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Steven Strogatz is the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University. Also the author of Sync and The Joy of x, he lives in Ithaca, New York.
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Obras por Steven Strogatz

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I liked this book. It went over a lot of semi familiar ground but with a fascinating level of detail that I'd not seen before. For example, Strogatz describes Archimedes letter about his "method" to his friend Eratosthenes and it could well be a letter written today by mathematicians sharing techniques. I loved it.
Here are some of the more interesting bits for me (Though impossible to summarise and the lack of ability to reproduce geometric figures an a review is a limitation).
Archimedes was trying to calculate the value of pi by taking ever more and more sided regular polygons inscribed in a circle. Thus the hexagon argument demonstrates n > 3......Of course, six is a ridiculously small number of steps, and the resulting hexagon is obviously a very crude caricature of a circle, but Archimedes was just getting started. Once he figured out what the hexagon was telling him, he shortened the steps and took twice as many of them. He did that by detouring to the midpoint of each arc, taking two baby steps instead of striding across the arc in one big step.... Then he kept doing that, over and over again. A man obsessed, he went from six steps to twelve, then twenty-four, forty-eight, and, ultimately, ninety-six steps, working out their ever-shrinking lengths to migraine-inducing precision.
.... arithmetically. By using a 96-gon inside the circle and a 96-gon outside the circle, he ultimately proved that pi is greater than 3 + 10/71 and less than 3 + 10/70....which reduced to the familiar 22/7 which is accurate for pi to 2 decimal places.
... The squeeze technique that Archimedes used (building on earlier work by the Greek mathematician Eudoxus) is now known as the method of exhaustion because of the way it traps the unknown number pi between two known numbers. The bounds tighten with each doubling, thus exhausting the wiggle room for pi........ How to measure the length of a curved line or the area of a curved surface or the volume of a curved solid — these were the cutting-edge questions that consumed Archimedes and led him to take the first steps toward what we now call integral calculus. Pi was its first triumph.

Why did the greeks use geometry rather than numbers? We no longer make this distinction between magnitude and number, but it was important in ancient Greek mathematics. It seems to have arisen from the tension between the discrete (as represented by whole numbers) and the continuous (as represented by shapes)......Sometime between the sixth and the fourth centuries BCE, somebody proved that the diagonal of a square was incommensurable with its side, meaning that the ratio of those two lengths could not be expressed as the ratio of two whole numbers......In modern language, someone had discovered the existence of irrational numbers. The suspicion is that this discovery shocked and disappointed the Greeks, since it belied the Pythagorean credo. If whole numbers and their ratios couldn't even measure something as basic as the diagonal of a perfect square, then all was not number. This deflating let-down may explain why later Greek mathematicians always elevated geometry over arithmetic......We've now calculated pi to twenty two trillion figures. Yet twenty-two trillion is nothing compared to the infinitude of digits that define the actual pi. Think of how philosophically disturbing this is. I said that the digits of pi are out there, but where are they exactly? They don't exist in the material world. They exist in some Platonic realm, along with abstract concepts like truth and justice.

Archimedes confesses that the Method led him to that proof [of the area in a parabola cut off by a straight line] and to the number 4/3 in the first place....What is the Method, and what is so personal, brilliant, and transgressive about it? The Method is mechanical; Archimedes finds the area of the parabolic segment by weighing it in his mind. He thinks of the curved parabolic region as a material object. And by a process of balancing an infinite number of "lines" in the section of the parabola against an infinite number of similar lines in a constructed triangle outside the parabola, Archimedes was able to show that the area of the parabola segment was 1/3 of the large triangle outside the parabola and the parabolic segment of interest must have 4/3 the area of the triangle constructed inside the parabola. Similar techniques are used today in Animation with computers........The more triangles we use and the smaller we make them, the better the approximation becomes.......When Avatar was released nearly a decade later, in 2009, the level of polygonal detail became more extravagant. At director James Cameron's insistence, animators used about a million polygons to render each plant on the imaginary world of Pandora.

Before the seventeenth century, motion and change were poorly understood. Not only were they difficult to study; they were considered, downright distasteful. Plato had taught that the object of geometry, was to gain "knowledge of what eternally exists, and not what comes for a moment into existence, and then perishes. According to Aristotelian teaching.......Earth sat motionless at the centre of God’s creation while the sun, moon, stars, and planets revolved around it in perfect circles....Although an Earth-centered cosmology seemed reassuring and commonsensical, the motion of the planets presented an awkward problem. The word planet means "wanderer." Mars is an example....and actually moves backwards when observed from the earth....but today we understand that retrograde motion is an illusion. It's caused by our vantage point on Earth as we pass the slower-moving Mars.

........Researchers in several different fields came up with a generalization of sine waves called wavelets. These little wavelets are more localized than sine waves. Instead of extending periodically out to infinity in both directions, they are sharply concentrated in time or space...... Wavelets suddenly turn on, oscillate for a while, and then turn off. As a practical example of their use, the FBI had to find a way to compress the [fingerprint] files without distorting them. Wavelets were ideal for the job. By representing fingerprints as combinations of many wavelets and by turning the knobs on them optimally using calculus, mathematicians from the Los Alamos National Lab teamed up with the FBI to shrink their files by a factor of more than twenty. It was a revolution for forensics.......I wonder what Fermat would have thought of this use of his ideas. He himself was never especially interested in applying mathematics to the real world. He was content to do math for its own sake.

Fermat had applied his embryonic version of differential calculus to physics. [to refreaction] No one had ever done that before. And in so doing, he showed that light travels in the most efficient way — not the most direct way, but the fastest. Of all the possible paths light can take, it somehow knows, or behaves as if it knows, how to get from here to there as quickly as possible. This was an important early clue that calculus was somehow built into the operating system of the universe....

In contrast to a mild power function like x or x squared, an exponential function like 2 to power x or 10 to power x describes a much more explosive kind of growth, a growth that snowballs and feeds on itself. Instead of adding a constant increment at each step as in linear growth, exponential growth involves multiplying by a constant factor.......For example, a bacterial population growing on a petri dish doubles every 20 minutes. If there are 1000 bacterial cells initially, after 20 minutes, there will be 2000 cells. After another 20 minutes, 4000 cells, and 20 minutes after that, 8000 cells, then 16,000, 32,000, and so on...... The lesson here is that we need to use powers of ten with care. They are dangerously strong compressors, capable of shrinking enormous numbers down to sizes we can fathom more easily......One special function stands out in importance...e. The important point about e is that an exponential function with this base grows at a rate precisely equal to the function itself. Let me say that again. The rate of growth of e to power x is e to power x itself. This marvellous property simplifies all calculations about exponential functions when they are expressed in base e.

Issac Newton updated it [the infinity principle of obtaining areas] by using symbols, not shapes, as his building blocks. Instead of the usual shards or strips or polygons, he used powers of a symbol x, like x and x to power 2 . Today we call his strategy the method of power series. Newton viewed power series as a natural generalization of infinite decimals. An infinite decimal, after all, is nothing but an infinite series of powers of 10 and 1/10.....By analogy, Newton suspected he could concoct any curve or function out of infinitely many powers of x.....The trick was to figure out how much of each power to mix in...... Rather than restricting his attention to a standard shape, like a whole circle or a quarter circle, he looked at the area of an oddly shaped "circular segment" of width x, where x could be any number from 0 to 1 and where 1 was the radius of the circle..... The advantage of using the variable x was that it let Newton adjust the shape of the region continuously, as if by turning a knob. A small value of x near O would produce a thin, upright segment of the circle, like a thin strip standing on its edge. Increasing would fatten the segment into a blocky region. Going all the way up to an x value of 1 would give him a quarter circle. ..... Through a freewheeling process of experimentation, pattern recognition, and inspired guesswork (a style of thinking he learned from Wallis’s book), Newton discovered that the area of the circular segment could be expressed by a power series:........ Isaac Newton had found a path to the holy grail. By converting curves to power series, he could find their areas systematically. The backward problem was a cinch for power functions, given the pairs of functions he had tabulated. So any curve that he could express as a series of power functions was every bit as easy to solve. This was his algorithm....... though he was not the first, more than two centuries earlier, mathematicians in Kerala, India, had discovered power series for the sine, cosine and arctangent functions........in his work on power series he behaved like a mathematical mash-up artist. He approached area problems in geometry via the Infinity Principle of the ancient Greeks and infused it with Indian decimals, Islamic algebra, and French analytic geometry.

The larger significance of Fourier's work [with sine waves] is that he took the first step toward using calculus as a soothsayer to predict how a continuum of particles could move and change. This was an enormous advance beyond Newton's work on the motion of discrete sets of particles. In the centuries to come, scientists would extend Fourier’s methods to forecast the behaviour of other continuous media, like the flutter of a Boeing 787 wing, the appearance of a patient after facial surgery, the flow of blood through an artery, or the rumbling of the ground after an earthquake........So why are sine waves so well suited to the solution of the wave equation and the heat equation and other partial differential equations? Their virtue is that they play very nicely with derivatives. Specifically, the derivative of a sine wave is another sine wave, shifted by a quarter cycle. That's a remarkable property. It's not true of other kinds of waves...... Being differentiated is a traumatic experience for most curves. But not for a sine wave. After its derivative is taken, it dusts itself off and appears unfazed, as sinusoidal as ever. The only injury it suffers —and it isn't even an injury, really — is that the sine wave shifts in time. It peaks a quarter of a cycle earlier than it used to...... If day-length data were a perfect sine wave an d we looked at its difference not from one day to the next but from one instant to the next, then its instantaneous rate of change (the derivative" wave derived from it) would itself be a perfect sine wave, shifted exactly a quarter of a cycle earlier.

Chladni patterns allow us to visualize standing waves in two dimensions. In our daily lives, we rely on the three-dimensional counterpart of Chladni patterns whenever we use a microwave oven. inside of a microwave oven is a three-dimensional space.
When youpress the start button, the oven fills with a standing-wave pattern of microwaves. Though you can't see these electromagnetic vibrations with your eyes, you can visualize them indirectly by mimicking what Chladni did with his sand.........For a standard microwave oven that runs at 2.45 GHz (meaning the waves vibrate back and forth 2.45 billion times a second), you should find that the distance between neighbouring melted spots [experiment done with cheese] is about six centimeters. Keep in mind, that's only the distance from a peak to a trough and hence is half a wave-length. To get the full wavelength, we double that distance. Thus the wavelength of the standing-wave pattern in the oven is about twelve centimeters....... Incidentally, you can use your oven to calculate the speed of light......Multiply the frequency of vibration (listed on the oven's door frame) by the wavelength you measured in your experiment, and you should get the speed of light or something pretty close to it.

The existence of the predictability horizon emerged from Poincare's work. Before him, it was thought that errors would grow only linearly in time, not exponentially; if you doubled the time, there’d be double the error. With a linear growth of errors, improving the measurements could always keep pace with the desire for longer prediction. But when errors grow exponentially fast, a system is said to have sensitive dependence on its initial conditions. Then long-term prediction becomes impossible. This is the philosophically disturbing message of chaos.....On the positive side, vestiges of order exist within chaotic systems because of their deterministic character. Poincaré developed new methods for analyzing nonlinear systems, including chaotic ones, and found ways to extract some of the order hidden within them. Instead of formulas and algebra, he used pictures and geometry. His qualitative approach helped sow the seeds for the modern mathematical fields of topology and dynamical systems..... Such a map is called a graph of a vector field. The little arrows on it are vectors showing that if the angle and velocity of the pendulum are currently here, this is where they should go a moment later.

Impossible to summarise all the information in this book. But there is certainly a lot there and I've even learned a lot more as I re-read parts for this review. An easy five stars from me.
… (mais)
booktsunami | 13 outras críticas | Jan 11, 2024 |
Really enjoyed this review of the history of calculus. Very approachable and gave a really interesting perspective on the history of "calculus" right back to the geometry of the Greeks.
thisisstephenbetts | 13 outras críticas | Nov 25, 2023 |
Mathematical education is in dire straits - this is a fact. But this book is not something I'd recommend for people who have always struggled to get into Math or for people who were good at Math but didn't necessarily like it.
The very issue with trying to please everyone is you end up pleasing no one. This book overextends itself in trying to court the seasoned amateur and the professional, not to mention trying to help people who loathe the field. Spoon-feeding the latter the absolute basics (what is multiplication, what is a circle, etc.) would hardly work as the basics are also the most tedious part of Math.
Paul Lockhart puts it aptly when he suggests that Math education needs to be fixed down-to-top - giving students time to come up with proofs by themselves and not simply giving them the absolute basics through flowery details. This is my opinion, of course, and if more people are interested in Maths because of this, then it's all the better - this is not how I would recommend starting your journey.
This is a breeze, so you could quickly finish this throughout a single afternoon. I'd recommend this if you have free time to whittle away, doubly if you're even slightly interested in the field.
… (mais)
SidKhanooja | 32 outras críticas | Sep 1, 2023 |
I thoroughly enjoyed this book, take it from an avowed and devoted nerd.

Please read in my blog page.

pw0327 | 13 outras críticas | Jul 22, 2023 |



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