What Shape is a Snowflake?

Book Description

The stripes of a zebra...the complexities of a spider's web...the waves of the ocean...and the shape of a snowflake. These and other natural patterns have been recognized by scientists for centuries. What do they have in common? They can all be accounted for mathematically.
In What Shape is a Snowflake? internationally acclaimed mathematician Ian Stewart shows how life on earth develops not simply from genetic processes, but also from the principles of mathematics. Starting with the simplest symmetrical patterns, each chapter looks at a different kind of patterning system and the key scientific issues that underlie it. Patterns can embrace chaos, fractals, dislocations, even statistical regularities, and are found in many things that at first seem irregular or featurless. A constant wind blowing over a flat expanse of sand, for example will develop ripples, which eventually lead to sand dunes that are often arranged in long parallel rows or other geometric forms. And the smooth surface of a growing organism will develop beautiful patterns, of spots, stripes and colors.
Beautifully illustrated, What Shape is a Snowflake? is an illuminating and engaging vision of how the apparently cold laws of mathematics find organic expression in the beauty of nature.

Customer Reviews:

An Illustrated Tour of Mathematical Patterns in Nature........2006-05-17

In "What Shape Is a Snowflake?: Magical Numbers in Nature", author Ian Stewart uses a quest to understand why snowflakes form in unique six-sided designs to take the reader on a tour of mathematical patterns in nature. "Snowflakes are a showcase for the mathematics of pattern formation," he says. "What Shape Is a Snowflake?" is an overview of the mathematics behind nature's patterns, from the microscopic to astronomical. Stewart starts by hinting at the depth and implications of his seemingly simple question about snowflakes, presenting a little history of mathematicians' efforts to understand patterns, and explaining the significance of symmetry.

Then he delves into the Why and How of patterns that manifest themselves in everything around us: big and small, living and non-living, spirals, wiggles, cycles, mirror symmetry, rotational symmetry, tiling patterns, spots and stripes, waves, lattices, and even patterns in time. When reading about patterns in living things, I could not help but doubt mathematical explanations of biological processes. But Stewart acknowledges this problem and makes the case that the principles underlying which patterns can and will occur may be governed by mathematics, though the patterns are coded in genetics.

The book's final section delves into some apparent inconsistencies in the links between mathematical laws and nature's patterns and mathematicians' continuing efforts to explain them with theories of bifurcation, symmetry-breaking, and fractal geometry. Finally, Ian Stewart answers that question about snowflakes -but not before he has posed a new question: What Shape Is the Universe? "What Shape is a Snowflake?" is a nice introduction to the mathematics of pattern formation for the layperson. It presents the ideas behind the patterns without mathematical formulae and with a great many color photographs and illustrations. It will pique the reader's interest in everything from ancient Pythagorean math to modern chaos theory by giving us a sense of what humans have learned about patterns and what continues to elude us.

Required reading for everyone learning math........2005-08-24

Rob Hardy's review is an excellent summation of this excellent book. "What Shape is a Snowflake?" is a book about the big picture, about the meanings behind and the connections between big ideas. This book is not about details of applying or calculating under these frameworks. As has been stated before, this is an excellently illustrated and formatted book. The pictures and text dance with each other, nicely balancing and building interest in each other.

As Mr. Stewart says, he sees mathematics and beauty as attached ideas and this book is an effort to show the beauty of mathematics. "Most people's mental image of mathematics is page upon page of complicated `sums' - not an especially beautiful sight. I sympathize, believe me. But that's arithmetic, not mathematics (I'm quite passionate about this). Those symbols on the page come no closer to the subject's true beauty than the staves and semiquavers of musical notation come to a Beethoven symphony." As such, this is definitely a book about mathematics and definitely not about arithmetic. There are many references in the book to original publications and theories, and there is a short section at the end for further reading so anyone who wants more detail has a place to start looking. For me this book provided a clear and concise description of the ideas at the foundation of various mathematical principles. Mr. Stewart focuses his book on patterns and their implications. He talks about the different dimensions, scale, and symmetry of patterns, he talks about bifurcation, fractals, chaos, randomness, complexity and phase transitions. He also showed how these ideas and principles thread their way through literally everything in the universe.

This book should be approachable for any child in junior high or high school. Additionally, I think it is an excellent introduction for any adult interested in understanding the world around us.

Mathematics deserves four colour.......2002-07-19

I must admit I was looking for more detail from this book than it contains. I was looking for more detail on hexagonal systems.
Instead there is less detail and less formal mathematics. I found it to be rather similar to other publications by Ian Stewart, such as the book Fearful Symetry which contains many of the same ideas.

Despite my personal desires I am glad to see that Ian has finally been granted lots and lots of expensive four colour illustrations with which to explain how interesting mathmatics really is.

I immediately found a use for it in the workshops I run for children. It is the best illustrated book Mr Stewart has yet produced.

A Universe Full of Mathematics.......2001-11-07

In _What Shape is a Snowflake? Magical Numbers in Nature_ (W. H. Freeman), Ian Stewart has managed to write a wonderfully comprehensive and colorful mathematical tour of the universe from top to bottom without putting a single equation into his book. In fact, there aren't really many numbers. He gets to show what happens when a mathematician looks at the infinite aspects of the world. He writes, "I am a mathematician. I experience these wonders through a mind that has spent a lifetime learning how to detect patterns, how to understand patterns, how to find new patterns... I stand on the shoulders (and lean on the elbows) of giants, on five thousand years of mathematical history that has been groping toward such understanding. I see what all humans see, and in a few respects perhaps I see more. I see clues to rules, laws, regularities."

The snowflake is key to his tour, and there is plenty to learn specifically from it, but since Stewart is keen to draw on patterns all over the place, the range of his book is amazing. In well connected chapters, looking closely at snowflakes takes him to the leafy patterns of frost on the window, the organization of leaves around spirals and Fibonacci numbers, the spiral of the nautilus shell, the stripes and amazing triangle patterns on other sea shells, the patterns of stripes on zebras and fish, the grooves in sand dunes and the lines of dunes themselves, the lines a sidewinder leaves in the sand, the synchrony of a millipede's legs and a horse's at different gaits, the oscillations of the legs of robots, the ups and downs of animal populations, the chaotic variations of weather and of the planets in the solar system, and the shape of the universe. It is clear that Stewart sees connections everywhere, and is only using the snowflake as an excuse to look at the foundations of physical laws, the nature of time, space, and matter, and why patterns in one field give clues to patterns in something entirely different. "I'm going on a journey in search of the snowflake's secret," he says, "and, with it, the deeper secrets of our astonishing universe. And you're coming with me." It's a beguiling invitation from a masterful guide.

Naturally a tour of this type, with all it encompasses, is not going to be long on detail, and anyway, one would have to start getting into equations for that. There is a useful list for further reading at the back of the book, for those who insist on stronger doses of such stuff. Stewart's book, however, is an exhilarating, accessible, vividly illustrated voyage through classic and current mathematical ideas. By the end of it, a reader will understand that the snowflake's shape is determined by phase transition, bifurcation, symmetry-breaking, chaos, fractals, and other complexities. Oh, and the book does eventually reveal what shape a snowflake is.

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WHAT SHAPE IS A SNOWFLAKE?
Ian. Stuart
Manufacturer: Weidenfeld & Nicolson
ProductGroup: Book
Binding: Paperback
ASIN: B000ORK4Y6

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Excellent book for touring kayakers

Could have had more detail.

"A picture is worth a thousand words..."

Great Beginners Guide!

Kayaking Made Easy, 2nd: A Manual for Beginners with Tips for the Experienced (Made Easy Series)
Dennis Stuhaug
Manufacturer: Globe Pequot
ProductGroup: Book
Binding: Paperback

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Kayaking (Paddling Basics , No 2)

Basic Kayaking: All the skills and gear you need to get started (Stackpole Basics)

ASIN: 0762701889

Book Description

The perfect step-by-step handbook for beginners, with sage advice on every aspect of flat-water kayaking, from selecting a boat to kayak sailing.

Customer Reviews:

Excellent book for touring kayakers.......2004-08-19

I mistakenly bought this book to help me get started with recreational kayaking. I found this fascinating, comprehensive, and well-written, but duhh -- I should have paid more attention to the subtitle: (Sea Kayaking How-To). I doubt I will ever get into kayaking on this level, but for people who want to know everything about heading off into the wild blue yonder in a kayak, it would be an excellent choice! I would *love* to find a Kayaking for Dummies book. From my point of view, most of the books out there are geared to outdoor gear-head types in good shape. I'm a 49-year-woman who wants to paddle the slow, spring-fed rivers of the Texas hill country and need something much more basic than this.

Could have had more detail........2004-06-07

Good introductory book. It could have had a lot more specific information on kayaks and gear.

"A picture is worth a thousand words...".......2002-05-21

A very useful book, but the author insists on taking two paragraphs to describe a technique or motion that could be conveyed in a single picture or diagram. The few pictures that are in the book are very indistinct. It's no mystery why authors do this. It's much easier and cheaper to type an extra dozen sentences than to have an artist develop an appropriate graphic. Too bad, though, as the section on paddling would have benefited from some simple illustrations. As it is, there are perhaps four or five actual diagrams and they really don't contribute much to the book.

Great Beginners Guide!.......2000-09-23

I found this book interesting, entertaining, and informative. The author is witty and his descriptions are excellent.

Rebecca "New Kayaker"

Customer Reviews:

excellent service.......2007-09-26

The service is faster I received the book in 3 days before the promise date..Very good service...

Very Useful For Beginners.......2007-05-15

This is a very useful book for beginners. The topics are contemporary, the pictures are engaging, and it's not overwhelming for new students who don't want to be scared away. I have book number 4.

Side by Side: Student Book 1, Third Edition.......2007-01-04

Excellent book for beginning English students. The material is presented logically, leads to great discussion topics and is useful for real life encounters.

I love "Side by Side".......2006-09-12

I was from China and started my life in the USA as a graduate student in Engineering. Although I took the TOEFL before I came to USA, my English is not good enough to express myself clearly and accurately. I decided to improve my English about 4 months ago and started to borrow some ESL books from local libraries.

About 2 weeks ago, I found four "Side by Side" books (Book 1 to Book 4) from a local library and borrowed them. Now I am working on the Book 3 and I have enjoyed every book. I believe that "Side by Side" Book 1 to Book 4 will make very good textbooks for ESL (or ESOL) programs.

SIde By Side Book 3.......2006-08-07

My wife and I have been using Side By Side since living in Japan as ESL teachers 1987-89. We used the first edition way back then. My wife still is active tutoring English to Japanese in our area of the USA and this is an excellent series for English learners. We would recommend it to anyone from any culture.

Book Description

City of Fortune is the highly-anticipated follow-up to The Crystal Chalice, the second volume in the Trinistyr Trilogy. Set in the popular Dragonlance world of Krynn, Dragonlance: The New Adventures follows a group of young companions as they band together for friendship, adventure and excitement.

In the lost city of Taeloc, Elidor’s hopes of renewing Vael’s health have been betrayed. Unwilling to give up on her, Elidor tracks the chalice to the Northern Ergoth city of Gwynneth, where danger awaits those with wishes and the Defiler still lurks in the shadows. Loyalties must be questioned at every turn. Who can Elidor trust?

City of Fortune is the highly-anticipated follow-up to The Crystal Chalice, the second volume in the Trinistyr Trilogy. Set in the popular Dragonlance world of Krynn, Dragonlance: The New Adventures follows a group of young companions as they band together for friendship, adventure and excitement.

The Crystal Chalice: Elidor Trilogy Volume II (Dragonlance: the New Adventure)

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The Crystal Chalice: Elidor Trilogy Volume II (Dragonlance: the New Adventure)
Ree Soesbee
Manufacturer: Mirrorstone
ProductGroup: Book
Binding: Paperback

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ASIN: 078693994X
Release Date: 2006-03-14

Book Description

As the Defiler’s curse holds Vael’s health hostage, Elidor searches for a way to rescue her without succumbing to the evil wizard’s demands.
Somewhere within the lost city of Taeloc, a crystal chalice may hold the answers to his prayers. A group of shady mercenaries offers to protect Elidor on the journey and give him the chalice, in exchange for his help in locating the city. Desperate to save his love, Elidor reluctantly agrees.
But nothing goes exactly as planned . . .

Crown of Thieves: Elidor Trilogy: Vol I (Dragonlance: The New Adventures)

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Great book!

Crown of Thieves: Elidor Trilogy: Vol I (Dragonlance: The New Adventures)
Ree Soesbee
Manufacturer: Mirrorstone
ProductGroup: Book
Binding: Mass Market Paperback

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ASIN: 0786938331
Release Date: 2005-10-27

Book Description

Elidor, the beloved elf thief, returns in this new trilogy to fight an undead king and the demons of his own past.

Crown of Thieves continues a new series of Dragonlance fantasy adventures written specifically for readers ages 10 and up. The series features a group of young companions who band together for friendship, adventure, and excitement during the golden age of the Dragonlance world.

Customer Reviews:

Great book!.......2007-01-04

My son has enjoyed all these books, I have purchased the other two series and this series is no exception! Great book for all ages!

Elidor (Odyssey Classics (Odyssey Classics))

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Elidor
A LITTLE TOO VAGUE
Elidor by Alan Garner
"Elidor" pleases
worth second (and third) looks

Elidor (Odyssey Classics (Odyssey Classics))
Alan Garner
Manufacturer: Odyssey Classics
ProductGroup: Book
Binding: Paperback

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ASIN: 0152056246

Book Description

A mechanical street map, a deserted slum, a church in ruins, and a football. Four ordinary things lead the Watson children on an extraordinary adventure to a magical land called Elidor. In pursuit of four ancient treasures, the forces of evil have crossed over into our world, and it falls to the Watson children to find the treasures, seal the bridge between worlds, and guard the strayed unicorn Findhorn . . . even though their heroism may cost them everything.

Customer Reviews:

Elidor.......2005-03-30

This is a very interesting book and my son and I enjoyed reading it together. It was a bit scary at times so I'd recommend an older reading age, perhaps 9 or 10 at the youngest.

The story itself was well put together and, like a lot of things in life, leaves us wondering what was going on in the part of the story we (and the main characters) are not privy to.

There is some sort of closure at the end when the children get a glimpse of Elidor and return the treasures but I would have liked some closure at the end regarding the children and what happens later. However if you are reading this with your children it makes for a great creative exercise in imagining what would happen next.

A LITTLE TOO VAGUE.......2004-12-05

While the four Watson children, Roland, Nicholas, David, and Helen are exploring a bombed out section of London, they are transported to the other dimensional world of Elidor. Upon arriving there, the kids are enlisted by Malebron the King to take back the Four Treasures: a spear, a sword, a stone, and a cauldron, to our world. It seems that the four cities of Elidor were constructed to guard these Treasures, but a darkness has come upon the land, and only one city still stands. The problem is that dark forces might come hunting for them, even in London.

While mildy entertaining and a trifle scary at the outset as the children have to deal with the unknown, the book gradually settles into an unsatisfying first draft of a tale. I say this because nothing is ever revealed. We never have an idea of who or what the darkness is that is threatening Elidor. We have no idea what the Four Treasures true power is. I mean, what was so important about them? We never find out who Malebron really is. I'm so sick of these storylines where the heroes are kept in the dark and some magical figure gives them a quest or task to complete but never tells them exactly how to do it or resorts to talking in riddles. I mean just say what you want these kids to do instead of babbling every time you talk to them! Oh yeah, and a unicorn shows up, having a power and significance that is never explained. There is just too much vagueness working against this novel for it to succeed. The parts in London when shadow figures start haunting the kid's house is pretty good, but that's about all there is here. Sorry book.

Elidor by Alan Garner.......2003-06-20

This Fantastic book is about 4 children who are brought into elidor. they meet malebron who gives them the four treasures of Elidor But by taking them back into their world, they stir up more trouble than they think!
Thrilling adventure!

"Elidor" pleases.......2002-11-29

"Elidor" is best described as a solid little fantasy story -- it's just not spectacular. While suspenseful and intriguing, it doesn't really have a good sense of the epic or the atmospheric. But author Alan Garner definitely gets points for his subtle use of Celtic mythology and managing to create a believable unicorn.

Four kids exploring a wrecked church accidently venture into another world, the mysterious realm of Elidor. There, young Roland encounters a mysterious wounded man who sends him into a castle, claiming that he can save all of Elidor. After Roland frees his siblings from a spell, the man gives them a stone, a spear, a sword, and a cauldron, and they are sent back to their own world to guard these items from evil forces.

But problems arise when the kids go home and hide the items: The power that they emanate is so intense that it disrupts electricity and radio signals, causing problems all over their town. They bury the objects -- but that's only a temporary measure. The strange situation grows stranger when armed warriors appear near the buried objects, and an Ouija board displays the name "Findhorn" and a picture of a unicorn...

One of the wonderful things about Garner's Alderly duology is that when strange people and things were encountered by our heroes, it made our world seem like almost a parallel universe. It made everything seem magic. That quality is somewhat lacking in "Elidor"; the opening chapters have a promising glimpse of Elidor, but unfortunately this is never exploited. The evil force is another problem; unlike in the Alderly books, the evil is never given a face or an identity, and so it seems a little difficult to really get upset about.

Despite this, the opening and final chapters show Garner's lyrical style, and all of it displays his keen sense of plot development. The descriptions of the electrical disruptions are almost surreal, and his descriptions of Elidor are outstanding. So is his usage of Celtic mythology, though to a lesser extent than his other children's books; the dialogue ranges from chirpy British-schoolchild conversation to the formal language of the Elidor inhabitants. And don't be afraid of the portrayal of the unicorn -- Garner shies away from all the cliches.

It's not really a lightweight read, but "Elidor" is a beautifully written little tale that will thrill fantasy readers. Quite nice.

worth second (and third) looks.......2002-01-07

Ok, I admit taht when I first read ELIDOR as probably an 11 year old I liked it the least of Garner's books to date. It was "ok" but the idea of a magical fantasy being set in the urban wasteland of post-war Manchester just didn't click. Maybe it is a comment on the last quarter century, maybe it is just a matter of growing up, but going back the book all these years later I found myself absolutely enthralled. Garner makes the transition from everyday life to the realm of Welsh mythology & back seamlessly and flawlessly. His characters are remarkably "real" -even those with "bit parts" like the parents of the protagonists. The language is lyrical, and there are scenes that will haunt you for years. My one complaint is that it is more a "novella" than a novel -at this re-reading I would gladly have kept going for another 200 pages, but what we get is great.

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ELIDOR
Alan Garner
Manufacturer: Collins
ProductGroup: Book
Binding: Mass Market Paperback
ASIN: B000P0ZEBK

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ELIDOR
Alan Garner
Manufacturer: Collins
ProductGroup: Book
Binding: Hardcover
ASIN: B000P0VNTW

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Elidor
Alan Garner
Manufacturer: William Collins Sons & Co., Ltd.
ProductGroup: Book
Binding: Paperback
ASIN: B000NOCWQ8

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Elidor
Alan Garner
Manufacturer: Ace Books, Inc.
ProductGroup: Book
Binding: Paperback
ASIN: B000MPCTDY

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Elidor
Alan Garner
Manufacturer: Collins
ProductGroup: Book
Binding: Unknown Binding
ASIN: B0000CMQ3G

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Elidor
Alan Garner
Manufacturer: NY Ballantine Books 1981.
ProductGroup: Book
Binding: Paperback
ASIN: B000J5HPV8

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