It wasn’t…

Worse than that, in the third grade we had to learn about “conversion of units.” I figured it was good source of test problems. So I couldn’t escape from learning it. I admit it was difficult in the beginning. “The strange rule” said if we are to measure with a bigger scale, then we had to divide the number by ten and vice versa. Why was 120 cm equal to 1.2 m? If we knew that it was 120 already why did we bother to say it was also 1.2 in another unit? I got confused whether I should multiply the number by ten or divide by ten? (At least it was easy to multiply or divide by ten instead of another number, so I kept silent.)

In time, I realized that people used the unit to measure almost anything. Length is measured in meters, mass is in kilograms, time is in seconds. Wherever there was quantity, there was also a base-unit associated with it. Of course I never asked what a “second” was, because our teacher said everyone accepted this unit of time. Since it was a world-accepted “standard measure,” I subconsciously got the impression that the universe had a clock* and that people calibrated their time accordingly. In the same way, one kilogram was an absolute quantity in my mind by which every other mass can be measured.

As we grew up, more and more types of measures and units entered our lives: Volt, Joule, Ampere, Newton, and many others. Dealing with the “old” units of length and time was a piece of cake then. However, my faith in “standard measures” as universal remained unchanged until high school.

I was quite surprised in high school when our chemistry teacher told us that our very fundamental units of measures were actually not absolute. They were not fundamental in the sense that they, too, were defined in terms of other quantities. In fact, there is a history of “what to define as a unit” and “how to measure it.”

Let’s take time, for instance. We measure years by days and days by hours. Have we ever thought about why a year is 365 days and one day is 24 hours? Can’t we divide a year into 400 days or a day into 25 hours? Is this an artificial choice or a natural timing? It all depends on how we define a year and a day. We can identify a year by a full rotation of the earth around the sun. Also we can distinguish the beginning of day and night clearly. These are definite intervals of time dictated through our observations, and so there is not much choice other than setting one year at 365 days. Is there a similar fact behind the relation of day to hours? Not at all! It was in ancient Egypt, around 2000 BC, that days for the first time were sliced into 24 pieces of time. In the age of Babylonians, however, a day was designed to be 60 hours. Perhaps the reason for such division of the day (into 24 or 60) hours was that 24 or 60 are nice numbers which are divisible by many integers; the same reason why a full-angle is 360 degrees instead of 2π.

In the Middle Ages, for Muslims, measurement of astronomical phenomena was a very serious affair. They were very concerned about accurate timing. Determining the changing time of the five daily prayers and the beginning and ending of the month of Ramadan was more than a custom, it was a religious duty. And such calculations required high precision. This precision was exemplified in year 1000 AD by the Muslim scholar al-Biruni who gave the times of the new moons in terms of days, hours, minutes, seconds, thirds, and fourths after noon Sunday.

In the West, the first accurate time measurements were made by Roger Bacon in thirteenth century. In 1657, Christian Huygens invented the pendulum clock, which uses swinging weights to keep time. Later, Hyugens and William Clement refined the design so that clocks were accurate up to seconds. Another problem with older clocks was that although they worked fine in the local region, they lost accuracy at different parts of the globe and were thus unsuitable for navigation. The reason for the lack of accuracy was that earth’s rotation around the sun on its axis (which definitely affects the period of the pendulums) was not uniform. Several adjustments were made in the nineteenth century for better accuracy by improving the design to compensate for thermal expansion of the metal rods and air drag, which globalized the measurement of time. In 1956, the “second” was redefined in terms of the earth’s revolution around the sun, according to data gathered in year 1900. As the scientists were not completely satisfied, they re-defined the second (as the atomic second) a decade later. In 1967, the Thirteenth General Conference on Weights and Measures defined a second of atomic time in the International System of Units as:

The duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.

Measuring the length was another important task for ancient peoples. Among the earlier civilizations, the most accurate system was developed by Indus Valley Civilization. While their contemporaries were using parts of the body for measurement, as early as 2600 BC, the Indus civilization had a much finer unit system that accounted even for millimeters. Most societies continued to use their own length scale until eighteenth century.

As early as seventeenth century, with the advances in the accurate measurement of time, pendulum motion was suggested to measure standard length. In the eighteenth century, there were two main approaches for measuring the standard unit of length. One suggested defining the meter as the length of a pendulum with a half-period of one second. The other suggested defining the meter as one ten-millionth of the length of the Earth’s meridian along a quadrant, which is the distance from the equator to the North Pole. In 1791, the French Academy of Sciences selected the choice based on the meridian. After several changes in the definition in 1960, the International Bureau of Weight and Measure organized the 11th CGPM (General Conference on Weights and Measure), during which the meter was redefined as 1,650,763.73 wavelengths of the orange-red emission line in the electromagnetic spectrum of the krypton-86 atom in a vacuum. The final decision came from the 17th CGPM as: “a meter is defined as 1/299,792,458 of a light-second.”

Figure 1. Historical International Prototype Meter bar, made of an alloy of platinum and iridium, was the standard from 1889 to 1960.

As for the measurement of mass, the situation is even more complicated since scientists cannot even agree on what mass is. There are mainly two different understandings of mass based on its features. One is called inertial mass (related to the quantity of a material); the other is gravitational mass (related to gravitational pull and acceleration). Whether these two concepts are equivalent or not is still in debate though in modern theories like Einstein’s general relativity, these two definitions are equivalent. We can, therefore, leave these philosophical discussions about the concept of mass to the scientists and go back to its measurement.

Just like the measurement of time and length, scientific mass measurement gained a boost after the French Revolution. At first, a gram, defined as the absolute mass of 1 cm3 of water at 0o C, was chosen as the standard. In 1799, scientists made a slight modification to the unit of mass by re-setting the definition at 4oC since it is the temperature at which water is most stable. Later, officials noticed that this unit was too small to be a standard of everyday commercial materials, which usually appear in large amounts. In 1889, the International Prototype Kilogram (IPK), made of an alloy of 90% platinum and 10% iridium (by weight), was designed to define the standard mass (Figure 2). After the first production, several more stable replicas of IPK have been produced to replace the older ones. Today every government who subscribes to this standard must have an exact copy of IPK, and these replicas must be returned to Paris periodically as they may get rusted or dirty with time.

Figure 2. Shown above is a computer-generated image of the International Prototype Kilogram (IPK). The IPK is made of a platinum-iridium alloy and is stored in a vault at the BIPM in Sèvres, France.

These facts may sound very odd to some, as it did to me when I first heard of them. I asked myself: If all these measures are defined in terms of something else, what is the point of defining them in the first place? For example, if we can agree to use the second as some interval of time, why do we bother to count the number of oscillations of Cesium. The answer is: We cannot agree unless we use a reference time which is geography-free, climate-free, and politics-free. Only then we will be sure that I, here in Western Pennsylvania, a person on the top of Everest, or a person in a submarine under the Pacific Ocean will call the same interval of time a “second.” In other words, the oscillation of the cesium isotope was believed to be free from all possible deficiencies that are results of physical location (Australia or America), environmental change (the Amazon Forests or the Sahara Desert), and politics.

In short, sand-clocks for measuring time (think of what kind of sand in what shape of glass tube) or the arm of a king as a length unit (imagine a king who seized the throne at 13 and died at 60), or weighing with iron cylinders (common in small grocery stores in some countries) is too unreliable, too unstable, too local, and of course, inaccurate to create a standard. Especially in this age of globalization, a consensus on measurement is absolutely necessary.

It seems a bit ironic that a simple-looking concept of science, measurement, could cause such controversy. A simple way to keep track of numbers that belong to different kind of quantities evolved into an area of serious research through time. Perhaps then, I should have not worried that much about my bad math grades on a subject which troubled the scientist themselves. After all, my grades were just my teacher’s own measurement.**

O. S. Caglayan has a PhD in mathematics. He is a freelance writer. He lives in Pittsburgh, Pennsylvania.

* This famous quotation attributed to Newton was opposed by Leibnizian view of time: “The universe is the clock.” The scientist as philosopher, Friedel Weinert, Springer; 1 edition (May 27, 2004)

** The author is indebted to his dear elementary school teacher Muazzez Ozalp for instilling in him the love of science.

References

1. G. J. Toomer. Ptolemey’s Almagest (Princeton, New Jersey: Princeton University Press, 1998)
2. The History of Time (Leofranc Holfrod-Strevens).
3. al-Biruni (1879). The chronology of ancient nations: an English version of the Arabic text “Vestiges of the Past”. London: W.H. Allen, 147-149. OCLC 9986841.
4. Matthew Bennett, Michael F. Schatz, Heidi Rockwood and Kurt Wiesenfeld, Proc. R. Soc. Lond. A 2002 458, 563-579.
5. Ian Whitelaw. A Measure of All Things: The Story of Man and Measurement, St. Martin’s Press, 2007.
6. http://physics.nist.gov/cuu/Units/meter.html
7. www.bipm.org/eng/home

As a relatively new and exciting science, chaos science grew very slowly during its infancy. Yet in the last decade, due to active research in many areas, it has became one of the hottest topics in academia as well as the popular science press. Many books have been published, and millions of Internet pages have been designed full of fractal pictures.1 Given that chaos is associated with disorder or confusion in a system or condition, why does it continue to attract so many people?

History of chaos

To understand chaos, one first has to understand the Newtonian worldview. Sir Isaac Newton’s (1642-1727) development of the calculus and laws of classical mechanics began a scientific revolution in seventeenth-century Europe that caused all subsequent scientists to view nature from a profoundly different perspective. Now that they finally could determine the dynamics of bodies by simple equations, they believed that they had found the ultimate eternal rules that shape the universe.

French physicist Pierre-Simon Laplace (1749-1827), who based his work upon Newton’s work, is credited with the following famous quotation (often referred to as Laplace’s Demon): ‘We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.’2

Laplace’s Demon states the idea of determinism, that the past completely determines the future. One can clearly see why determinism was so attractive to scientists at that time. However, in Laplace’s word everything was predetermined: no chance, no choice, no uncertainty. A solid, inevitable destiny had frozen the events in every corner of the past and is continuously spreading out to the future to do same there. Determinism apparently invokes the idea that whole universe is like a clock. God set it in motion at the beginning of creation and then removed Himself, for everything had been planed before. The ideas that there was no place for free will and that God could not interfere killed the belief in a soul and, consequently, in spirituality. Philosophers and scientists have discussed this for many years. Determinism affected many philosophies and triggered the major ideological movements of during eighteenth century, especially in Europe.

Toward the end of the 1800s, mathematicians and scientists began encountering some very difficult equations, some of which we know today are unsolvable. The most troublesome are various nonlinear differential equations. Even though it looks like such a simple and totally deterministic system, the problem of three bodies attracting each other with purely gravitational forces (e.g., the sun, Earth, and moon triple) turns out to be missing an exact solution. At first, such problems were cast-off as special cases and largely ignored.

One reason for this also might come from the fascinating world of quantum mechanics, which dazzled even the great physicists, and the lack of fast computers at that time. When these equations finally were studied in detail, a fundamental change that would ultimately overthrow determinism began to occur in mathematics and science. An indication of the science that would be come to be known as ‘chaos’ began to appear.

Why does chaos interest people?

In contrast to its common usage, chaos does not actually mean disorder or confusion. By definition, it should occur in well-defined orderly systems. However, most natural physical systems often can exhibit an unpredictable or intractable behavior in the long run, even though the system is defined by clear-cut orderly mechanisms. In this sense, chaos can be defined as unpredictability rather than disorder.

For example, meteorologists use 12 sets of well-defined equations to forecast the weather. They relate such atmospheric parameters as pressure, temperature, flow speed, and time to each other. One can make a computer program that calculates the parameters’ final values by taking any initial conditions as the run’s starting point. In principle, therefore, if we know the initial temperature, pressure, and time values that describe today’s weather conditions, it is possible to derive tomorrow’s weather conditions by running a computer program, which is nothing more than a chain reaction of numerical iterations.

However, in practice, initial conditions cannot be measured exactly and so contain a degree of uncertainty. But since the system’s governing laws are known, one may estimate the effect of errors on future results. Hence, instead of giving the exact results, one can provide an approximate range of possibilities. This range can still be very useful, provided that the deviations do not stray too far from the actual values. In addition, knowing how the error grows in the system might help us understand and control the systems. But if we apply these error estimates to weather forecast equations, we will encounter a large problem, for errors grow exponentially in such systems. Even a tiny deviation at the beginning can create huge deviations from the actual values. It also can provide unrelated or nonsensical results.

An example of chaotic systems

This numerical behavior was first observed by the meteorologist Edward Lorenz, a pioneer in modern chaos work. Fascinated by the results he obtained, in the early 1960s he gave an interesting metaphor to explain the situation of high sensitivity to initial conditions: A butterfly’s slight wing movement (i.e., a little deviation from the initial conditions) can change the future in a way that causes some chain reaction that ultimately result in a hurricane.

Such systems that exhibit a very high sensitivity to initial conditions are called chaotic systems. Chaos comes from the mathematical properties hidden in the equations defining the system, and such unpredictability cannot be removed. Even if the measurements’ quality could be improved by minimizing errors, chaos never disappears from a chaotic system.

One may suppose that chaos occurs in complicated systems, such as weather forecast systems having 12 sets of equations. But even much simpler systems, such as billiards, can exhibit a very high degree chaos. A usual billiard system consists of many balls and a rectangular shaped table. I challenge master billiard players by requesting them to play the game in a stadium-shaped table. I am sure that they will find it difficult to do so, because such billiard tables would be chaotic systems.

If a system is defined as chaotic, this does not necessarily mean that its behavior is totally undefined all the time. As in stadium billiards, a ball has to be inside the billiards, so it should be somewhere on the table, even though sometimes we cannot foretell its exact position because of chaos. Besides, if you send the ball with a velocity perpendicular to a straight side, it will bounce back and forth between the two sides forever. Therefore, depending on which initial conditions are taken, chaotic systems also can show characteristics of regular motion.

A 3-body problem (in general n-body problems) such as the sun, Earth, and moon system, is a chaotic system. But since we can predict the motions of these celestial objects with great precision for many years in the future, why do we call this system chaotic? This triple system possesses a very special set of conditions: distance between bodies, their masses, and their velocities. These parameters cause it to exhibit near-regular behavior. It is analogous to the example of stadium billiards given above, for this triple system bounces back and forth between the table’s sides.

Conditions for chaos

Chaos also can be caused by other factors than just uncertainties measured in the initial conditions. Scientists generally define hypothetical systems by isolating them from the outside world in order to simplify them as much as possible. However, as even objects in the real world that are far apart interact with each other, no system in the real world can be isolated. Given this, a closed (isolated) system might be defined as a fluctuating approximation to its real counterpart, which is changing in an unpredictable manner all the time. In short, even though we would know the exact initial conditions, the actual system could be chaotic due to changes in the approximate system. In this sense, many physical systems have an inclination toward being chaotic.

Due to its maximum complexity, the universe is the largest chaotic system. Observable regular patterns in special parts of that system repeat themselves in time. While the rest of the flows are unpredictable, they are not totally irregular, abrupt, or disordered. Just like whirls in a flowing river, they are in a kind of free motion searching for convenient conditions in which to give birth to organized structures.

The future of chaos

Chaos gives today’s scientist a new worldview, for Newton’s concrete, cold, and deterministic one has been shaken by the uncertainty principle of quantum mechanics. No one ever thought that the Newtonian worldview could be replaced. However, now scientists are more likely to be open to chance and choice than their predecessors. The question is whether chaos theory will cause large revolutions in how we understand the universe. However, the existing excitement, expanding research and growing number of articles, and its numerous applications from economy to biology, seem to indicate that a surprise improvement might not be so far off. ‘

Footnotes

1. Fractal: A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature.
2. ‘Chaos and Fractals: Laplace’s Demon.’ Online at: www.pha.jhu.edu/ ldb/seminar/laplace.html.