Geometry & Topology Books

The Fractal Geometry of Nature

Benoit B. Mandelbrot

Amazon Price: $36.00 By: W. H. Freeman Amazon Marketplace: 21 new & used starting at $31.87

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Customer Reviews:
Total reviews: 13 Average rating: 4.0 of 5

A seminal work 5 out of 5 stars.
19 of 21 people found this review helpful.

Very few books have so many quotes as this one. I am not sure if there is much left to be said, but I know this. For those professionals who still think that fractals are "spurious solutions coming from the discretization of differential equations", should take a closer look to this book. Not only won't harm, but also will show many interesting features about the nature of fractals and the "fractality" of nature, besides the fact that many of them come from *difference* equations, which are not necessarily related to the discretization of a differential equation. This book is based on serious work from many well-reputed mathematicians before Mandelbrot, e.g., Haussdorff, Lyapunov and some others. Although the book does talk about the mathematics behind fractals (wouldn't be so much a book of mathematics if it didn't, but also a philosophical one) and the necessity of coining some new mathematical terms, it also contains so much about history of mathematics, the path that leads towards fractals. As I said, the book is many times quoted, but (without trying to point a firing, accusing finger), there is a difference in quoting a book because it is famous, and another actually reading it, and having enlightenment for our own sake. Certainly I think is a "must-have-it" for most mathematicians, for many physicists, philosophers of science and engineers, but also it wouldn't be a bad guest in the library of any layman, provided the layman overcomes for some minutes the initial "classical" fear to mathematics. I would say this layman won't regret it at all. Mandelbrot does explain most of the concepts practically "ab initio", from the very scratch, including etymology and history as I previously said. One little thing against this book though: it doesn't have so many color plates as some other books on the subject, but it does have all the needed graphics to grasp the concepts.

Editorial Review:

Clouds are not spheres, mountains are not cones, and lightening does not travel in a straight line. The complexity of nature's shapes differs in kind, not merely degree, from that of the shapes of ordinary geometry, the geometry of fractal shapes.

Now that the field has expanded greatly with many active researchers, Mandelbrot presents the definitive overview of the origins of his ideas and their new applications. The Fractal Geometry of Nature is based on his highly acclaimed earlier work, but has much broader and deeper coverage and more extensive illustrations.

The (Mis)behavior of Markets

Benoit Mandelbrot, Richard L. Hudson

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Customer Reviews:
Total reviews: 54 Average rating: 3.5 of 5

Editorial Review:

Benoit B. Mandelbrot, one of the century's most influential mathematicians, is world-famous for making mathematical sense of a fact everybody knows but that geometers from Euclid on down had never assimilated: Clouds are not round, mountains are not cones, coastlines are not smooth. To these classic lines we can now add another example: Markets are not the safe bet your broker may claim. In his first book for a general audience, Mandelbrot, with co-author Richard L. Hudson, shows how the dominant way of thinking about the behavior of markets-a set of mathematical assumptions a century old and still learned by every MBA and financier in the world-simply does not work. As he did for the physical world in his classic The Fractal Geometry of Nature, Mandelbrot here uses fractal geometry to propose a new, more accurate way of describing market behavior. The complex gyrations of IBM's stock price and the dollar-euro exchange rate can now be reduced to straightforward formulae that yield a far better model of how risky they are. With his fractal tools, Mandelbrot has gotten to the bottom of how financial markets really work, and in doing so, he describes the volatile, dangerous (and strangely beautiful) properties that financial experts have never before accounted for. The result is no less than the foundation for a new science of finance.

Flatland: A Romance Of Many Dimensions

Edwin A Abbott

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Customer Reviews:
Total reviews: 159 Average rating: 4.5 of 5

Thinking W A Y Outside the Box 5 out of 5 stars.
0 of 0 people found this review helpful.

Yes, many young people have been required to read Flatland against their wills. Yes, many people have missed the real point of the book. This book stretched the mind and imagination in ways that are fun and challenging. The author might not have been entirely serious in writing the book, but nonetheless provided serious food for thought.

I believe Flatland is an excellent (and quick) reading experience for minds in the formative stage, a stage I recommend maintaining throughout life. The book's theological implications were the most important to me. I had always wondered where heaven might be, how God can see inside us, and what the spirit is made of. I do not know if extrapolating the Flatland concepts into a fourth (or fifth) physical dimension reflects ultimate reality, but it provides a sufficiently possible and plausible explanation to remove rationalist objections.

The 3-D sphere that intersects the plane of reality provided the "Aha" moment. The sphere embodied perfection and could mysteriously appear and disappear. Explaining the view from above the plane to a flat square is as difficult as explaining the spiritual realm to a person unable to envision beyond the world seen with the eye. A greater-dimensional being floating above the plane can see inside the geometric shapes, reach inside their skins without intersecting their boundaries, think far more complex thoughts, and take them out of their limited reality to a better place they could not have imagined. If a Flatland person had no thickness, he would have no volume by our reckoning, and therefore no real existence. If there is a spiritual dimension and a person has no thickness in that direction at all, then he may not really exist either.

We have learned to adjust to modern concepts of reality that are no longer Euclidean and Newtonian. Perhaps we need a view of creation that is not limited by unfounded presumptions of limited dimensionality. After you ponder the concepts of Flatland and extrapolate them to your life, I wonder what new thought may form.

Editorial Review:

This highly original and entertaining short novel (which has been in print continually since its original publication in 1884) tells the story of A. Square, an inhabitant of the two-dimensional world Flatland. Touching on themes of humanity's insatiable quest for truth, authority's tendency to squash radical ideas born from this quest, and the necessity of curiosity, "Flatland" is an odd and charming little book whose impact far surpasses its concise prose.

Geometry

Jurgensen, Ray C. Jurgensen

List Price: $67.88 By: Houghton Mifflin Company Amazon Marketplace: 19 new & used starting at $4.74

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Customer Reviews:
Total reviews: 31 Average rating: 3.5 of 5

A good introductory textbook 4 out of 5 stars.
7 of 7 people found this review helpful.

This book is not Euclid's "The Elements". The Elements, a compilation of many mathematicians' work, was written for mathematicians and their math-dedicated young disciples. Its opening proposition and complex figure reveals that the reader was expected to have a substantial degree of preexistent geometry knowledge.

To reach high school students who have only completed Algebra I, Jurgensen et al labels some theorems postulates to avoid getting bogged down, but so do the other high school geometry textbooks.

This book fulfills its mission to introduce students to a formal system of mathematically analyzing planar figures and objects. If it is carefully read and digested, every problem, including the most-challenging C and C-starred exercises, is solvable. Problems that require the student to write a half-dozen solution/proof statements, and remember (or refresh themselves by re-reading) material from prior sections, and even prior chapters, are commonly encountered (always for the C-level problems).

A small number of exercises require a dozen or so statements in which a student establishes one line of argument, then introduces another, and finally combines them, which is to say, students must be able to organize their thoughts into coherent streams, and then engage in translating visuo-spatial information into mathematical-language statements.

Is this hard? Yes. It takes not only intelligence, but serious effort to fully benefit from studying Jurgensen. Real math isn't something you breeze through. As a student encountering the material for the first time, you may frequently have to read a passage or statement several times, thinking, "I don't get this," then maybe even sleep on it, and then you'll have Eureka moments, "Oh yes! Why didn't I see this before? This is so obvious."

For anyone who wants to think about pursuing university studies in mathematics, engineering, and the sciences, acquiring diligent, persevering work habits is essential. Mathematics is not social studies. For some students the challenge is enjoyable, for others it is painful and a "waste of time". For teachers who find that their students can only consistently answer A-category problems, hit or miss on Bs, and can't even begin to tackle Cs, this book will probably not be satisfactory.

A larger question is when should Euclidean geometry be taught? Should it be taught at all?

I think it is disruptive, and counterproductive to teach algebra I, leave the subject for a full year to study geometry, then jump back to algebra. The trouble is that the "classical" curriculum of the 19th century only taught one course in algebra and then one course in geometry. But as mathematics education was extended, particularly during the Cold War, geometry held its place as the second course in the timetable, but was followed by additional algebra, pre-calculus and calculus, with insufficient thought given to the utility and efficiency of this particular sequence.

Measures such as focusing on analytical geometry (coordinate) exclusively, or nearly so, or teaching algebra II with trigonometry before Euclidean geometry, have been tried in some schools. I know of two schools that no longer even have a "Geometry" titled class. These and other alternatives seem to be working, according to people who are using them.

One thing I would point out is that most state high school graduation requirements today specify the completion of three mathematics units, and many universities require or recommend such. They DO NOT say "three mathematics courses, including geometry". So, for example, if a student is doing well in algebra, I'd say to him or her, stick with it. Go on to algebra II (preferably with trigonometry), precalculus and calculus. Learn about plane figures and solids from a modern perspective. (I can still remember generating the volume of a sphere using integration, and drawing a nicely shaded 3D sphere and discs thirty-something years ago in my second semester of calculus. I thought, "Wow, this is cool!")

So, I think schools should be flexible in their math curricula, and realize there is not a college or university in the country that will ever look at an applicant's transcript and say, "This student took AP Calculus, but there's no 'Geometry' here. Rejection."

If this book is to be used, I would recommend a couple stratagems. One is to for the student to try to prove its theorems before examining the authors' proofs. It's not hard to cover them up with a sheet of paper. This encourages students to acquire conceptual knowledge through active pursuit.

Secondly, students using Jurgensen or most other popular geometry books should subscribe to hotmath.com, where odd-numbered exercises solutions are presented in a thoughtful way, starting with a hint, then progressively revealing steps as the student feels the need to examine them for more help.

There was initially a mixed reception among educators to Hotmath, with some teachers enthusiastically signing up to provide solutions, while others were discomfited by an external knowledge resource that they felt undermined their ability to maintain traditional authority-control over what their students learned.

The disagreement was resolved. The American Mathematical Society invited Hotmath's president to give a lecture at one of its regional conferences several years ago, which was well received. All the leading textbook publishers got on board, which is to say, they recognized that the college-level promulgation of student solutions manuals starting two decades ago had proven to be heuristically sound, and the principle was applicable to college-preparatory mathematics.

For even-numbered exercise homework assignments, students can usually tackle the neighboring odd-numbered ones, check the Hotmath solutions, then apply the same principles to their homework problems.

Geometry

Ron Larson

Amazon Price: $64.66 List Price: $95.08 Usually ships in 24 hours By: Houghton Mifflin Company Amazon Marketplace: 34 new & used starting at $49.19

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Customer Reviews:
Total reviews: 7 Average rating: 3.5 of 5

Good In Some Ways; Weak In Others 3 out of 5 stars.
6 of 7 people found this review helpful.

Our school uses this book for all Geometry classes. The book is quite thorough, but serves the teacher more than the students. The students for the most part don't read it; just use it to find the assigned homework problems.

One glaring weakness is on page 306 where Postulate 7 is proven from Postulate 5 in problem 24. After hammering into my students that postulates cannot be proven, there goes the book proving a postulate!

Must have when you get text book 5 out of 5 stars.
2 of 9 people found this review helpful.

This is a must have for students that purchased the text book, gives them an opportunity to practice what they learn in the theory.

Geometry textbook 5 out of 5 stars.
1 of 1 people found this review helpful.

Hello to all Amazon shoppers. This book was shipped very rapidly and arrived in perfect condition. I was extremely pleased by the speedy delivery.

school supplies 4 out of 5 stars.
1 of 10 people found this review helpful.

Order arrived 2 days later than expected, but I was very pleased with the price I paid and the book was in excellent condition

Weak Explanations and Fails to Challenge Even the Average High School Student 2 out of 5 stars.
0 of 0 people found this review helpful.

As a long time mathematics tutor and teacher I know this book very well. I don't think the material is presented or explained in a way that is especially helpful for young people. As a tutor I have to constantly reintroduce the topic and/or try to stay ahead of the student's class. Beyond that, the students are asked to do only the simplest of proofs. Additionally, a new topic will be introduced and then no problems appear in the exercise portion of the section to help the student test and practice his or her understanding of the newly introduced topic (and of course, those problems invariably will show up on the chapter exam and the final).

Moreover, I think the book just fails the kids. It seems to omit certain standard concepts by being "accessible" and undemanding of even the most minor critical thinking skills. I believe that both of these shortcomings will leave the student unprepared for the challenging problems on standardized tests and on college entrance exams. Not to mention any sort of subsequent advanced work in high school and college. Another thing about the Larson book is that the answers to many of the problems are so arithmetically peculiar that the student has no feeling that maybe they actually got the right answer. Good problems reassure the student that they are on the right track. Also, once a new concept or definition is introduced it is never repeated.

Overall, I think that the more capable students will be shortchanged and misled into thinking that they know more than they actually do and the less capable student might pass geometry but will perform poorly on college entrance exams and be unable to successfully progress in mathematics if they need to do so.

Sir Cumference and the First Round Table: A Math Adventure

Cindy Neuschwander, Wayne Geehan

Amazon Price: $7.95 List Price: $7.95 Usually ships in 24 hours By: CHARLESBRIDGE PUBLISHING - Model: CB1570911525 Amazon Marketplace: 41 new & used starting at $3.86

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Features:

• Made with the Best Quality Material with your child in mind.
• Top Quality Children's Item.

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Customer Reviews:
Total reviews: 18 Average rating: 4.5 of 5

Great Way Of Sneaking In Math! 5 out of 5 stars.
9 of 9 people found this review helpful.

My daughter has always had a problem learning Math but reading this book (along with the others in this series) has helped her immensely! The books themselves are a bit young for her but the concepts in them (Pi, Geometry, etc) are explained in a way I think she needed.

I would recommend these to anyone who has a child with problems in math concepts.

Led to instant recall of proper geometric terms 5 out of 5 stars.
4 of 4 people found this review helpful.

My kids (9 and 6) , who are homeschooled, loved this story and it resulted in them being able to instantly recall the proper names of geometric elements and classes (e.g., radius, circumference, obtuse, acute). This is probably due to the clever visual and contextual associations provided. We bought another book in the series right afterward with the same results. Plan to get them all.

Sir Cumference and the First Round Table: A Math Adventure 5 out of 5 stars.
3 of 3 people found this review helpful.

Very cute story and a nice way of using word play to describe geometric vocabulary. Great for an introduction to the concept and as a review for older children.

Editorial Review:

Assisted by his knight, Sir Cumference, and using ideas offered by his wife and son, King Arthur finds the perfect shape for his table.

The Golden Ratio: The Story of PHI, the World's Most Astonishing Number

Mario Livio

Amazon Price: $10.17 List Price: $14.95 Usually ships in 24 hours By: Broadway Amazon Marketplace: 111 new & used starting at $2.00

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Customer Reviews:
Total reviews: 82 Average rating: 4.0 of 5

Editorial Review:

Throughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, Mario Livio tells the tale of a number at the heart of that mystery: phi, or 1.6180339887...This curious mathematical relationship, widely known as "The Golden Ratio," was discovered by Euclid more than two thousand years ago because of its crucial role in the construction of the pentagram, to which magical properties had been attributed. Since then it has shown a propensity to appear in the most astonishing variety of places, from mollusk shells, sunflower florets, and rose petals to the shape of the galaxy. Psychological studies have investigated whether the Golden Ratio is the most aesthetically pleasing proportion extant, and it has been asserted that the creators of the Pyramids and the Parthenon employed it. It is believed to feature in works of art from Leonardo da Vinci's Mona Lisa to Salvador Dali's The Sacrament of the Last Supper, and poets and composers have used it in their works. It has even been found to be connected to the behavior of the stock market!

The Golden Ratio is a captivating journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras who believed that this proportion revealed the hand of God; astronomer Johannes Kepler, who saw phi as the greatest treasure of geometry; such Renaissance thinkers as mathematician Leonardo Fibonacci of Pisa; and such masters of the modern world as Goethe, Cezanne, Bartok, and physicist Roger Penrose. Wherever his quest for the meaning of phi takes him, Mario Livio reveals the world as a place where order, beauty, and eternal mystery will always coexist.


From the Hardcover edition.

What's Your Angle, Pythagoras? A Math Adventure

Julie Ellis

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Customer Reviews:
Total reviews: 5 Average rating: 4.5 of 5

Unfortunate Anachronisms 3 out of 5 stars.
70 of 77 people found this review helpful.

This is generally a good book about the Pythagorean Theorem. I was disappointed, however, that someone did not catch the gross anachronisms before publication. In the book young Pythagorus travels to Alexandria, Egypt. However, Pythagoras was born (as the book points out) around 569 BC. This predates Alexander the Great by more than 200 years. Of course Alexandria would not have existed before Alexander the Great. Also as Pythagoras' ship approaches Alexandria, you can see the great lighthouse, one of the Seven Wonders of the Ancient World, in the background. The lighthouse wasn't built, however until around 271 BC - even after Alexander's death. I know it's just a fictional children's book, but come on. Our children deserve a little better research.

Pythagoras made easy! 5 out of 5 stars.
21 of 22 people found this review helpful.

This is probably the most enjoyable way I have come across to teach Pythagorean Theorem to my children. It's one of those special books which children will read without realising they are learning a mathematical concept. Highly recommended to teachers and parents grappling with this sometimes difficult topic!

Editorial Review:

In ancient Greece, young Pythagoras discovers a special number pattern (the Pythagorean theorem) and uses it to solve problems involving right triangles.

Sir Cumference and the Dragon of Pi: A Math Adventure

Cindy Neuschwander

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Customer Reviews:
Total reviews: 12 Average rating: 4.0 of 5

A wonderful way to remember pi 5 out of 5 stars.
11 of 11 people found this review helpful.

I feel the age range given on the website should not be ages 4-8. This is appropriate for 8 and up. That said, my 5 year old enjoys it as just a fairy tale and I would not attempt any "teaching" to him. He listens as I read it to my older daughters and maybe it will seep in so when he's old enough to begin measuring, he will easily grasp the concepts.

This is a wonderful book - good illustrations and a clever story. The author works the math part in without it being cumbersome. A few times through this book and your child will have a good grasp of pi. At the least, it will be a good memory trigger if he/she needs help in class.

Great Book 5 out of 5 stars.
1 of 3 people found this review helpful.

This is a great book to teach kids about the number pi. This is a book that makes math fun and gets kids interested in learning math.

Clever with Much Kid Appeal 5 out of 5 stars.
0 of 0 people found this review helpful.

This is great. My nine yr. old niece loved it, and she was able to understand the concept. Clever and good learning approach. Even her little brother was fascinated by the pictures, so I'm sure he will make use of it when he gets older. The artwork is really good, classic in style but with a lot of individuality. I recommend this for a range of ages because kids will grow into the book, and it is a wonderful way to learn.

Editorial Review:

When Sir Cumference drinks a potion which turns him into a dragon, his son Radius searches for the magic number known as pi which will restore him to his former shape.

Geometry the Easy Way

Lawrence S. Leff

Amazon Price: $10.17 List Price: $14.95 Usually ships in 24 hours By: Barron's Educational Series Amazon Marketplace: 95 new & used starting at $1.83

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Customer Reviews:
Total reviews: 20 Average rating: 4.0 of 5

This book is a real help 5 out of 5 stars.
7 of 7 people found this review helpful.

30 years after taking Geometry in High School, I started back to college recently to get my degree. Problem: I had forgotten almost everything I had learned 30 years ago, and not only that, but 30 years ago I hadn't really paid much attention and didn't learn the material well anyway. I discovered this book "Geometry the easy way" and it has been excellent. It is extremely well-written and the subject is made simple for almost anyone who desires to learn Geometry, including High School students. Obviously, Mr. Leff knows his subject well, and knows how to explain it the best way possible. I give it my highest recommendation, and I almost never write these reviews, so that's saying a lot from me.

BTW, I tried "Algebra the Easy Way" and was extremely disappointed. I thought that book was terrible because it tried to tell a fictional story along with the material as a way of making it simpler, but it failed miserably with this tactic, at least with me.

Editorial Review:

These books are ideal student self-help supplements. They offer valuable overviews of course work and extra help with difficult subject areas. Covers the "how" and "why" of geometry. Includes hundreds of examples and exercises with solutions. Includes more than 700 drawings, graphs,and tables.