The Mathematical Mechanic: Using Physical Reasoning to Solve Problems

by Mark Levi

Everybody knows that mathematics is indispensable to physics--imagine where we'd be today if Einstein and Newton didn't have the math to back up their ideas. But how many people realize that physics can be used to produce many astonishing and strikingly elegant solutions in mathematics? Mark Levi shows how in this delightful book, treating readers to a host of entertaining problems and mind-bending puzzlers that will amuse and inspire their inner physicist. Levi turns math and physics upside down, revealing how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways lengthy mathematical calculations never can. Did you know it's possible to derive the Pythagorean theorem by spinning a fish tank filled with water? Or that soap film holds the key to determining the cheapest container for a given volume? Or that the line of best fit for a data set can be found using a mechanical contraption

Genres: Mathematics, Science, Physics, Nonfiction, Engineering

200 pages, Hardcover

First published January 1, 2009

About the author

Librarian Note: There is more than one author in the Goodreads database with this name.

This is 1^ Mark Levi

2^Mark Levi - Serbian for Marc Levy
3^Mark Levi - Photographer
4^Mark Levi - Cook book

Reviews

[Joe · Rating 2 out of 5]

My assumption going into this book was that the author would use physical reasoning to make higher mathematical concepts clearer and more accessible. It's a great idea: Gauss' Law in 2D can be imagined as a spreading puddle of oil, of course!

Instead you get a mess of contrived analogies that are literary Rube Goldberg devices at best and not applicable at worst. The author would spend pages upon pages setting up these exercises and, though some were well done (like where to park in a drive-in to maximize your movie field of view), most were too ambitious to succeed.

What really killed me though was that after each hypothetical mechanical system was laboriously hammered together to elucidate some rather simple math concept, the author would provide a more rigorous mathematical proof which was always so much more valuable than the brittle physical analogies.

All in all, it's an interesting idea and a book worth borrowing from the library.

[Fernando Pestana da Costa · Rating 5 out of 5]

A book full of insights about the interplay of mathematics and physics, or, more precisely, about how reasonings based on elementary principles of balances of forces, conservation of energy, and the like, can provide illuminating arguments (almost rigorous proofs, really) of a very large number of mathematical results of very different natures, some geometric, some algebraic, some analytic. Some of them are really impressively elegant: such as the "fish tank" proof of the Pythagorean theorem, based on the obvious principle that if the water in a tank at rest is not disturbed it shall remain undisturbed... (pp. 9-11), the proof of the theorem on the polygon of least area circumscribed around a given convex set and its generalizations (pp. 34-40), the fastest descent problem (pp. 104-6), or the dual cones theorem and the Gauss-Bonnet formula via an ingenious mechanical "device". A very enjoyable and enlightening book!

[Hridoy Ranjan · Rating 5 out of 5]

This is quite an interesting book in my collection. While in other books we solve physics problems and to do that we take help from mathematics, this book is different. It tries to prove many mathematical theorems with the help of physical laws like conservation of energy, etc. The proofs are very clever and satisfying to learn.

There are a total of 11 chapters. In each chapter, the author discusses different kinds of math problems. For example, Chapter 2 discusses the Pythagorean theorem using the concept of mass and energy conservation laws. In Chapter 3, he discusses the concept of maximum and minimum and solves the problems using physics only. Each chapter comes with a new flavor.

Out of the many different books that I have read, this is the most unique one. It not only helps you learn the concepts but also helps you see the world differently. After learning each proof, you will feel smart. I highly recommend giving it a try

[Joy Ghosh · Rating 5 out of 5]

"abstractness is the price of generality"

This is a very elegant book of mathematics. It portrays unique ways of understanding mathematical theorems from principles of physics. Sometimes higher level maths becomes very abstract to visualise but physics helps us to fully grasp the picture underlying any mathematical trickery. There are very few books which do this elegantly and "mathematical mechanic" is one of them

[Mi Lia · Rating 5 out of 5]

Most mathematical physics books use mathematics to solve physical problems.

This book goes the other way round following the steps of Archimedes. It uses physics to solve ("prove") mathematical problems (theorems). So I'd call it a physical mathematics book.

An amazingly insightful little book that anyone interested in mathematics, physics and how those two sister fields are interconnected !

[Rory · Rating 2 out of 5]

I get the idea, but where is the value in this? Yes, there are a couple of good concepts, but most are too heavily constructed to be intuitive, leaving me just wanting to spend the time learning more applicable mathematical proofs. I was bored and ended up plowing through the pages to find something of actual value. This being said, if this is your thing, I commend you.

[Jacob Hartzer]

What a delightful book for introducing new ways of looking at mathematical problems. Many time more difficult, but sometimes shockingly easier, mechanical solutions bring the untenable world of mathematics into the physical thanks to the creativity of Mark Levi.

[Robert · Rating 3 out of 5]

A super intriguing idea.

Unfortunately I didn't know enough physics to fully appreciate it.

[Nilendu Misra · Rating 3 out of 5]

Cool concept, like proving Pythagorean theorem with a fishtank, and good mind expanding model — interlace physical proof for abstract mathematical stuff. Or, as the author says “Math, Russian style”!

[Maurizio Codogno · Rating 2 out of 5]

Niente da fare. Riponevo molte speranze su questo libro, tanto che me l'ero preordinato sei mesi prima che uscisse l'edizione in brossura. Invece è stato una delusione... ma iniziamo dal principio.

L'idea di Mark Levi è semplice: invece che usare la matematica per dimostrare le proprietà fisiche, lui ha usato le proprietà fisiche per dimostrare le proposizioni matematici, a partire dal teorema di Pitagora in poi. Come scrivevo, l'idea non è male, ma purtroppo io devo avere un blocco mentale per quanto riguarda la fisica, e quindi leggevo quelle pagine e non capivo nulla (a parte che se devo usare tutti quei congegni senza attrito, quelle molle di lunghezza a riposo zero e via discorrendo, tanto vale che mi metta a parlare di circonferenze senza spessore, no?). A essere del tutto onesti, ci sono due capitoli che almeno per me hanno avuto un certo valore: quello sui problemi di massimo e minimo, con l'idea di costruire una serie di computer analogica per risolvere i vari problemi, e quello sull'elettricità, con la derivazione delle leggi fondamentali a partire da quelle dei fluidi incompressibili. Anche l'appendice finale mi potrebbe essere utile se solo dovessi fare un po' di fisica, ma per fortuna non è il caso...

Insomma, a me non è piaciuto. Magari a voi però sì.

[Yi · Rating 2 out of 5]

The concept is great - to solve math problems or proof theorems using physical reasoning. And some examples using mechanics are really good. Science centres use some of the examples too.
However, some so-called physical examples, are made to fit the theorem. The author might have covered this point in the introduction, but I just don't like doing things just for doing them. One such example is on Page 111, computing integral of sin(t)dt from 0 to x with a pendulum. Any one knowing what these symbols mean should be able calculate this in less than 30 seconds, if they are fluent; otherwise, I don't think they would bother.
Some examples are never intuitive to me or rather, they might sound intuitive, but really needs rigorous mathematical proofs before I would admit it. One such example is on Page 78, and the author does address my comments here.

This book was recommended or referenced by one of youtube channels or the blogs I follow.

[Denise · Rating 3 out of 5]

What I took the time to understand, I really liked. My husband uses math for work on a daily basis, in fact, he was a math major in college. He was talking to a fellow math enthusiast and this book was mentioned. It sounded really interesting so I purchased a copy. Now, during my college years I did manage to make it through Geometry, Trig, and Calculus so even if I had to dust away a lot of cobwebs, I understood basically what the author was presenting. I like the physical reasoning because I'm a highly visual person and thinking about how fish tanks on poles can link to Pythagorean Theorem is pretty cool. Unfortunately, during the last decade, I've lost so much that trying to get through each section became a hefty undertaking. I've put the book back on my to-read shelf and hope to get through the rest of it at a later date.

[Shifting Phases · Rating 3 out of 5]

A delightful tour through various fields of math. Levi develops intuitive explanations for mathematical concepts by performing thought experiements. The theme that draws the book together is that the experiments rely on the reader's familiarity with real-world objects such as springs and fish tanks. The book does not focus on generalizable problem-solving techniques, so it wasn't especially helpful to me for use in class. But I found it fun that you can use a fishtank to demonstrate the Pythagorean theorem.

[Steve Cox · Rating 3 out of 5]

Good concept, but repetitive.

[Jeffrey · Rating 5 out of 5]

Awesome giggly fun

[Joshua Niehus · Rating 4 out of 5]

Fun and enlightening!

Wish i had this book back in my undergrad physics days...

[Bob · Rating 5 out of 5]

Mathematical proofs turned into physical models. Makes you really think about what the proof really means. Quite interesting! And difficult.

[Mario E. · Rating 1 out of 5]

It was really bad. The concepts were not for the novice to physics or math. I was expecting it to be as clear as Stephen Hawkings books on astrophysics. But....