Math II (562417), страница 4
Текст из файла (страница 4)
You can stretch, bend or even shrink this doughnut. What's important to a topologist is not how long or wide the doughnut gets - only that the resulting figure still has a hole.
A perfect Mobius strip can also be stretched or distorted. And as long as it is not cut the wrong way, it remains one-sided. (To learn more about topology, check out Dr. Weeks's Web site: www.northnet.org/weeks/TorusGames/ TorusGames.htm)
Who thinks up this stuff?
Nov. 17, 1790, was an important time in history. The French Revolution was being fought. George Washington was serving his first term as president of the United States. And August Ferdinand Mobius had just been born in Saxony (Germany).
Until he was 13, Mobius was home-schooled. His father, a master dancing instructor, had died when he was 3. His mother was a descendent of religious reformer Martin Luther. In 1813, Mobius left for Gottingen University. There he studied under brilliant mathematician Karl Friedrich Gauss.
Mobius was a patient man. He liked to work alone solving math problems. Unfortunately, mathematicians at that time were poorly paid. To earn a living, Mobius became an astronomer and professor. By 1848, he was director of the Leipzig Observatory in Germany.
But mathematics remained his real love. In 1858, Mobius was given credit for discovering the one-sided strip.
Historians, however, don't all agree that this breakthrough should have been named after him. Months before Mobius announced his finding, another mathematician, Johann Benedict Listing, produced an unpublished paper. In it he described Mobius's strip. Biographers suggest that Mobius and Listing were unaware of each other's work.
Was Mobius the first person to think of this shape? The mystery may never be solved. Ten years after announcing his discovery, Mobius died.
Making paper strips is fun. But is there any practical use for Mobius's band? You may be surprised to hear that the answer is yes.
Back in 1949, an inventor patented an abrasive belt that used the principle of one-sidedness. Another 1952 patent showed a conveyor belt that could transport hot objects - and it, too, used Mobius's concept.
And what's it good for, anyway?
Pamela Clute is a mathematics professor at the University of California at Riverside. She points to a number of other applications, including fan belts, typewriter ribbons, conveyor belts, even exercise equipment. Think of it: If the conveyor belt at the grocery checkout had a half-twist in it (and it probably does), the surface of the belt would last twice as long.
Even astrophysicists are interested in the Mobius strip.
Researchers at the University of Warwick (Rhode Island) have examined an electromagnetic region near the earth they call a tail. Scientists think the tail's charged particles follow a path that looks like a Mobius strip. Mobius may have even answered a bigger question: What is the shape of the universe? Some scientists think the universe may be like one giant Mobius strip!
An old saying goes, "There are two sides to every story." Do you suppose it was written by someone who'd never heard of a Mobius strip?
A piece of paper that has only one side
If you don't believe paper can be one-sided, prove it for yourself! You'll need: notebook paper, tape or a glue stick, scissors, and a pencil. Make the simple one first, at left. Then try the one on the right.
Step 1. Cut a strip, lengthwise, from a piece of notebook paper. Make it about two inches wide.
Step 1A. Cut another strip, same as the first. This time, fold it into thirds, lengthwise.
Step 2. Twist one end of the strip halfway (180 degrees). Keeping the paper twisted, tape or glue the ends together. The resulting band should NOT look like a simple round cylinder.
Step 2A. Now unfold it. Twist and tape (or glue) the paper into a Mobius strip as before. The folds will be your guides for cutting the Mobius strip.
Step 3. Using a pencil, put a dot halfway between the two edges of the strip. Beginning at the dot, draw a line lengthwise down the middle of your Mobius strip. Don't lift the pencil off the paper. Keep drawing until you reach the dot again. See? The strip has only one side!
Step 3A. You're ready to cut again. Use the folds as a guide for your scissors. But before you start, think. What will happen? Look again at the figures at the bottom of the page. Which one do you think the cut-in-thirds Mobius strip will look like?
Step 4. Look at your Mobius strip. What will happen if you cut it in half, lengthwise? Think about it, and look at the figures at the bottom of the page. Choose the one you think the cut-in-half Mobius strip will look like. Now carefully cut your strip in half. Did you guess correctly?
Step 4A. Start cutting. What happened? Did you guess correctly? One-sided figures sure don't behave the way two-sided figures do! Are you ready to try to predict what will happen if you cut a Mobius strip in fourths?
What do you think it will look like?
After you've cut the Mцbius strips above, they'll look like one of these four figures. Which ones? (Answers on page 18.)
THE SUPPRESSED DRAWING: PAUL DIRAC'S HIDDEN GEOMETRY
Source: Representations, Fall2000 Issue 72, p145, 22p, 4bw
Author(s): Galison, Peter
Purest Soul
FOR MOST OF THE TWENTIETH century, Paul Dirac stood as the theorist's theorist. Though less known to the general public than Albert Einstein, Niels Bohr, or Werner Heisenberg, for physicists Dirac was revered as the "theorist with the purest soul," as Bohr described him. Perhaps Bohr called him that because of Dirac's taciturn and solitary demeanor, perhaps because he maintained practically no interests outside physics and never feigned engagement with art, literature, music, or politics. Known for the fundamental equation that now bears his name--describing the relativistic electron--Dirac put quantum mechanics into a clear conceptual structure, explored the possibility of magnetic monopoles, generalized the mathematical concept of function, launched the field of quantum electrodynamics, and predicted the existence of antimatter.
In this paper I will explore the meaning of drawing for Dirac in his work. In the thirteen hundred or so pages of his published work between 1924 and World War II, aside from a few graphs and a diagram in a paper that he coauthored with an experimentalist, Dirac had practically no use at all for diagrams. He never used them publicly for calculation, and I know of only two, almost trivial, cases in which he even exploited a figure for pedagogical purposes. His elegant book on general relativity contained not a single figure; his famous textbook on quantum mechanics never departed from words and equations.(n1) If anything, diagrams appear to be antithetical to what Dirac wanted to be "visible" in his thinking. Dirac was known for the austerity of his prose, his rigorous and fundamentally algebraic solution to every physical problem he approached. (Even his fellow physicists found his ascetic style sometimes to be too terse--in response to questions, he would repeat himself verbatim; other physicists sometimes complained that his papers lacked words.) Now it is not the case that diagrams are simply absent from physics. To cite one famous example, there is the famous diagrammatic-visual reasoning of theorists like James Clerk Maxwell who insisted that full understanding would only come when joined to imagined, visualizable machines running with gears, straps, pulleys, and handles. Maxwell wanted objects described and drawn that could, in the mind's eye, be grasped with the hands and pulled with the muscles. Similarly visual were Einstein's thought experiments, his use of hurtling trains, spinning disks, and accelerating elevators. Dirac's papers contain none of this. Not even schematic diagrams appear in his writings, visualizations of the sort that Richard Feynman introduced to facilitate calculation and impart intuition about colliding, scattering, splitting, and recombining particles.(n2)
It would seem, then, that the corpus of Dirac's work would be the last place to look for pictures. But in the Dirac archives something remarkable emerges. I was astonished, for example, to find these comments penned by Dirac as he prepared a lecture in 1972: "There are basically two kinds of math [ematical] thinking, algebraic and geometric." This sounds like the theoretical twin of a contrast I have long pursued between laboratory methods that yielded images (analogous here to Dirac's geometric thinking) and those methods predicated on the logical or statistical compilations of data points (analogous to Dirac's algebraic thinking).(n3) So I was intrigued. Given Dirac's austere public predilection for sparse prose, crystalline equations, and the complete absence of diagrams of any sort, I assumed that in the next sentences he would go on to class himself among the algebraists. On the contrary, he wrote in longhand,
A good mathematician needs to be a master of both.
But still he will have a preference for one rather or the other.
I prefer the geometric method. Not mentioned in published work
because it is not easy to print diagrams.
With the algebraic method one deals with equ[ations] between
algebraic quantities.
Even tho I see the consistency and logical connections of
the eq[uations], they do not mean very much to me.
I prefer the relationships which I can visualize
in geometric terms.
Of course with complicated equations one may not be able to
visualize the relationships e.g. it may need too
many dimensions.
But with the simpler relationships one can often get
help in understanding them by geometric pictures.(n4)
These pictures were not for pedagogical purposes: Dirac kept them hidden. They were not for popularization--even when speaking to the wider public, Dirac never used the diagrams to explain anything. Astonishing: across the great divide of visualization and formalism that has, for generations, split both physics and mathematics, we read here that Dirac published on one side and worked on the other.
The poverty of print technologies in and of itself seems rather insufficient as an explanation for the privacy of Dirac's diagrams, but in another (undated) account his characterization may be more apt: "The most exciting thing I learned [in mathematics in secondary school at Bristol] was projective geometry. This had a strange beauty and power which fascinated me." Projective geometry provided this Bristolean student new insight into Euclidean space and into special relativity. Dirac added, "I frequently used ideas of projective geometry in my research work in later life, but did not refer to them in my published work because I was doubtful whether the average physicist would know enough about them to appreciate them."(n5) Lecturing in Varenna, also in the early 1970s, he recalled the "profound influence" that the power and beauty of projective geometry had on him. It gave results "apparently by magic; theorems in Euclidean geometry which you have been worrying about for a long time drop out by the simplest possible means" under its sway. Relativistic transformations of mathematical quantities suddenly became easy using this geometrical reformulation. "My research work was based in pictures--I needed to visualise things--and projective geometry was often most useful--e.g, in figuring out how a particular quantity transforms under Lorentz transf[ormation]. When I came to publish the results I suppressed the projective geometry as the results could be expressed more concisely in analytic form."(n6)
So Dirac had one way of producing his physics in his private sphere (using geometry) and another of presenting the results to the wider community of physicists (using algebra). Nor is this a purely retrospective account. For there remains among his papers a thick folder of geometrical constructions documenting Dirac's extensive exploration of the way objects transform relativistically. These drawings are not dated but on their reverse sides are writings dated from 1922 forward. None of these drawings were ever published or, as far as I can tell, even shown to anyone (figs. 1 and 2).
The question arises: how ought we to think about Dirac's "suppressed" geometrical work? Dirac himself saw projective geometry as key to his entrance into a new field: "One wants very much to visualize the things which we are dealing with."(n7) Should one therefore split scientific reasoning, as Hans Reichenbach did, between a "logic of discovery" and a "logic of justification"? For Reichenbach there were some patterns of reasoning that were, in and of themselves, sufficient for public demonstration. Other procedures, more capricious and idiosyncratic, could not count as demonstrations though they might serve the acquisition of new ideas.(n8) This distinction saturates the philosophy of science of the postwar era. In Karl Popper's hands it helped to ground his demarcation criterion between science and non-science: only scientific theories, in the context of justification, were falsifiable, only in the realm of the justifiable was there anything dignified of the word logic. "My view," Popper wrote, "may be expressed by saying that every discovery contains 'an irrational element', or 'a creative intuition', in Bergson's sense."(n9) By contrast, Gerald Holton took the private-scientific domain to have a sharply articulable structure that can be characterized by commitments to particular thematic pairs (such as continuum/discretum or waves/particles). According to Holton, this rich, three-dimensional space of private thought is then "projected" onto the plane of public science (defined by the restricted axes of the empirical and the logical). In this empirical-analytic public plane, much of the private dynamic of science is necessarily lost.(n10) Recent work in science studies has either denied the force of the Reichenbachian distinction, or maintained the public/private distinction in other terms. For example, Bruno Latour, in his early work with Steve Woolgar, characterized private science by a different grammar: the private is filled with modifiers, modal qualifications that slowly are filtered out until only a public, assertoric language remains.(n11)
Certainly the common view of drawing as preparation would fit this sharp separation of public and private. Private sketches, in virtue of their schematic and exploratory form, would count as the precursors to the completed painting; private scientific visualization and sketches would, without requiring rigor, precede the public, published scientific paper. In such a picture the interior is psychological, aleatory, hermetic, and unrigorous while the exterior is fixed, formally constrained, communicable, and defensible. One thinks here of Sigmund Freud for whom the visual was primary, preceding and conditioning the development of language. To the extent that primitive reasoning is supplanted by language, the pictorial, unconscious form of reason is of a different species from that of conscious, logical, language-based thought.
For some analysts of science, the advantages of the radical public/private distinction is that it brought the private into a psychological domain that opened it up to studies of creativity. For others, the separation permitted a more formal analysis of the context of justification through schemes of confirmation, falsification, or verification. For those who saw published science as merely the last step of private science, the distinction helped shift the balance of interest toward "science-in-the-making" and away from the published end product.
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