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Announcement of Philosophia Naturalis #2

Saturday, September 30, 2006

Philosophia Naturalis #2 will be published on Thursday, October 12 at Nonoscience -- check there for any further information.

Make it easy on yourself -- don't put off sending your suggestions till the last minute. Submitting an article to the carnival takes very little time (assuming it's already written!) So do it now, and then you won't have to keep thinking about it.

Also, there is now a blog just for information about the carnival. It's called... Philosophia Naturalis. You can check there for announcements of future editions and any other pertinent information, including how to submit an article.

One good way to keep track of what's going on is to add the RSS feed to your favorite reader/aggregator. Then you'll get announcements whenever a new edition is posted.

U.S. science education lags, study finds

Wednesday, September 27, 2006

U.S. science education lags, study finds
Science education in U.S. elementary and middle schools is overly broad and superficial, according to a government report issued on Thursday that also faults science curricula for assuming children are simplistic thinkers.

"All children have basic reasoning skills, personal knowledge of the natural world, and curiosity that teachers can build on to achieve proficiency in science," said the report from the National Research Council, one of the National Academies.

Part of the problem is that state and national learning standards for students in elementary and middle schools require children to memorize often-disconnected scientific facts, the report said.

Hoo boy. I feel a rant coming on. About what's wrong with U. S. science education.

I guess I could write a book on this, but I'll try to be briefer than that, since I don't have the time tonight and since I expect many people who read science blogs have given this much thought and have detailed opinions on this. (Many are probably, in fact, science teachers at some level.)

Basic premise: science education in the U. S. sucks, and indeed a large part of the educational system sucks. (Some earlier comments on this are here.)

Deficiences in U. S. science education are often noted in connection with the notorious failure of many in the U. S. public to either understand or accept the science of evolution. For instance, this post from Cosmic Variance discusses a study published recently in Science about how the U. S. was almost dead last in a comparative study with 32 European countries in the percentage of respondents surveyed who agreed with the assertion that “Human beings, as we know them, developed from earlier species of animals.”

I have no problem with the teaching of facts, not even "often-disconnected facts". Facts make up the basic stuff of science. Kids need to learn all kinds of basic facts -- from important chemical elements, to classifications of animals, to standard parts of living cells. Definitions of key scientific terms also need to be learned. I realize that science teachers struggle even to get kids to learn such facts.

But of course, learning facts isn't enough. Learning how the facts and definitions are related in coherent general theories is also important. I'm not going to try to suggest how to improve the teaching of facts, definitions, and theories. That would take much too long, and I'm certainly no expert on teaching.

Rather than get into the tactics of teaching science, I want to just make some points about strategy.

First, teachers have to arouse curiosity in their students. Because curiosity is, in my opinion, what drives the whole scientific enterprise. Other things like practical applicability of scientific knowledge are useful byproducts, but certainly not primary motivators to students (and people in general) for learning and doing science. If a student isn't really curious to understand how an amazing thing such as a living cell (for example) works, the chances of learning much about it go way down.

Second, teachers have to educate students about some basic thinking skills -- what is sometimes called "critical thinking", including basic logic, "scientific method", and so forth. This is necessary in order for students to understand why some kinds of plausible theories are better, while others are just not even "scientific".

Third, teachers have to convey to students why science is important. This is a matter of philosophy (axiology, to be precise), because the importance of science has to be compared with that of various other endeavors that people undertake -- from raising a family to the "fine arts" to ... whatever. Doing this is, of course, a tall order, as it entails having some grounding in philosophical traditions of our culture (and of the cultures of others).

Thinking about this philosophical angle, I have to wonder whether there's not a basic problem in the intellectual culture of the U. S. that impedes the teaching of science. It seems quite likely to me that there is too much emphasis in U. S. traditions on "pragmatism" and the glorification of "what works" -- as opposed to other things that satisfy human aesthetic senses and curiosity. What I'm talking about here is part of what people such as (for instance) the American historian Richard Hofstadter wrote about in his book Anti-intellectualism in American Life.

This kind of cultural tradition certainly has not made it impossible for U. S. scientists to have become extremely successful at discovering and creating basic (as well as applied) science over the past century. Instead, the effect of U. S. cultural tradition is manifested -- I would suggest -- in the lagging results of science education among the general public.

If there's any merit to this suggestion, the outlook for actually making much improvement in U. S. science teaching is not especially promising, at least on a time scale measured in units less than decades.


Additional information:

On Not Becoming a Scientist - ScienceNOW article

Taking Science to School: Learning and Teaching Science in Grades K-8 - the actual NRC report


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Shhhh! Maybe if we don't talk about it, it will go away

Journal: Agency Blocked Hurricane Report
A government agency blocked release of a report that suggests global warming is contributing to the frequency and strength of hurricanes, the journal Nature reported Tuesday.

The National Oceanic and Atmospheric Administration disputed the Nature article, saying there was not a report but a two-page fact sheet about the topic. The information was to be included in a press kit to be distributed in May as the annual hurricane season approached but wasn't ready.

May? As in eight months from now? But is anyone really surprised?
"The document wasn't done in time for the rollout," NOAA spokesman Jordan St. John said in responding to the Nature article. "The White House never saw it, so they didn't block it."

Anyone think it wouldn't have been blocked if it got as far as the White House? In any case, it got blocked at a lower level. Since when is it necessary to get W. H. approval for scientific findings? Oh, wait. Never mind. If that's not a law already, the W. H. will be sending legislation to Congress real soon.
NOAA Administrator Conrad Lautenbacher is currently out of the country, but Nature quoted him as saying the report was merely an internal document and could not be released because the agency could not take an official position on the issue.

However, the journal said in its online report that the study was merely a discussion of the current state of hurricane science and did not contain any policy or position statements.

In fact, there's been a lot of news about global warming and hurricanes just this month. Here are various news reports from about two weeks ago:

  1. September 11, 2006 - Humans 'causing stronger storms'
  2. September 11, 2006 - Humans affect sea warming in hurricane zones
  3. September 11, 2006 - A Human Spin on Hurricanes
  4. September 12, 2006 - Human Activities Found To Affect Ocean Temperatures In Hurricane Formation Regions
  5. September 12, 2006 - Human activities are boosting ocean temperatures in areas where hurricanes form
  6. September 12, 2006 - Report links global warming, storms

Memo to W. H.: The cat's already out of the bag.

And actually, it's been a rather busy month in terms of news about global warming. For instance, here are some articles on the melting of arctic sea ice, and its effect on polar bears (among other things):

  1. September 13, 2006 - Arctic sea ice shrinks, a sign of greenhouse effect
  2. September 14, 2006 - Winter Arctic sea ice in drastic decline
  3. September 14, 2006 - Arctic sea ice diminished rapidly in 2004 and 2005
  4. September 14, 2006 - Warming Climate May Put Chill On Arctic Polar Bear Population
  5. September 14, 2006 - 'Drastic' shrinkage in Arctic ice
  6. September 14, 2006 - Arctic ice: it's melting - Scientists say wintertime loss of polar ice is growing along with a continuing summertime pattern and is strong evidence of global warming
  7. September 15, 2006 - Polar bears drown, islands appear in Arctic thaw
  8. September 15, 2006 - Arctic Ice Meltdown Continues With Significantly Reduced Winter Ice Cover

Yeah, OK, so it's getting hotter. But that doesn't mean it's caused by humans. Maybe the Sun is just putting out more heat? Nope:

  1. September 13, 2006 - No Sunshine for Global Warming Skeptics
  2. September 13, 2006 - Don't Blame the Sun
  3. September 14, 2006 - Changes In Solar Brightness Too Weak To Explain Global Warming
  4. September 14, 2006 - Study clears Sun of climate change

And what about all the methane -- an even more potent greenhouse gas than CO2 -- that's being released from thawing permafrost in Siberia:

  1. September 6, 2006 - Siberia's pools burp out nasty surprise
  2. September 6, 2006 - Melting lakes in Siberia emit greenhouse gas
  3. September 6, 2006 - Study Says Methane a New Climate Threat
  4. September 7, 2006 - Greenhouse Gas Bubbling From Melting Permafrost Feeds Climate Warming
  5. September 7, 2006 - Methane bubbles climate trouble
  6. September 7, 2006 - Melting permafrost spews out more methane
  7. September 8, 2006 - Siberian lakes burp "time-bomb" greenhouse gas
  8. September 13, 2006 - Greenhouse gas bubbling from Siberian permafrost

Should we be worried about this even if the W. H. isn't?

World has 10-year window to act on climate: expert
A leading U.S. climate researcher said on Wednesday the world has a 10-year window of opportunity to take decisive action on global warming and avert a weather catastrophe.

NASA scientist James Hansen, widely considered the doyen of American climate researchers, said governments must adopt an alternative scenario to keep carbon dioxide emission growth in check and limit the increase in global temperatures to 1 degree Celsius (1.8 degrees Fahrenheit).

"I think we have a very brief window of opportunity to deal with climate change ... no longer than a decade, at the most," Hansen said at the Climate Change Research Conference in California's state capital.

If the world continues with a "business as usual" scenario, Hansen said temperatures will rise by 2 to 3 degrees Celsius (3.6 to 7.2 degrees F) and "we will be producing a different planet."

Does the name "James Hansen" ring any bells? It should:
Hansen, who heads NASA's Goddard Institute for Space Studies, has made waves before by saying that President George W. Bush's administration tried to silence him and heavily edited his and other scientists' findings on a warmer world.

Yep. Same old same old. W. H. trying to shut up the scientists. We wrote about it back in February (about 2/3 down the page).

Perhaps the W. H. knows something we don't. Like, maybe, a little bit of global warming is no big deal, compared to the nuclear war they're planning to start with Iran -- real soon now.

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Hottest topics in physics

Tuesday, September 26, 2006

Which research topics in physics are currently the "hottest"? It depends on how you measure "hot". In this paper: An extension of the Hirsch Index: Indexing scientific topics and compounds, it's done by modifying a citation indexing technique used to measure which scientists are most influential. Instead of counting how often an author's paper is cited by others, the technique counts how often papers are cited that have specific topics mentioned in the abstract.

Here are the highest-scoring topics, in decreasing order:

  1. carbon nanotubes
  2. nanowires
  3. quantum dots
  4. fullerenes
  5. giant magnetoresistance
  6. M-theory
  7. quantum computation
  8. teleportation
  9. superstrings
  10. heavy fermions
  11. spin valves
  12. spin glass
  13. porous silicon
  14. quantum critical point
  15. geometrical frustration
  16. quantum information

Here's a news article with a good summary: Hottest topic in physics revealed.


General relativity passes cosmic test - Einstein's theory holds in extreme gravitational fields.

Tuesday, September 19, 2006

Chalk up another win for "orthodox" scientific theories. General relativity has again passed stringent tests.

General relativity survives gruelling pulsar test — Einstein at least 99.95 percent right
An international research team led by Prof. Michael Kramer of the University of Manchester's Jodrell Bank Observatory, UK, has used three years of observations of the "double pulsar", a unique pair of natural stellar clocks which they discovered in 2003, to prove that Einstein's theory of general relativity - the theory of gravity that displaced Newton's - is correct to within a staggering 0.05%. Their results are published on the 14th September in the journal Science and are based on measurements of an effect called the Shapiro Delay.

The double pulsar system, PSR J0737-3039A and B, is 2000 light-years away in the direction of the constellation Puppis. It consists of two massive, highly compact neutron stars, each weighing more than our own Sun but only about 20 km across, orbiting each other every 2.4 hours at speeds of a million kilometres per hour. Separated by a distance of just a million kilometres, both neutron stars emit lighthouse-like beams of radio waves that are seen as radio "pulses" every time the beams sweep past the Earth. It is the only known system of two detectable radio pulsars orbiting each other. Due to the large masses of the system, they provide an ideal opportunity to test aspects of General Relativity.

The large mass of the pulsars and their proximity to each other is the key thing, resulting in a very strong gravitational field. The binary pulsar system provides an opportunity to check the validity of general relativity under conditions that are more extreme than any studied before.

Shapiro delay can be described as an apparent change in the speed of light in a strong gravitational field. It occurs because spacetime itself is warped in the field, which effectively forces light to travel a larger distance. Since the radio-frequency beam from each pulsar sweeps across Earth at a very precise frequency (22.8 milliseconds for one, 2.8 seconds for the other), like an exceptionally accurate clock, it is possible to predict exactly when the beam should be seen. Any departure from this prediction would be due to the Shapiro delay. The orbital period of the two pulsars around each other is about 2.4 hours. During this period, the distance between the pulsars varies, so the mutual gravitational fields vary correspondingly. This allows the theoretical delay time to be calculated, and the observations match the prediction very well.

This effect is distinct from the time dilation which occurs in a large gravitational field. The dilation causes time intervals to appear to lengthen. So the period of rotation of each pulsar appears to change, and the spectrum of radio waves from each object is redshifted, as the pulsars experience a change in the gravitational field. Here again, the observations match the predictions of general relativity very well.


Additional information:

Pulsars' Gyrations Confirm Einstein's Theory
News article about the research from Science. (Subscription required for full access.)

Tests of General Relativity from Timing the Double Pulsar
The actual research paper from Science. (Subscription required for full access.)

Millisecond Pulsars as Tools of Fundamental Physics
A review paper by Kramer posted to the arXiv in May 2004. It explains the underlying physics in some detail. It also describes the binary pulsar system, which had only recently been discovered.

The Confrontation between General Relativity and Experiment
An expository paper by Clifford M. Will that reviews the status of experimental tests of general relativity and of theoretical frameworks for analyzing them.

General relativity passes cosmic test - Einstein's theory holds in extreme gravitational fields.
News article at Nature.com news. (Subscription required)


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Friday galaxy blogging

Friday, September 15, 2006

Blogging cats is common. Some people do orchids. I do galaxies.

The Eternal Life of Stardust Portrayed in New NASA Image (8/31/06)
A new image from NASA's Spitzer Space Telescope is helping astronomers understand how stardust is recycled in galaxies.

The cosmic portrait shows the Large Magellanic Cloud, a nearby dwarf galaxy named after Ferdinand Magellan, the seafaring explorer who observed the murky object at night during his fleet's historic journey around Earth. Now, nearly 500 years after Magellan's voyage, astronomers are studying Spitzer's view of this galaxy to learn more about the circular journey of stardust, from stars to space and back again.

Large Magellanic Cloud - click for 720×900 image

Philosophia Naturalis #1

Thursday, September 14, 2006

Welcome to the première edition of a new science blog carnival. Since the idea and objectives of the carnival have been explained here, I won't repeat those comments now. Let's get right down to business.

It's been 50 years since "artificial intelligence" emerged as a recognized subject of enquiry in computer science (see here). And for even longer than that, a few individuals -- such as Alan Turing -- have speculated about the possibility that computers could "think". AI has been controversial the whole time. Turing proposed a test, the Turing Test, which he offered as a criterion for computers having the ability to think. There has been plenty of skepticism that a computer could actually pass the test anytime soon. And it has also been argued that even if a computer could pass it, that would not mean the computer could really think. Scott Aaronson of Shtetl-Optimized takes on one form of this argument in Alan Turing, moralist.

Reaching back somewhat further in time, Scott also channels Aesop with this fable of The physicists and the wagon. Not to be outdone, Dave Bacon, the Quantum Pontiff, fires back with a tale of The Computer Scientist and the Abominable Approximation. The dueling fabulists are really talking about quantum computers and the differing approaches of physicists and computer scientists.

Writing in eSkeptic Phil Molé argues against the possibility of AI from a different point of view. In A.I. Gone Awry he describes three different approaches that have been tried for implementing AI, and he explains why he thinks none of them can work.

Another example of philosophical controversies that are actively roiling the waters of modern science is the pitched battle going on in high-energy physics over string theory. Despite the theory's mathematical elegance, after several decades of active study, physicists have not yet come up with any clear way to test the theory, and many are now skeptical that such a test will turn up any time soon, if ever. Peter Woit of Not Even Wrong is one of the better-known skeptics, and Lee Smolin is another. Both have recently published books on the subject. In The Trouble With Physics Woit reviews Smolin's book.

The nice thing about science, the largest part of it anyway, is that most controversies are eventually resolved by an accumulation of evidence that strongly favors one hypothesis over all others. This seems to be happening now, finally, regarding the issue of "dark matter" in cosmology. Just a few weeks ago, new evidence was reported that strongly supports the existence of dark matter as an explanation for many puzzles about the seemingly anomalous behavior of visually observable matter. Sean Carroll at Cosmic Variance gives a comprehensive summary in Dark Matter Exists of what was found and what it means.

Having established that dark matter is real, the question remains: What is it? Another writer at Cosmic Variance, Mark Trodden, addresses this question in Identifying Dark Matter. This is a problem in particle physics rather than cosmology per se, though cosmological observations will almost certainly contribute to pinning down the answer. Basically, physicists are pretty sure that the standard model of particle physics isn't complete. There must be other particles and possibly other forces beyond those we know about. Not only are they needed to extend the standard model, but they're natural candidates for the "stuff" of dark matter.

There's a "standard model" of cosmology too -- it's called the Big Bang. Though this model also has its skeptics, the present evidence in its favor is quite strong. And the recent findings about dark matter further bolster the theory. Actually, a big reason that skeptics of the Big Bang persist is that there are many misconceptions about the theory. Jon Voisey at The Angry Astronomer clears up some of them in The Big Bang – Common Misconceptions.

One aspect of the Big Bang about which there remains some uncertainty is the distance scale and correspondingly the amount of time that has elapsed since the initial singularity. Rob Knop at Galactic Interactions explains in Is the Universe a couple of billion years older than we thought? how astronomers try to measure distances. He then writes about recent research that raises questions about what we thought we knew regarding the cosmological distance scale -- and consequently how the figure of about 13.7 billion years for the length of time since the big bang could be 15% too low.

This length of time, otherwise known as "the age of the universe", is closely related to another quantity, the "Hubble constant". It's not really a constant, by the way, since it changes over the lifetime of the universe. It's more accurately called the Hubble parameter, and it expresses the rate at which distant galaxies are receding from us as a function of their distance, which is the key fact that Edwin Hubble discovered about 80 years ago. The value of this parameter at the present time is symbolized by H0, and it's usually expressed in units of kilometers per second per megaparsec. If you unwind that, the units are proportional to the reciprocal of time, and so 1/H0 has units of time. To a first approximation, this is the "age of the universe". (A decade or so ago, some astronomers thought H0 was larger than today's best consensus value. So it was feared that the universe could not be as old as the oldest known stars. This was unsettling to many people.) The upshot of the new distance measurements is that H0 might be even smaller than the current consensus, making the universe even older. Mollishka at {mollishka's title goes here} points out here that measurements of H0 still aren't that exact.

Although scientists haven't thought for quite a long time that Hubble's "constant" is really constant, there are other important "constants of nature" whose actual constancy has only recently (less than a decade) been seriously studied experimentally. Accounts of such investigations show up from time to time in the popular media, such as here. Reacting to that story, Rob Knop in Are the fundamental constants changing? writes to point out that possible variations in constants such as the speed of light, if they exist at all, are likely to be quite minute, even over the entire lifetime (14 or 15 billion years) of the universe. So it's undestandable that research that claims to have found such variations is very controversial. (In a subsequent correction, Rob notes that there is a more credible piece of research on this than the example he first chose.)

Based on Rob's remarks, Chad Orzel of Uncertain Principles picks of the conversation. In two longish articles (part 1, part 2) he delivers a detailed explanation of what is involved in trying to investigate possible variations in a dimensionless constant such as the "fine structure constant". This latter number, which is suspiciously close to 1/137, plays a very large role in quantum physics. Among other things, it affects atomic spectra, and so can in principle be studied by examining spectra from very distant quasars. It depends in turn on four other physical constants -- the speed of light, the charge of an electron, Planck's constant, and the permittivity of free space. Any one of these, or even perhaps all, could be changing very slightly with time... so this is a rather interesting question.

Head spinning yet? If not, perhaps you'd like to contemplate the question of the direction of time. This is a matter of deep concern to everyone: Why the heck do we keep getting older instead of younger? Alejandro Satz at Reality Conditions writes about the direction of time by way of a review of Huw Price's 1996 book Time's Arrow and Archimedes' Point. According to Alejandro, Price describes several well-known asymmetries ("arrows") of time, in spite of the time symmetry of practically all physical laws. The book then goes on to consider possible explanations for the asymmetries. Alejandro is planning to continue his review in the "future" -- or is that the "past"?

Winding down now, let's turn to simpler subjects, like math.

Perhaps you've heard anecdotes like this: Harried math grad student stumbles into class late. She sees a couple of assertions written on the blackboard and assumes they are homework. A week later she returns and hands the professor a written-out proof of one of the assertions. Professor is flabbergasted, since the assertion in question was a famous unsolved problem. Evidently that's not just an Urban legend. Apparently it's even happened more than once, as Luis Alberto Sanchez Moreno of Astronomer. In the wild. explains.

With the apparent resolution of the Poincaré conjecture much in the news lately, many folks are interested in learning more about the fundamentals of topology. (Well, we can dream, can't we?) One of the most basic concepts is that of a "continuous function". Fortunately, we have "Paranoid Marvin" of Antopology ready with a good primer: Continuity Introduced.

Mathematics, of course, is not entirely about higher dimensions and similar abstractions. It has applications just about everywhere. I recall seeing a report that came out in the past year of someone, probably a lonely grad student, who came up with a mathematical model of sexual attraction, but I don't have time to go looking for it just now. However, mathematical biology is currently an active field, and Deepak Singh is enthusiastic about a paper that uses finite element modeling to study a biological process. He suggests that "in a few years, those interested in studying protein structure and function will require a healthy training in multiscale modeling (quantum chemistry, molecular dynamics, coarse grained simulations, continuum dynamics), bioinformatics, and mathematical modeling."

And now for something completely different... Did you know that there was an area in the upper midwest of the U. S. between lakes Superior and Michigan that was unglaciated even in the depths of the last ice age? I didn't. It's called the Driftless Area, and "Pascal" of Research at a snail's pace (this is about glaciology, see?) explains how it happened.

And that concludes today's opening performance of Philosophia Naturalis. We'll be back here again in just four short weeks, on Thursday, October 12. Watch this blog for further details. Or just go back to the original announcement for information on how you can suggest articles for inclusion here -- which I hope you will consider doing, because, well, sharing your interests is a Good Thing.

Update 9/30/06: The next edition will be posted at Nonoscience on October 12.

The Poincaré conjecture, II

Monday, September 11, 2006

Note: There are a few special mathematical symbols used in the following. Some versions of Microsoft's "Internet Explorer", maybe every one, for all I know, can't display them and show square boxes instead. Don't ask me why. MS IE sucks. Use Firefox, Netscape, or some other real browser.

In the previous post on this subject we looked, from an intuitive point of view, at what the conjecture (now, apparently, a theorem) is about. In a nutshell, it's really quite simple: What properties does an abstract topological object need to have in order to be "equivalent" to a sphere?

We provided a list of properties that have long been known to do the trick in the case of a special kind of sphere, the 2-sphere S2, a 2-dimensional surface (like a soccer ball) in ordinary 3-space.

It's natural, in fact inevitable (at least for a mathematician) to wonder whether a similar result applies in higher dimensions. The next dimension up involves the 3-sphere S3, a 3-dimensional object in "ordinary" 4-space.

At this point, most non-mathematicians start feeling queasy and weak in the knees because of all the traditional hype and mystery often associated with "the 4th dimension". In particular, how on Earth can anyone really understand what's going on with that, when admittedly it's barely possible, at best, to visualize what happens in a space of more than 3 dimensions?

Rest assured, most mathematicians can't really visualize it either. Instead, there are alternative ways to reason about spaces of more than 3 dimensions, and they don't depend much on "visualizing". All it takes is some straightforward reasoning, which we will try to explain. (Quite possibly the mathematicians who do this for a living, and actually prove theorems about such things, have a much better than average capacity for visualizing. But that's not very much needed to understand the general ideas.)

Another thing that most non-mathematicians wonder about, besides just grasping what's going on, is how on Earth it is possible to mathematically prove assertions about higher-dimensional topology -- or even about topology in more "ordinary" spaces of 2 or 3 dimensions. Even for people who've managed to cope with calculus in a 2-dimensional "cartesian" coordinate system, it's daunting to imagine how one ever works with stretchy objects on "rubber sheets".

Here's the trick: You start by making the right definitions of the things you're dealing with. This inevitably requires introducing technical language, and a series of very precise -- some would say painfully fastidious -- definitions.

First, a word of caution. Some of what follows is rather "explicit" and possibly not suitable for folks who have little patience with technical "mumbo jumbo". If that's you... sorry. The aim here is to expose a few technical ideas. If this kind of thing isn't your cup of tea, don't say you weren't warned. If you do have patience, I hope you'll find this educational. On the other hand, if you've studied topology already, you may not find much new here.

The first piece of teminology we need is "manifold". We mentioned that in the previous article. It applies to the particular type of topological objects that we need to deal with. This class of objects includes spheres of all dimensions (such as Sn, for n = 1, 2, 3, ....). It also includes all the objects for which it is even meaningful to ask whether they are topologically "equivalent" to a sphere. And we're not going to get to a more explicit definition of manifold in this article, either, since we need some more basic definitions first.

Defining a "sphere" (or an "n-sphere") is relatively easy, and we actually did it in the previous article for n = 1 or 2. To wit, the 2-sphere S2 is defined as the set of points in 3-space, specified by their coordinates (x,y,z), which satisfy the equation x2 + y2 + z2 = r2. That's a little open-ended, since it actually defines a sphere of radius r for some positive number r. Since we're only concerned with topological properties (which ignore stretching), we may assume r = 1.

In order to state the appropriate definition for n-space, where n is any positive integer, we first have to be more precise about "n-space". When n = 1, what we are taking about is the real number line, denoted by ℝ. The generalization for larger n is denoted by ℝn and called n-space. It consists of points specified by n real numbers in an ordered list called an n-tuple, such as (x1, x2, ..., xn). Very likely you have encountered this called a "vector" or "n-vector", etc. Frequently in mathematics, different terms are used for the same thing. Don't be too distracted by the variations in terminology. Similarly, ℝn is often called, redundantly, Euclidean n-space. Don't let it throw you. (Such Euclidean spaces are critical to defining what a manifold is, but we're not quite ready for that yet.)

Given all that, we can define an n-sphere Sn as the set of points in (n+1)-space ℝn+1 where the (n+1)-tuple (x1, x2, ..., xn+1) satisfies the relationship x12 + ... + xn+12 = 1 among its coordinates. Rigorously, it needs to be proven that this set of points actually constitutes a manifold, but that's not very hard to do, once one has the definition of manifold.

The last major term that's not yet been made more precise is "equivalent". As in the question, "when are two manifolds equivalent?" Now, "equivalent" can have many different meanings in mathematics. The notion of "equivalence" can be formalized in what mathematicians call "category theory". Category theory is a very interesting topic, but saying much more about it here would take us way too far afield. Let it suffice to say here that category theory deals with collections of objects that are all of the same "kind" (or formally, of the same "category") and correspondences between two objects of the same kind. The objects are often (but not necessarily) defined set-theoretically as sets of elements (e. g. the points of a topological object), and the correspondences are called "maps" (or "functions", "arrows", etc.). So as not to get too abstract about this now, in the context of topology one thinks of objects as sets of points and the maps as ordinary functions that specify the correspondence between the points of one object and the points of another.

I suppose most readers are comfortable with the notion of "function", but if not, then in the context of topology you can think of a function that maps one topological object to another as a transformation that twists, stretches, or bends one object into a different shape (the second object). This is how one makes more precise the idea of topology as "rubber sheet geometry".

However, not just any map or function between objects is worthy of consideration, as far as topology is concerned. Colloquially, one says that only maps that do not "rip" or "tear" objects need be considered. A little more formally, what is required is that the admissible functions preserve the essential "structure" of the objects. "Structure" is deliberately vague, because it depends entirely on the "kind" (category) of object under discussion. In particular, when talking about manifolds, preserving their "structure" necessarily entails not "tearing" them. In the context of topological objects (including manifolds) all this can be made more precise by defining additional concepts (e. g. "open" sets, "continuous functions").

But I won't further strain your patience by going into more detail about that now. All that remains to do is explain what "equivalent" means in the topological context. Two topological objects ("spaces" or manifolds) X and Y are said to be "equivalent" if there are admissible functions f:X→Y and g:Y→X such that for any element x of X and y of Y, g(f(x)) = x and f(g(y)) = y. In the notation of function "composition", these conditions can be written as g∘f = IdX and f∘g = IdY. IdX and IdY are the "identity" functions on X and Y respectively. They are the trivial functions which take every point of X or Y to itself. (The function f is the "inverse" of g, and vice versa.)

When such a pair of admissible functions exists for the objects X and Y, topologists say that the objects are "homeomorphic". This is what "equivalent" means in this case. Objects that are homeomorphic to each other are the "same" as far as topology is concerned. Each of the functions in an inverse pair, like f and g, is called a homeomorphism.

Finally, in this terminology, the problem that the Poincaré conjecture addresses is to give sufficient conditions, for a given positive integer n, on an n-dimensional manifold X, so that X is homeomorphic to Sn.

Yes, we really needed all that terminology in order to state precisely what it means to say that X can be twisted, stretched, or bent so that it becomes a sphere, without tearing it. What we gain from going to all this trouble is the technical machinery needed to actually prove or disprove such a statement. There's no free lunch.

In the next installment, we'll describe the additional machinery that's needed to state what conditions must be imposed on a manifold X so that it is equivalent (homeomorphic) to a sphere.

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Cassiopeia A - The colorful aftermath of a violent stellar death

Saturday, September 9, 2006

Cassiopeia A - The colorful aftermath of a violent stellar death
A new image taken with NASA's Hubble Space Telescope provides a detailed look at the tattered remains of a supernova explosion known as Cassiopeia A (Cas A). It is the youngest known remnant from a supernova explosion in the Milky Way. The new Hubble image shows the complex and intricate structure of the star's shattered fragments.

Cassiopeia A - Click for 1280×921 image

Physical sciences and technology blog carnival: Philosophia Naturalis

Tuesday, September 5, 2006

In this edition of the Tangled Bank I raised the issue of a blog carnival focused on the physical sciences and technology. It's been a month since then, longer than I hoped, but now the planets seem to be aligned just right.

The carnival will be named Philosophia Naturalis. That's Latin for Natural Philosophy, which Wikipedia describes as "a term applied to the objective study of nature and the physical universe that was regnant before the development of modern science." The name is also a reference to Isaac Newton's 1687 Philosophiae Naturalis Principia Mathematica (although Newton didn't originate the term).

Just as the Tangled Bank focuses its attention on the life sciences and medicine, Philosophia Naturalis will take the physical sciences and technology as its focus. The physical sciences include physics, astronomy, cosmology, chemistry, mathematics, computer science, and Earth sciences. And just as medicine is applied life science, technology is applied physical science, including such topics as nanotechnology, robotics, artificial intelligence, alternative energy, and quantum computing. We also intend to be generous about considering "borderline" topics for inclusion.

Why another carnival? Surprisingly, there seem to be no carnivals out there having this focus. There needs to be a way for people interested in any of the physical sciences and advanced technology to easily read new articles in these fields -- articles that have been judged to be especially noteworthy. There also needs to be a way for people who write about these topics to bring their work to the attention of a wider audience.

The first edition will be published right here, on Thursday, September 14. Future editions will appear at least once a month, and more often if participation warrants. Volunteers to edit and host future editions are most welcome.

But the project needs your help right now. It needs you to submit suggestions for articles to be included. This may be your own writing. However, since only links and brief quotations will be published in Philosophia Naturalis, anyone other than the copyright owner can also send in suggestions of especially good articles they've found.

We're looking for the best articles published on the Web within the past few months that fall within the topic focus. Articles may range from introductory tutorials for a wide audience to more specialized pieces that still may be interesting to educated people with an active curiosity. These need not be blog articles. They could also be any good, short science writing that first appeared in a print publication and has been posted on the Web by its copyright holder for general access.

We have some submissions already for the first edition of the carnival, but we need more. All you have to do is send a short email message -- no later than Wednesday, September 13 for the first edition -- to carnival@scienceandreason.net, with a link to the article and a few words why you're recommending it. Please put "Philosophia Naturalis" somewhere in the subject line.

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