Mathematics In The Modern World (vignette).docx

Mathematics in the Modern World “Vignette from NATURE’S NUMBERS of Ian Stewart”

By: Alegarbes , Jaya C. BS Industrial Engineering 1

Submitted to: Mr. Maulana Salatun

Stewart's goal in Nature's Numbers is to equip the reader with a mathematician's eye for a sightseeing trip through the mathematical universe. The interrelations of nature's patterns, structures, and processes form the underlying theme of the book. Nobody doubts that nature provides numerous examples of beautiful shapes and symmetries. Stewart argues, full circle, that patterns of form and motion reveal deep regularities in the world around us: sixfold symmetry of snowflakes led Kepler to conjecture that all matter is composed of atoms; patterns of waves and dunes provide clues to the laws of fluid flow; and tiger stripes and hyena spots provide a key to understanding the processes of biological growth. These are just a few of the many fascinating examples Stewart provides. Also explored are nature's most recently discovered patterns, patterns in apparent randomness, and the resulting theories of chaos and fractals. Ian Stewart has put together a very interesting novel for those are particularly intrigued by physics and also short delve into mathematics. Professor Ian Stewart explains new and unsuspected structures in the world around us. "We live in a universe of patterns where every night the stars move in circles across the sky." This was taken from the first two lines of the book and what an opening few lines they were. After reading those lines it was difficult not to be curious to wonder what Mr. Stewart has to say for himself. The first chapter states the uncontroversial idea that nature is full of patterns. It then leads on to mathematics, linking the two aspects very nicely indeed. He shows a very insightful idea in the second chapter. There is a diagram which shows a computer model of the evolution of the human eye. There are in total 1829 steps, where each step in the computation corresponds to approximately 200 years of biological evolution. It is something I had previously never heard of. About halfway through the book there is a chapter on broken symmetry. This was very clear and well written and anyone could understand this section. He talks about mirror images and tries to justify it with simple evidence. Towards the end of the book there is a section which I found rather interesting. It was the formation of a detached drop. It starts as a bulging droplet hanging from a surface then producing a narrow neck and then eventually developed into a spherical drop. So much about for the whole book, now further detailed discussions will be given on the next pages about my selected top three topics out of nine chapters Ian Stewart had given in his book.

I.

Chapter 8: DOES GOD PLAY DICE?

The chapter of “Does God play dice?” is an overall summary of what we call the Mathematics of Chaos. Now what is chaos? Chaos theory usually involves the study of a range of phenomena exhibiting a sensitive dependence on initial conditions. From chaotic toys with randomly blinking lights to wisps

and eddies of cigarette smoke, chaotic behavior is generally irregular and disorderly; other examples include weather patterns, certain neurological and cardiac activity, the stock market, and certain electrical networks of computers. Although chaos often seems totally ‘random’ and unpredictable, it actually obeys strict mathematical rules deriving from equations that can be formulated and studied. Today, there are several scientific fields devoted to the study of how complicated behavior can arise in systems from simple rules and how minute changes in the input of nonlinear systems can lead to large differences in the output; such fields include chaos and cellular automated theory.

II.

Chapter 7: THE RHYTHM OF LIFE

To sum up, I think this chapter more like have presented the relation of Mathematics through biological motion of legged organisms especially animals. Most of this chapter is about gait analysis, a branch of mathematical biology that grew up around the questions "How do animals move?" and "Why do they move like that?" To introduce a little more variety, the rest is about rhythmic patterns that occur in entire animal populations, one dramatic example being the synchronized flashing of some species of fireflies, which is seen in some regions of the Far East, including Thailand. Although biological interactions that take place in individual animals are very different from those that take place in populations of animals, there is an underlying mathematical unity, and one of the messages of this chapter is that the same general mathematical concepts can apply on many different levels and in many different things. Nature respects this unity, and makes good use of it.

III. Chapter 4: THE CONSTANTS OF CHANGE When delving into the chapter deeply, somehow it discusses a part of our history where famous mathematicians and scientific discoverers and their contributions were introduced and argued upon altogether (mostly Isaac Newton’s contributions were demonstrated and were made as examples). It was also highlighted that nature, the creation of higher beings, is by definition perfect, and ideal forms are mathematical perfection, so of course the two go together. And perfection was thought to be unblemished by change. And the universe may appear to be a storm-tossed ocean of change, but Newton and before him Galileo and Kepler, the giants upon whose shoulders he stoodrealized that change obeys rules. Not only can law and flux coexist, but law generates flux. Today's emerging sciences of chaos and complexity supply the missing converse: flux generates law. Further explanations proceeds to the creation and uses of Calculus by giving diagrams that includes acceleration, the sequence of velocities, and the likes. How can we explain this constant that is hiding among the dynamic variables? When all else is flux, why is the acceleration fixed? One attractive explanation has two elements. The first is that the Earth must be pulling the ball downward; that is, there is gravitational force that acts on the ball. It is

reasonable to expect this force to remain the same at different heights above the ground. Indeed, we feel weight because gravity pulls our bodies downward, and we still weigh the same if we stand at the top of a tall building. Of course, this appeal to everyday observation does not tell us what happens if the distance becomes sufficiently large-say the distance that separates the Moon from the Earth. That's a different story, to which we shall return shortly. The second element of the explanation is the real breakthrough. We have a body moving under a constant downward force, and we observe that it undergoes a constant downward acceleration. Suppose, for the sake of argument, that the pull of gravity was a lot stronger: then we would expect the downward acceleration to be a lot stronger, too. Without going to a heavy planet, such as Jupiter, we can't test this idea, but it looks reasonable; and it's equally reasonable to suppose that on Jupiter the downward acceleration would again be constant-but a different constant from what it is here. The simplest theory consistent with this mixture of real experiments and thought experiments is that when a force acts on a body, the body experiences an acceleration that is proportional to that force. And this is the essence of Newton's law of motion. The only missing ingredients are the assumption that this is always true, for all bodies and for all forces, whether or not the forces remain constant; and the identification of the constant of proportionality as being related to the mass of the body. As further mathematical equations and issues were illustrated, I think that this chapter has called upon a thing about God as also a reason for mathematics. It has touched a part of the omnipresent being that although it was already frequently known and said that nature is well-surrounded by mathematics it does not suggest that nature is mathematics – that (as the physicist Paul Dirac put it) "God is a mathematician." Maybe nature's patterns and regularities have other origins; but, at the very least, mathematics is an extremely effective way for human beings to come to grips with those patterns.