The Joy of Fourier repost Author: arundhati Date: Robert J.

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DMCA Information. Ready to exercise those brain cells? Humans have been having fun with mathematics for thousands of years. Along the way, they've discovered the amazing utility of this field—in science, engineering, finance, games of chance, and many other aspects of life. This course of 24 half-hour lectures celebrates the sheer joy of mathematics, taught by a mathematician who is literally a magician with numbers.

Professor Arthur T. Benjamin of Harvey Mudd College is renowned for his feats of mental calculation performed before audiences at schools, theaters, museums, conferences, and other venues. The world of mathematics contains some of the greatest ideas of humankind—ideas comparable to the works of Shakespeare, Plato, and Michelangelo. These mathematical ideas can add texture, beauty, and wonder to your life. Most importantly, you don't have to be a mathematician to have access to this world.

## More Joy of Mathematics: Exploring Mathematics All Around You

Professor David Spiegelhalter tries to pin down what chance is and how it works in the real world. A blend of wit and wisdom, animation, graphics and gleeful nerdery is applied to the joys of chance and the mysteries of probability, the vital branch of mathematics that gives us a handle on what might happen in the future.

Each date was considered to carry a burden by a god being auspicious or not and was passed along cyclically. The alternating cycles of days, months and years were linked to endlessly recurring cycles. It was composed of twenty day cycles, each named after a god. Associated with each god's glyph was a number from 1 to As a result each day of the day calendar had a unique label composed of a god's glyph and a number which took days before a repetition would occur.

Consequently there was a 1-to-1 correspondence between the the gods' glyphs and the sequential repetition of the numbers 1 through Somewhat analogous to astrology, the Maya associated each of these days with characteristics that were deemed favorable or unfavorable days depending on the symbols. They believed the nature of the day you were born on characterized you and your entire life. The solar calendar called haab was composed of 18 months of 20 days each. The five remaining days of the day solar calendar were named Uayeb, and were considered lifeless and useless days.

Individuals did very little during this five day period and their actions, during this time, were restricted by their religious beliefs. The solar and ceremonial calendars were used concurrently consequently a day would take both dates into account. Considering both systems as composing single manner of dating, the entire system would repeat every 52 years. The combination of calendars was called the Calendar Round. This meant for a span of 52 years each day had a unique label.

Thus, each day for thousands of years had a unique name or label described using the names from the three calendars. Venus' synodical revolution varies between days over a span of five revolutions. This averages to They calculated corrections for this discrepancy to be 24 days for revolutions which made the error only one day in years! Naturally tied into all these calendars and names of dates is an intriguing Maya lore and religion, which I urge the reader to explore. According to the Maya calendars, our present cycle began August 12, B.

The Maya numeration system was very likely influenced by their calendars. But their place value did not follow a strict base 20, i. Here we note the use of the 18 and the 20 which is present in their solar and ceremonial calendars. They used a symbol for zero which was used as both a place holder and quantity. No one knows. But the objects of the universe display chirality -handedness. For example, a sphere or a rectangle is achiral. Chiral objects are those objects which cannot be superimposed on their mirror images.

A screw, a tree and a hand are examples of things that are chiral, i. Molecules have been discovered that form from the right and from the left. We find right and left spirals in plants, shells, bacteria, The right handed spiralof the umbilical chord, antlers, bone formawistena. The left handed spiral of the honeysuckle. Some areas have a preponderance of one chiral over the other.

For example, there are many more right handed people than left, and the same is true of right handed helices of seashells, plants and bacteria. Yet, amino acids are mainly left handed, and helical neutrinos are only left handed. There are some elements that form both ways depending on conditions that exist or mutations that take place. Scientists are studying chirality on micro and macro levels. They are even analyzing the chirality of the four forces that govern the universe gravity, electromagnetic force, strong nuclear force, and weak nuclear force.

Will they discover that nature creates a balance of mirror symmetric forms?

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Will chirality unlock more secrets of the universe? Wanting control over our lives, we seek elaborate and costly equipment and methods for prediction. But has our ability for long range predictions of various phenomena been reliable?

Do natural phenomena follow a predictable pattern? Do they repeat outcomes in a cyclical fashion? Various sciences for years have relied on the existence of patterns to formulate theories and make general laws. But the theory of chaos is shaking up the scientific world.

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Physicists and other scientists of the traditional schools of thought are beginning to look at the theory of chaos more seriously. It may be very distressing to admit that from the very simple events of our world to complicated and major events of the universe the formulas and laws that have evolved over the centuries may not always describe what will take place.

Scientists of all disciplines are having to sharpen their mathematical skills and awareness to understand and recognize when chaos may present itself in their particular field.

### More Joy of Mathematics: Exploring Mathematics All Around You

What he essentially discovered was that simple minute differences in the initial data could result in enormous differences in the outcomes, i. In weather forecasting this has come to be called The Butterfly Effect, - using the analogy that the fluttering of the wings small changes of a butterfly in one part of the world could start small air turbulences that could multiply and result in a full scale hurricane in another part of the world.

Technically it has come to be described as sensitive dependence on initial conditions. The occurrence of small variations in the general picture of weather predictions are continually taking place and their long range global effect is not predictable since it was not feasible to register all possible changes, regardless of how simple or small. These can be considered small errors in information input which can multiply over and over, and thus produce a chaotic event.

An artist's renditionof the Lorenz attractor. Lorenz created the first picture of the science of chaos when he graphed the results of his experiment in three dimensions. The result was a 3-D spiral-like curve that never intersected or repeated itself. It came to be known as the Lorenz attractor. It can be graphed in multi-dimensional space. The strangeattractor changes constantly, endlessly looping but never exactly repeating itself.

Lorenz's work was published in in The Journal of Atmospheric Sciences. Unfortunately, at that time, scientists in other fields either did not have access to it or chose to ignore it.

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It was not until the 's that other mathematicians and scientists began discovering similar results, especially when computers were used for modeling information. The studies were from a wide spectrum of fields that seemed to be totally unrelated, yet the results were astonishingly similar-the strange attractorsurfaced over and over. Although the variations follow the pattern of a strange attractor, the nature of the strange attractor makes it impossible to predict future outcomes 2. The one important aspect of this double spiral-type curve was that it never intersected itself.

Its path was formed from infinite variations generating this orderly aesthetic curve, which does not repeat or cross. In pursuit of the chaos theory an entirely new type of scientific experimentation has evolved where mathematics is the major means of exploration in a laboratory harbored inside a computer. The chaos theory will require scientists in all fields to, develop sophisticated mathematical skills, so that they will be able to better recognize the meanings of results.

## [PDF.16ir] More Joy of Mathematics: Exploring Mathematics All Around You

Mathematics has expanded the field of fractals to help describe and explain the shapeless, asymmetrical,and randomness of the natural environment. Add to this the recent findings in the chaos theory, and we find mathematicians are on the threshold of the discovery of order out of chaos. In addition, the chaos theory was enhanced by Mandelbrot s work with fractals - usually computer generated pictures of a system with infinite variations which may be either geometric or random and whose detail is not lost when magnified. These geometric objects endlessly repeat themselves on an ever shrinking scale, each reproduction is a miniature version of the original e.

Bear in mind that millions of years in the life of the universe is a small amount of time. They first appeared in the 's as toys and magic devices. The creation of the hexa-hexa flexagon in by Arthur H. Stone actually rekindled mathematical interest in these intriguing figures.