.

Concha de Náutilo by Domínio Público Museu do Amanhã

In nature, this type of symmetry marks the growth rhythm in the development of several species - and is also perceptible to the naked eye, fitting into the rules that determine the conception of "beauty" in art. The greatest example of the materialization of the golden spiral in nature is perhaps the nautilus, a prehistoric mollusk that still has living 'relatives' in the Pacific Ocean.

Fóssil de Ammonide by H. Zell, Wikimedia Commons Museu do Amanhã

The nautilus is a surviving species of the archaic subclass of nautoloids which appeared at the beginning of the Paleozoic - long before the dinosaurs and even before the appearance of the first terrestrial animals. The subclass of ammonoids included the extinct species of ammonites - still much appreciated by fossil aficionados - that also displayed the golden proportion in their shells.

Symmetric Fern by Unknown Author Museu do Amanhã

Same, but different?

We can see several other forms of symmetry in nature. There is a form of bilateral symmetry, like the reflection of an image in a lake that can be divided into two identical parts; and it can also be radial when the image forms around a central point and "radiates" to all sides, such as an open flower or a yellow dandelion. Symmetry also manifests itself in complex forms such as fractals, in which a structure looks similar to the whole on any scale. Also, in the case of sounds and waves of the same frequency, we can say with certainty that sounds and lights are also symmetrical. In the natural world, symmetries are not completely perfect and harbor some visible imperfections.

Spider Web by Unknown Author Museu do Amanhã

According to IME-USP professor Eduardo Colli,: "our eye looks for symmetries, even if these are not perfect in nature. In fact, the greatest beauty in the symmetries of nature lies in these little imperfections.”

Butterfly by Unknown Author Museu do Amanhã

Bilateral beauty

One of the main symmetries in nature is bilateral. We see how one side of the body of a plant or animal is a very close copy of the other, as if it were a plane, able to split the image into two sides - or two almost perfectly reflected images. Not infrequently, this morphology has a clear function: for example, it would be very difficult for a bird to fly straight if its wings weren't the same size.

Oranges by Unknown Author Museu do Amanhã

Spheres, spheres

An object is spherically symmetrical if it can be cut into two equal halves - regardless of the direction of the cut, as long as it passes through its center. Fruits like oranges and some lemons have a shape that is very close to being spherical.

Jupiter by NASA / SwRI / MSSS / Kevin M. Gill Museu do Amanhã

For Plato, the sphere was the most symmetrical and homogeneous form that existed. And therefore the most beautiful and perfect form of all. He said that the Cosmos had a spherical shape - as well as the celestial bodies, like the planet Jupiter we see in the image.

Hubble Views Stellar Genesis in the Southern Pinwheel by NASA Museu do Amanhã

According to Graham Smith, a mathematician from IM-UFRJ, "today's physicists believe that the so-called 'cosmological constant' is positive, which means that at the scale of the universe, the cosmos really could be a sphere. It would be a fourth-dimensional space-time sphere - but even so it seems that Plato was not that wrong after all!”

Radial Flower by Unknown Author Museu do Amanhã

Radial shapes

A body is radially symmetrical if you can cut it several times and generate equal pieces. Or if it is possible to "rotate" it around a central axis and get a circle effect. The main difference compared to spherical shapes is that in the case of spheres there is no "up" or "down" side, as in a more or less flat plane. In radial forms, these sides exist.

Conifer Pine by Unknown Author Museu do Amanhã

Take this conifer pine, for example: viewed from above, it has radial symmetry; when seen from the side, it has a more or less spherical symmetry.

Bolinopsis infundibulum by NOAA Photo Gallery Museu do Amanhã

A mixture of symmetries

There are shapes or species that combine more than one type of symmetry. Bi-radial species, for example, combine radial and bilateral symmetries. These are not very common in nature, and perhaps one of the best representatives of this type of format are the comb jellies. Resembling jellyfish, these marine animals have symmetrical opposite sides, but each side is different from its adjacent one. What does that mean? If it were a geometric figure, a comb jelly could easily be represented by a rectangle: the top and bottom sides are the same. However, these differ from the right and left sides (which are also the same). If all sides were exactly the same, the figure would no longer be a rectangle, but a square.

Menger Sponge by Niabot, Wikimedia Commons Museu do Amanhã

Breakable Shapes

“Fractal", a term coined by the French mathematician Benoît Mandelbrot in the mid-1970s, comes from the Latin word "fractus", or "broken”. This explains the logic of a fractal's geometry: it is a structure with a symmetrical scale. Any part of a fractal, no matter how small, has the same shape as the whole figure. A good example is the cube you see, better known as Menger's Sponge. The figure is named in honor of the Austrian mathematician Karl Menger, who in the last century studied the topology of geometric objects.

Menger Sponge in the Making by Niabot, Wikimedia Commons Museu do Amanhã

You can create a Menger Sponge by removing the central part of a cube and repeating the process a few times on an increasingly smaller scale.

Romanesco Broccoli by Unknown Author Museu do Amanhã

Probably the best representation of fractal forms in nature is the Roman cauliflower.

Estrela RS Puppis by NASA, ESA Museu do Amanhã

Symmetries in another dimension

Not all the symmetries we know happen in the spatial dimension, in the form of geometric figures or in forms found in nature. Symmetries also exist in the natural world in other ways that we can see, hear, and feel. Light and sound, for example, behave as a wave - and we can say that these are symmetrical when their wavelength is regular. Its symmetry does not occur in space the way a geometrical figure visibly does- its pulsation, light and sound are symmetrical in time. Some stars, for example, have regular variations in brightness, or pulsations. RS Puppis, located near the center of our Milky Way, is one of these: its frequency of pulsation is approximately 40 days.

Dandelion by Unknown Author Museu do Amanhã

Symmetries are everywhere all the time. Just look around to see that they surround us. In addition to endowing our daily life with more grace and beauty, they also have many functions of which we are unaware. Nature hides numbers, equations, and proportions that can be unraveled by anyone who is curious enough. As the celebrated physicist Richard Feynman once said, "knowledge of science only enriches the excitement, mystery, and admiration" for nature. It does not take away its beauty.

Museum of Tomorrow, 2018 Chairman of the IDG Board of Directors: Fred Arruda CEO: Ricardo Piquet General Curator: Luiz Alberto Oliveira Executive Director: Henrique Oliveira Director of Programming: Adriana Karla Rodrigues Director of Fundraising: Renata Salles Director of Scientific Development: Alfredo Tolmasquim Manager of Content: Leonardo Menezes Research and Writing: Meghie Rodrigues Editing: Emanuel Alencar

Consultants: Eduardo Colli (Department of Applied Mathematics, Institute of Mathematics and Statistics - University of São Paulo) and Graham Andrew Craig Smith (Institute of Mathematics, Federal University of Rio de Janeiro)

Photos and videos: Phillip Maiwald, Marie-Lan Nguyen, H. Zell, Niabot (Wikimedia Commons), NASA/SwRI/MSSS/Kevin M. Gill, NOAA Photo Gallery, NASA, ESA, G. Bacon (STScI), the Hubble Heritage Team (STScI/AURA)-Hubble/Europe Collaboration, and H. Bond (STScI and Pennsylvania State University)

Mauá Square

Museu do amanhã, lives on hold, consumption slaves, recycling and solutions, port, culture and art, the epoch of humans, overconsumption and waste, humans from the port, the port and its hills, the most dangerous species on the planet.

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

• View all journals

Pure mathematics articles from across Nature Portfolio

Pure mathematics uses mathematics to explore abstract ideas, mathematics that does not necessarily describe a real physical system. This can include developing the fundamental tools used by mathematicians, such as algebra and calculus, describing multi-dimensional space, or better understanding the philosophical meaning of mathematics and numbers themselves.

Latest Research and Reviews

Various distance between generalized Diophantine fuzzy sets using multiple criteria decision making and their real life applications

• Murugan Palanikumar
• Nasreen Kausar
• Fikadu Tesgera Tolasa

AI-driven research in pure mathematics and theoretical physics

Advances in artificial-intelligence-assisted mathematical investigations suggest that human–machine collaboration will be an integral part of future theoretical research.

• Yang-Hui He

High accuracy solutions for the Pochhammer–Chree equation in elastic media

• Mostafa M. A. Khater
• Suleman H. Alfalqi

Calibration estimation of population total using multi-auxiliary information in the presence of non-response

• Anant Patel
• Menakshi Pachori

Optimizing decision-making with aggregation operators for generalized intuitionistic fuzzy sets and their applications in the tech industry

• Muhammad Wasim
• Awais Yousaf
• Hamiden Abd El-Wahed Khalifa

A new multi-attribute group decision-making method based on Einstein Bonferroni operators under interval-valued Fermatean hesitant fuzzy environment

• Jasni Mohamad Zain

News and Comment

The real value of numbers

• Mark Buchanan

Machine learning to guide mathematicians

• Fernando Chirigati

A different perspective on the history of the proof of Hall conductance quantization

• Matthew B. Hastings

e is everywhere

From determining the compound interest on borrowed money to gauging chances at the roulette wheel in Monte Carlo, Stefanie Reichert explains that there’s no way around Euler’s number.

• Stefanie Reichert

Imagination captured

Imaginary numbers have a chequered history, and a sparse — if devoted — following. Abigail Klopper looks at why a concept as beautiful as i gets such a bad rap.

• Abigail Klopper

Prime interference

• David Abergel

Quick links

• Explore articles by subject
• Guide to authors
• Editorial policies

• Entertainment
• Environment
• Information Science and Technology
• Social Issues

Home Essay Samples Science Mathematical Predictions

The Power of Mathematics: Unveiling its Influence on Nature and Phenomena

Table of contents

Introduction.

*minimum deadline

Cite this Essay

To export a reference to this article please select a referencing style below

Related Essays

Need writing help?

You can always rely on us no matter what type of paper you need

*No hidden charges

100% Unique Essays

Absolutely Confidential

Money Back Guarantee

By clicking “Send Essay”, you agree to our Terms of service and Privacy statement. We will occasionally send you account related emails

You can also get a UNIQUE essay on this or any other topic

Thank you! We’ll contact you as soon as possible.

Home / Essay Samples / Life / Beauty / Beauty Of Mathematics: The Reasons Why Math Is Beautiful

Beauty Of Mathematics: The Reasons Why Math Is Beautiful

• Category: Life , Science
• Topic: Beauty , Mathematics in Everyday Life

Pages: 2 (1009 words)

Views: 5218

• Downloads: -->

--> ⚠️ Remember: This essay was written and uploaded by an--> click here.

Found a great essay sample but want a unique one?

are ready to help you with your essay

You won’t be charged yet!

Black Hole Essays

Volcano Essays

Bioethics Essays

Machu Picchu Essays

Hindi Essays

Related Essays

We are glad that you like it, but you cannot copy from our website. Just insert your email and this sample will be sent to you.

By clicking “Send”, you agree to our Terms of service  and  Privacy statement . We will occasionally send you account related emails.

Your essay sample has been sent.

In fact, there is a way to get an original essay! Turn to our writers and order a plagiarism-free paper.

samplius.com uses cookies to offer you the best service possible.By continuing we’ll assume you board with our cookie policy .--> -->