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.
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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
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Fun teaching resources & tips to help you teach math with confidence
Math in Nature: 5 Stunning Ways We See Math in the World
Have you ever stopped to look around and notice all the amazing shapes and patterns we see in the world around us? Mathematics forms the building blocks of the natural world and can be seen in stunning ways. Here are a few of my favorite examples of math in nature , but there are many other examples as well.
The Fibonacci Sequence:
Named for the famous mathematician, Leonardo Fibonacci, this number sequence is a simple, yet profound pattern.
Based on Fibonacci’s ‘rabbit problem,’ this sequence begins with the numbers 1 and 1, and then each subsequent number is found by adding the two previous numbers. Therefore, after 1 and 1, the next number is 2 (1+1). The next number is 3 (1+2) and then 5 (2+3) and so on.
What’s remarkable is that the numbers in the sequence are often seen in nature .
A few examples include the number of spirals in a pine cone, pineapple or seeds in a sunflower, or the number of petals on a flower.
The numbers in this sequence also form a a unique shape known as a Fibonacci spiral, which again, we see in nature in the form of shells and the shape of hurricanes.
Fractals in Nature:
Fractals are another intriguing mathematical shape that we seen in nature. A fractal is a self-similar, repeating shape, meaning the same basic shape is seen again and again in the shape itself.
In other words, if you were to zoom way in or zoom way out, the same shape is seen throughout.
Fractals make up many aspects of our world, included the leaves of ferns, tree branches, the branching of neurons in our brain, and coastlines.
Learn more about fractals and how we see and apply them in our world today at the Fractal Foundation .
Hexagons in Nature:
Another of nature’s geometric wonders is the hexagon. A regular hexagon has 6 sides of equal length, and this shape is seen again and again in the world around us.
The most common example of nature using hexagons is in a bee hive.
Bees build their hive using a tessellation of hexagons. But did you know that every snowflake is also in the shape of a hexagon?
We also see hexagons in the bubbles that make up a raft bubble. Although we usually think of bubbles as round, when many bubbles get pushed together on the surface of water, they take the shape of hexagons.
Concentric Circles in Nature:
Another common shape in nature is a set of concentric circles. Concentric means the circles all share the same center, but have different radii. This means the circles are all different sizes, one inside the other.
A common example is in the ripples of a pond when something hits the surface of the water. But we also see concentric circles in the layers of an onion and the rings of trees that form as it grows and ages.
If you live near woods, you might go looking for a fallen tree to count the rings, or look for an orb spider web, which is built with nearly perfect concentric circles.
Math in Outer Space:
Moving away from planet earth, we can also see many of these same mathematical features in outer space.
For instance, the shape of our galaxy is a Fibonacci spiral. The planets orbit the sun on paths that are concentric. We also see concentric circles in the rings of Saturn.
But we also see a unique symmetry in outer space that is unique (as far as scientists can tell) and that is the symmetry between the earth, moon and sun that makes a solar eclipse possible.
Every two years, the moon passes between the sun and the earth in such a way that it appears to completely cover the sun. But how is this possible when the moon is so much smaller than the sun?
Because of math.
You see, the moon is approximately 400 times smaller than the sun, but it is also approximately 400 times further away.
This symmetry allows for a total solar eclipse that doesn’t seem to happen on any other planet.
Isn’t nature amazing??
Want to know even more about these topics and explore them more deeply with your kids? Try my math enrichment curriculum: Math in Nature .
See what your kids will explore in this short video:
This curriculum, designed for grades 3-6 , provides hands on lessons to look at math in the real world and also practice important math skills.
It includes picture book lists for each topic, a detailed teacher manual , student handouts for the lessons, ‘fun fact’ summary pages and a list of math art projects to go along with each theme.
Buy Math in Nature HERE
I also encourage you to grab the FREE set of math in nature posters to show your kids math in the real world. Use these to decorate your math space and invite discussions and excitement about the beauty of math.
Simply enter your email below to receive these posters. You will also receive a special offer for my Math in Nature curriculum , as well as math teaching tips and other freebies and offers.
I hope this gives you some fun new math ideas to learn and explore along with your kids!
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The mystique of mathematics: 5 beautiful maths phenomena
Photo: Unsplash
Pattern and symmetry – with a touch of surprise – may be the mathematical formula for what we find beautiful.
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 favourite 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 recognise 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 learnt 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 centre 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 centre 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 3D 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. Dr Thomas Britz
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.”
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Sherry Landow News & Content Coordinator Tel: (02) 9065 4039 Email: [email protected]
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Mathematics in Nature
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Mathematics
Mathematics in Nature: Modeling Patterns in the Natural World
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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
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NOVA: The Great Math Mystery
Airs Wednesday, March 28, 2018 at 9 p.m. on KPBS TV + Sunday, April 1 at 9 p.m. on KPBS 2
Is math invented by humans, or is it the language of the universe?
Underlying civilization’s stunning advancements and the technological wonders of the modern age is something deep and mysteriously powerful: mathematics.
But where does math come from and why does it work so well to explain our physical world? Is it humankind’s clever trick, or the deeply embedded language of the cosmos?
In "The Great Math Mystery," NOVA and a colorful cast of the world’s top mathematicians, physicists, and engineers embark on a mathematical mystery tour — a provocative exploration of math’s astonishing power as it has evolved over the centuries — to ponder a profound question: is math an invention or a discovery?
Whether we think we’re good with numbers or not, we all use math in our daily lives.
NOVA sheds fascinating light on how math works in our brains and why it works so well.
Math is essential in art, architecture, and music. It is evident everywhere in nature and the patterns of our world — from the spiral in the center of a sunflower, to the swirl of a nautilus shell or the whirlpool of a galaxy.
Math reveals the secrets behind the elliptical orbits of the planets and the electromagnetic waves used in the first wireless radio transmissions.
It predicts the discovery of the Higgs boson and the successful landing of rovers on Mars.
NOVA spins through stunning examples of math’s power that permeate history, drawing on Pythagoras , Galileo , Sir Isaac Newton , Albert Einstein , and more — and taps into the world’s top modern-day minds, who continue to puzzle over whether math’s tremendous potency is the hidden language of the universe or all in our heads.
Mario Livio , a notable astrophysicist who helps operate the Hubble Space Telescope from Space Telescope Science Institute , has spent much time wrestling with such questions and sees an uncanny accuracy in the way mathematics can reveal the secrets of the universe, making it seem an inherent part of nature.
Livio shows viewers through his Hubble research how centuries old, simple mathematical equations and groundbreaking insights on falling bodies and gravity by early scientists such as Galileo and Newton still permeate the universe as laws of science that can be applied to distant galaxies.
NOVA also looks at various patterns in nature and whether plants and animals have a fundamental ability to perceive numbers.
The film looks at the intriguing series of numbers known as the Fibonacci sequence.
Statistically, these occur frequently in botany, in the spirals on the bottom of pine cones, in the two sets of spirals in the seeds on sunflower heads, in the number petals on various kinds of daisies (e.g., 3, 5, 8, 13, 21, 34, 55).
In another segment, Duke University’s Elizabeth Brannon shares the latest research on how well lemurs understand quantity and the abstract essence of numbers using fascinating computer recognition tests.
NOVA also looks at Pi (“π”) as just one example of a vast interconnected web of mathematics that seems to reveal an often hidden and deep order to our world.
Pi doesn’t only have to do with circles, but appears in probability calculations and in many phenomena with wave-like characteristics.
Pi tells us which colors should appear in a rainbow and how middle-C should sound on a Piano; Pi shows up in apples in the way cells grow into spherical shapes or in the brightness of a supernova.
The affinity between music and math is also examined in the film in a sequence featuring acclaimed jazz bassist Esperanza Spalding , who demonstrates on her upright bass the discovery by 6th century Greek scholar Pythagoras related to simple ratios in vibrating strings of varying lengths.
To him, these simple mathematical relationships produced the most beautiful, harmonious musical tones.
To look at what math does to the brain, NOVA studies the MRI images of a teen math prodigy who scored a perfect 800 on the SAT for Math at age 11 and compares the neuron activation in his parietal lobes with that of kids with average performance scores.
To illustrate the amazing predictive power of mathematics, NOVA selects examples that yielded highly tangible and relatable results, which are borne out all around us in the daily devices we use that require invisible waves of energy to communicate: televisions and TV remotes, radios, cell phones, satellites, baby monitors, Wi-Fi, garage door openers and GPS.
All of these are a result of a set of mathematical equations published in the 1860s by a Scottish mathematical physicist, which then inspired the work of Guglielmo Marconi , who later ushered in the era of the wireless telegraph and radio at the turn of the century.
NOVA also travels to CERN , home to the world’s largest and most powerful particle accelerator, located deep underground in Switzerland, to show viewers where the Higgs boson, the so-called God particle, was confirmed — another significant discovery based on mathematical predictions.
Many of the world-class mathematicians and physicists featured in the film acknowledge that math feels more like an inherent truth that they discover, rather than an invention — including British mathematician and Barnard College professor Dusa McDuff ; mathematician and Columbia University professor Maria Chudnovsky ; theoretical physicist and University of Maryland professor Jim Gates ; and Christophe Golé, mathematician and Smith College professor .
Others, like physicist and MIT professor Max Tegmark , go a step further, arguing that mathematics is our physical reality.
Max perceives a real world that doesn’t just have some mathematical properties, but only mathematical properties.
As a dynamic and clever visual to demonstrate show viewers how Max might perceive the real physical world and the essence of reality to be purely mathematical, NOVA creates a special animation version of Max, “shrinking” him down and “inserting” him into a computer video game.
Several others fall more into the Einstein camp — who thought math to be a product of human thought, even though he wondered how it did so well in explaining the universe as we see it.
The late Eugene Wigner, Nobel laureate in physics , spoke of “the unreasonable effectiveness of mathematics” in describing the physics of the universe, calling it “a wonderful gift…we neither understand nor deserve.”
It is an opinion shared by particle physicist and Stanford University professor Savas Dimopoulos who calls mathematics “a servant far more capable than [we] are” , one that if you "ask nicely...will carry you all the way to the truth.”
Adam Steltzner , the lead landing engineer of NASA’s Mars Curiosity Rover Project , demonstrates that “unreasonable effectiveness” using hydraulic boom lifts and custom-built ramps to show viewers how the mathematical expression of Galileo’s Renaissance discovery about the law of falling bodies is the same relationship used hundreds of years later by him and his team to land the “Curiosity” rover on the surface of Mars.
But there are also some experts — such as computer scientist/physicist Stephen Wolfram and physicist/engineer Derek Abbott —who point out the limitations of mathematics in explaining complex areas for which the models and prediction methods are “reasonably ineffective” — including meteorology and weather forecasting beyond a few days, human psychology, parts of biology, such as the interaction of neurons on the brain, and financial systems like the stock market.
Even Adam Steltzner admits that while engineers see math as a powerful tool, they are also keenly aware of its shortcomings.
When the elegance of math meets the messiness of reality, engineers shift into the domain of approximation rather than absolutes.
Precision may be sacrificed for practicality if it garners results, making mathematics appear a very human invention.
For many, the answer may be that there are elements of both discovery and invention.
In the end, the puzzle may never be completely resolved and the question remains a great math mystery.
This episode premiered April 2015 on PBS.
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A NOVA production for WGBH Boston . Senior Executive Producer is Paula S. Apsell .
More From Forbes
No, the universe is not purely mathematical in nature.
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The idea that the forces, particles and interactions that we see today are all manifestations of a ... [+] single, overarching theory is an attractive one, requiring extra dimensions and lots of new particles and interactions. Many such mathematical constructs exist to explore, but without a physical Universe to compare it to, we're unlikely to learn anything meaningful about our Universe.
At the frontiers of theoretical physics, many of the most popular ideas have one thing in common: they begin from a mathematical framework that seeks to explain more things than our currently prevailing theories do. Our current frameworks for General Relativity and Quantum Field Theory are great for what they do, but they don't do everything. They're fundamentally incompatible with one another, and cannot sufficiently explain dark matter, dark energy, or the reason why our Universe is filled with matter and not antimatter, among other puzzles.
It's true that mathematics enables us to quantitatively describe the Universe, it's an incredibly useful tool when applied properly. But the Universe is a physical, not mathematical entity, and there's a big difference between the two. Here's why mathematics alone will always be insufficient to reach a fundamental theory of everything.
One of the great puzzles of the 1500s was how planets moved in an apparently retrograde fashion. ... [+] This could either be explained through Ptolemy's geocentric model (L), or Copernicus' heliocentric one (R). However, getting the details right to arbitrary precision was something that would require theoretical advances in our understanding of the rules underlying the observed phenomena, which led to Kepler's laws and eventually Newton's theory of universal gravitation.
About 400 years ago, a battle was unfolding about the nature of the Universe. For millennia, astronomers had accurately described the orbits of the planets using a geocentric model, where the Earth was stationary and all the other objects orbited around it. Armed with the mathematics of geometry and precise astronomical observations — including tools like circles, equants, deferents, and epicycles — the precise mathematical description of the heavenly bodies' orbits matched what we saw spectacularly.
The match wasn't perfect, however, and attempts to improve upon it either led to more epicycles or, in the 16th century, Copernicus' heliocentrism. By placing the Sun at the center, explanations of retrograde motion became simpler, but the fits to the data were worse. When Johannes Kepler came along, he had a brilliant idea that sought to solve everything.
By having each planet orbit on a sphere which was supported by one (or two) of the five Platonic ... [+] solids, Kepler theorized that there must be exactly six planets with precisely defined orbits.
He noticed that there were six planets total, if you included Earth but not Earth's Moon. He also noticed that mathematically, there were only five Platonic solids: five mathematical objects whose faces are all equal-sided polygons. By drawing a sphere inside and outside each one, he could "nest" them in a way that fit the planetary orbits extremely well: better than anything Copernicus had done. It was a brilliant, beautiful mathematical model, and arguably the first attempt at constructing what we might call "an elegant Universe" today.
But observationally, it failed. It failed to even be as good as the ancient Ptolemaic model with its epicycles, equants and deferents. It was a brilliant idea, and the first attempt to argue — from pure mathematics alone — how the Universe ought to be. But it just didn't work.
What came next was a stroke of genius that would define Kepler's legacy.
Kepler's three laws, that planets move in ellipses with the Sun at one focus, that they sweep out ... [+] equal areas in equal times, and that the square of their periods is proportional to the cube of their semimajor axes, apply just as well to any gravitational system as they do to our own Solar System.
He took his beautiful, elegant, compelling model that disagreed with observations, and threw it away. Instead, he went and dove into the data to find what types of orbits would match how the planets actually moved, and came away with a set of scientific (not mathematical) conclusions.
- Planets didn't move in circles around the centrally located Sun, but rather in ellipses with the Sun at one focus, with a different set of parameters describing the ellipse of each planet.
- Planets didn't move at a constant speed, but rather moved at a speed that varied with the planet's distance from the Sun, in such a way that planets sweep out equal areas in equal times.
- And finally, planets exhibited orbital periods that were directly proportional to the long axis (the major axis) of each planet's ellipse, raised to a specific power (determined to be 3/2).
There are four known exoplanets orbiting the star HR 8799, all of which are more massive than the ... [+] planet Jupiter. These planets were all detected by direct imaging taken over a period of seven years, and obey the same laws of planetary motion that the planets in our Solar System do: Kepler's Laws.
This was a revolutionary moment in the history of science. Mathematics wasn't at the root of the physical laws governing nature; it was a tool that described how the physical laws of nature manifested themselves. The key advance that happened is that science needed to be based in observables and measurables, and that any theory needed to confront itself with those notions. Without it, progress would be impossible.
This idea came up again and again throughout history, as new mathematical inventions and discoveries empowered us with new tools to attempt to describe physical systems. But each time, it wasn't simply that new mathematics told us how the Universe worked. Instead, new observations told us that something beyond our currently understood physics was required, and pure mathematics alone was insufficient to get us there.
We often visualize space as a 3D grid, even though this is a frame-dependent oversimplification when ... [+] we consider the concept of spacetime. In reality, spacetime is curved by the presence of matter-and-energy, and distances are not fixed but rather can evolve as the Universe expands or contracts.
By the early 1900s, it was clear that Newtonian mechanics was in trouble. It could not explain how objects moved near the speed of light, leading to Einstein's special theory of relativity. Newton's theory of universal gravitation was in similarly hot water, as it could not explain the motion of Mercury around the Sun. Concepts like spacetime were just being formulated, but the idea of non-Euclidean geometry (where space itself could be curved, rather than flat like a 3D grid) had been floating around for decades among mathematicians.
Unfortunately, developing a mathematical framework to describe spacetime (and gravitation) required more than pure mathematics, but the application of mathematics in a particular, tweaked way that would agree with observations of the Universe. It's the reason why we all know the name "Albert Einstein," but very few people know the name "David Hilbert."
Instead of an empty, blank, three-dimensional grid, putting a mass down causes what would have been ... [+] 'straight' lines to instead become curved by a specific amount. The curvature of space due to the gravitational effects of Earth is one visualization of gravitational potential energy, which can be enormous for systems as massive and compact as our planet.
Both men had theories that linked spacetime curvature to gravity and the presence of matter and energy . Both of them had similar mathematical formalisms; today an important equation in General Relativity is known as the Einstein-Hilbert action. But Hilbert, who had come up with his own, independent theory of gravity from Einstein, pursued bigger ambitions than Einstein: his theory applied to both matter and electromagnetism as well as gravity.
And that simply didn't agree with nature. Hilbert was constructing a mathematical theory as he thought it ought to apply to nature, and could never get out successful equations that predicted the quantitative effects of gravity. Einstein did, and that's why the field equations are known as the Einstein field equations, with no mention of Hilbert. Without a confrontation with reality, we don't have physics at all.
Electrons exhibit wave properties as well as particle properties, and can be used to construct ... [+] images or probe particle sizes just as well as light can. Here, you can see the results of an experiment where electrons are fired one-at-a-time through a double-slit. Once enough electrons are fired, the interference pattern can clearly be seen.
This almost identical situation came up again just a few years later in the context of quantum physics. You couldn't simply fire an electron through a double slit and know, based on all the initial conditions, where it would wind up. A new type of mathematics — one rooted in wave mechanics and a set of probabilistic outcomes — was required. Today, we use the mathematics of vector spaces and operators, and physics students hear a term that might ring a bell: Hilbert space .
The same mathematician, David Hilbert, had discovered a set of mathematical vector spaces that was enormously promising for quantum physics. Only, once again, its predictions didn't quite make sense when confronted with physical reality. For that, some tweaks needed to be made to the math, creating what some call a rigged Hilbert space or a physical Hilbert space. The mathematical rules needed to be applied with certain specific caveats, or the results of our physical Universe would never be recoverable.
The pattern of weak isospin, T3, and weak hypercharge, Y_W, and color charge of all known elementary ... [+] particles, rotated by the weak mixing angle to show electric charge, Q, roughly along the vertical. The neutral Higgs field (gray square) breaks the electroweak symmetry and interacts with other particles to give them mass. This diagram shows the structure of particles, but is rooted in both mathematics and physics.
Today, it's grown very fashionable in theoretical physics to appeal to mathematics as a potential way forward to an even more fundamental theory of reality. A number of mathematical-based approaches have been tried over the years:
- imposing additional symmetries,
- adding extra dimensions,
- adding new fields into General Relativity,
- adding new fields into quantum theory,
- using larger groups (from mathematical group theory) to extend the Standard Model,
along with many others. These mathematical explorations are interesting and potentially relevant for physics: they may hold clues as to what secrets the Universe might have in store beyond what's presently known. But mathematics alone cannot teach us how the Universe works. We will obtain no definitive answers without confronting its predictions with the physical Universe itself.
Visualizing the multiplication of the unit octonions, of which there are 8, requires thinking in ... [+] higher-dimensional spaces (left). The multiplication table for any two unit octonions is also shown (right). Octonions are a fascinating mathematical structure, but offers non-unique solutions to a myriad of possible applications.
In some ways, it's a lesson that every physics student learns the first time they calculate the trajectory of an object thrown into the air. How far does it go? Where does it land? How long does it spend in the air? When you solve the mathematical equations — Newton's equations of motion — that govern these objects, you don't get "the answer." You get two answers; that's what the mathematics gives you.
But in reality, there's only one object. It only follows one trajectory, landing in one location at one specific time. Which answer corresponds to reality? Mathematics won't tell you. For that, you need to understand the particulars of the physics problem in question, as only that will tell you which answer has a physical meaning behind it. Mathematics will get you very far in this world, but it won't get you everything. Without a confrontation with reality, you cannot hope to understand the physical Universe.
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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.
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- The Nature of Mathematics Pages: 2 (502 words)
- Should Science and Mathematics Be Taught in English Pages: 5 (1222 words)
- My Interest For Science, Mathematics And Computer Science Pages: 3 (776 words)
- A Literature Review of Goals and Self-Efficacy Beliefs in Mathematics Pages: 10 (2980 words)
- How Poetry and Mathematics are Intertwined in the Use of Ambiguity Pages: 14 (4102 words)
- The Mozart Effect and Mathematics Pages: 2 (572 words)
- The Beauty of Mathematics Is Cold and Austere Pages: 3 (614 words)
- Unlocking the Beauty and Power of Mathematics: A Perspective Shift Pages: 3 (799 words)
- Continuity: the Concept and Application of Continuity in Mathematics (Part I) Pages: 4 (1187 words)
- Continuity: the Concept and Application of Continuity in Mathematics (Part III) Pages: 3 (818 words)
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Mathematics and the Nature of Knowledge—An Introductory Essay
- First Online: 11 March 2018
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- Jens Erik Fenstad 4
Part of the book series: Logic, Argumentation & Reasoning ((LARI,volume 15))
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This book is a collection of essays on mathematics and the nature of knowledge. We claim that the mathematical sciences, mathematics, statistics and computing, are almost everywhere. In this introductory essay we present in brief our argument why these sciences are essential for human thought and action. The main body of the text presents further examples from the natural and social sciences, and also from language and art.
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Jens Erik Fenstad
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Fenstad, J.E. (2018). Mathematics and the Nature of Knowledge—An Introductory Essay. In: Structures and Algorithms. Logic, Argumentation & Reasoning, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-72974-9_1
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