The Remarkable World of Special Functions

The world of mathematical special functions is as intriguing and interesting as any domain, say in physics, or the natural world. The trouble is that one usually doesn't appreciate these functions enough while taking Calculus or Linear Algebra or Differential Equations, and one must wait until a research focus brings them to prominence.

Let's take the case of the Bessel functions. In solar physics one very key equation (Helmholtz) for which the (axially symmetric- in cylindrical coordinates r, z, t) Bessel function solution is:

B_z (r) = B_o J_o(a r)

B_t(r)) = B_o J_1(ar)

t = theta

J_0(a r) is a Bessel function of the first kind, order zero and J1 (ar) is a Bessel function of the first kind, order unity. The Bessel functions are defined (cf. Menzel, 'Mathematical Physics', 1961, p. 204):

J_m(x) = (1/ 2^m m!) x^m [1 – x^2/ 2^2 1! (m + 1) + x^4/ 2^4 2! (m +1) (m + 2) -…

-(1)^j x^(2j) / 2^(2j) j! (m + 1) (m +2 )……(m + j) + …]

for m = 0 and m = 1 forms one gets:

J_o(x) = 1 - x^2/ 2^2 (1!)^2 + x^4/ 2^4 (2!)^2 - x^6/ 2^6 (3!)^2 + ......

J_1(x) = x/ 2 - x^3/ 2^3*1! 2! + x^5/ 2^5 *2!3! - x^7/ 2^7 *3!4! - .....


The equations in B_z, B_t, with the special Bessel functions at root, are critical in describing the respective magnetic fields for a magnetic tube.

For a cylindrical magnetic flux tube (such as a sunspot represents viewed in cross-section) the “twist” is defined:

T(r) = (L * B_t(r))/ (r * B_z (r))

Where L denotes the length of the sunspot-flux tube dipole. If the twist value exceeds 2(pi) then the magnetic configuration may be approaching instability and a solar flare.

Then there is the Gamma function (call it 'G') for which:

G(a) = (a - 1 )!

where 'a' is a positive integer.

Thus, for a = 3:

G(3) = (3 - 1)! = 2! = 2*1 = 2

One can also make use of a recursion formula:

G(a + 1) = a G(a)

For example: G(4) = G (3 + 1) = 3 G(3) = 3 (2) = 6

Check this from the earlier formula: G(a) = (a - 1)!

G(4) = (4 - 1)! = 3! = 3*2*1 = 6

Now, there is also the Beta function, call it B(u,v) which can be expressed in terms of the Gamma functions G(u), G(v)

Thus:

B(u,v) = G(u) G(v)/ G(u + v)

Consider the Beta function B(3, 4):

B(3, 4) = G(3) G(4)/ G(3 + 4) = (2) (6)/ G(7) = 12 / G(7)

where: G(7) = (7 - 1)! = 6! = 6*5*4*3*2*1 = 720

so B(3,4) = 12/ 720 = 1/60

Why all the fuss about Beta and Gamma functions here?

Well, if any readers happened to have caught the PBS series 'The Elegant Universe', they'd have seen string theorist Brian Greene scribbling the Euler equation for string theory on a blackboard:

B(p, q) = G (p) G(q)/ G(p + q)

and arriving at the specific string theory form:

Beta([1- alpha(s)][1 - alpha(t)] =

Gamma (1 - alpha(s)) Gamma (1 - alpha(t))/ Gamma(2 - alpha(s) - alpha(t))

Special functions can indeed be loads of fun and have wide applications. In a future foray I'll look at fractional Gamma functions.

When Your Brother Turns into a Religious Loon

Fourteen years ago, my brother Mike ceased all contact with me after a family dispute. He insisted he would never write, or speak to me again. I didn't ask questions, or inquire why, I merely put it out of my mind. If he wanted nothing more to do with me, the feeling was mutual.

Three days ago, imagine my surprise when an envelope arrived with his name and address displayed prominently. Before opening the envelope I thought to myself: 'After fourteen years with no contact at all, what would he finally have to say to me, if anything?"

I opened it, and there was a Christmas card - with a full manger scene on the front (though everyone in the family knows I am an atheist) and with the words written inside by Mike's own hand: "May the Peace and Love of our Lord and SAVIOR. Jesus Christ, be with you!"

To me, if after 14 years this was his form of contact, he could as well have remained incommunicado. I mean, you don't say hoo, boo or hello to your brother in that long a time then you send a religious card when you know he's an atheist? That's like spitting in someone's beer.

Also inside was his business card which featured his web address for "Pastor Mike's" Net church. I went to his site and checked out various links wherein I saw a number of references to me painted as his "self-proclaimed atheist brother". Clearly, I was being expediently used as a stage prop for his site, to contrast his being "saved" and my being "under Satan's spell". At various other links he implored the faithful to pray for me lest Satan take me.

To take the wind out of his sails, I posted an atheist article I'd recently written about the God-Man myth - and its being plagiarized by Christians from earlier pagan sources- into his "Guest Book".

He then replied hours later, saying:

"I originally planned to delete my atheist brother's messages but decided to leave them so everyone can behold Satan's hold on him!"

Eventually, the article must have proven too Satanically powerful since he removed it as well as a more recent reply to his response. In addition, he password protected the guest book because - in his words :

" I WILL NOT allow my Lord's name to be blasphemed on this site !! , nor allow it to be used as a ' soap box ' for Satan and his demons of deception !!"

At which point I really did believe Mike to be certifiably insane. To reinforce this perception, he further added:

"Due to my atheist brother's spewing his anti - Christ rhetoric in the "Guest Book " section , I was forced to 'password protect ' it , as I REFUSE to give Satan (via this brother) , an opportunity to destroy and pollute any more souls with his 'venom' "

And to that was appended his extended "Open letter" to me recounting his "justified" reasons for locking me out of his site, even to the extent of disallowing a response to his off the wall "Open letter".

Following this segment of his missive:

"I also go by GOD'S LAW - NOT 'MANS LAW ' !! I realize that is a foreign concept to Non-Believers . I am the Lord's servant and messenger!"

I rendered a final response in an e-mail to him, which read in part:

"You're a putz, 'Pastor Mike, and always will be. You say one thing and do another, and I am not even sure you buy half the BS you've reeled out. "God's law" - "Man's law" - that's all diversion. Red herrings. At issue here is how true you are to yourself as opposed to being a pompous, posturing ass. My reading of this letter is you can dish it out, but you can't take it. You have to password protect everything in order to make sure Pastor Mike and his Thought Police encounter no "demonic words" to entrap wayward minds. Rather than provide people a chance to weigh and measure different world views and perceptions, you opt to censor.

Of course, whether Inquisitor or Witch hunter, this approach is always the fallback for the doctrinaire.

As for your “contact” via the Xmas card, it clearly had an ulterior purpose: to try to get us to your deluded “salvation”. You “ceased ignoring” us for that reason alone, not because you genuinely deigned in your heart of hearts to renew viable contact – but because it satisfied your subtext and agenda to exclusiveky renew it on YOUR terms! You wanted contact under the proviso it was understood by me that "God talk" would be included. In effect: "Hello again, and btw, take this spit in your eye, bro!"

As for "needing something" - yes I DO! I need you to remove ALL references about me from your site. Do that and we are squared away. I don't need or want to be a tool" to advance your religious pseudo-bullshit.

And also, don't bother to ever contact me again! I don't want to hear you again, or see anything from you again, ever. "

Hopefully, my religiously -blinded and reason-challenged brother will endorse that plea, and in the process make life easier for both of us. What a pity that one can choose one's friends but not one's family.

The Fed needs to get tough

More and more the economic news is dominated by the sub-prime mess which now threatens to work its way into the fabric of the entire economy. The reason is that the products have been bought by so many institutions, and purchased in confidence. We now know that this confidence was based more on a fiction or mirage than anything else.

In The Financial Times yesterday there appeared one of the most stirring and scariest analyses ever: 'Out of the Shadows: How Banking's Hidden System Broke Down, by Gillian Tett and Paul J. Davies. It noted the "plethora of opaque institutions and vehicles" that have emerged this past deacde in American and European markets. The authors also noted how the esoteric products, namely SIVs (structured investment vehicles) and CDOs (collateralized dent obligations) have been create by a second tier, hidden "banking system" which has effectively taken the loans of banks and repackaged them as these obscure products.

Few people had a clue what these products embodied, but they were packaged as bonds and promptly given the highest bond ratings ("AAA") to ensure investor confidence. As the sub-prime mess has unfolded, we've seen all those bonds were overrated and probably not worth much more than the "junk bonds" of the Michael Millken era.

We now know that the toxic current of this unregulated junk effluent has permeated every nook and cranny of finance, and the other shoe is still waiting to drop. In an article some weeks ago, the FT estimated the liability cost will come to over $1.5 TRILLION when all the ledgers are finally tallied. The pain will be widespread and strike every kind of insitution, from pension plans to insurance companies - which was sold a bill of goods an invested in good faith in these parlous products.

What happened? It is the end and culmination of a process called securitization which actually began after the passage of the Bank Holding (De-regulation) Act of 1984, which sped the way to speculative excesses resulting in travesties such as the S&L scandal in the late '80s.

These included forming a new type of bond known as "collateralized mortgage oblgations" - which turned out to be the "daddy" of today's CDOs. These sprung up in the financial gardens like toxic weeds along with other entities, IOs (inverse only strips) and reverse floaters - promptly branded as "TOXIC WASTE" by the bond traders themselves (License To Steal: The Secret World of Wall Street Brokers and the Systematic Plundering of the American Investor, by T. Harper, page 211).

The CMOs represented the ideal way for mainline banks to dispose of risky mortgage loans they no longer wanted - and didn't wish to deal with. Why deal with such loans, which could always "come a cropper" with defaults on the loans and the bank losing - when one could dispatch them to a secondary entity on Wall Street that repackaged them into thousands of separate loans, and sold them as "bonds" - wherein the CMOs were bundled.

As the CMOs proliferated and no one saw any force or intent of regulation, the geniuses on "the Street" eventually came up with more aggressive concepts - esecially after 12 or so Greesnpan fed cuts made a ton of cheap money available soon after 2001.

Why package just ordinary loans as CMOs, when sub-prime loans could be packaged by the tens of millions into CDOs, or SIVs? The profits would be enormous! Of course, so would the risks, but as usual, Maul Street never took these into account.

The Fed now is caught in one hell of a crunch, between the proverbial rock and hard place. On the one hand it has to bear in mind the inflation risk, and on the other a recession.

My take from reading numerous financial articles is that the former now is much more a threat and in any case, some manner of recession is needed as "medicine". Better to take it now, than later when the pain will be a hundred times more.

The Fed's course is therefore clear: they have to cease from now pandering to the markets, especially the bond market (which continually factors in rate cuts into future bond issues as a method of extortion) and let them sink to the level that reflects the real value of the bogus instruments they circulated. Not to do so is to invite a second bubble on the back of the first, and delay, compound the pain.

This means NO MORE RATE CUTS! Indeed, I predict by early next year - possibly as soon as March - they will have to reinstitute rate increases. The crack money that fed the sub-prime mania has to be shut off. Yes, the withdrawal symptoms will be horrible - but this is the price that will have to be paid for the Street's recklessness, greed and "irrational exuberance" that makes the 1927 peddling of "investment trusts" piddling by comparison.

It is time to take the medicine, sooner rather than later.

Astrology: Astronomy's Specious Sister

In a vain attempt to confer legitimacy on astrology, its proponents, purveyors and defenders often vigorously assert that astrology is astronomy’s older “sister” and paved the way for the latter, genuine science to emerge as what it is today. Maybe, but that doesn’t mean we retain the pseudo-science any more than we retain the use of hornets and wasps to remove toxins from wounds, as opposed to relying on current medicine!

Astrology buffs will also resort to listing names of the rich and famous who have relied (or currently rely) on horoscopes. After citing so many luminaries (including the late President Reagan and the British Royal Family), along with impressive anecdotal accounts of astrology-enriched lives, I am implored to be "realistic". "If such important people as these can embrace astrology, how can it possibly be wrong? How in the world can you possibly deny its validity?"

Actually, it's very easy, because it doesn't really work! Astrology is bogus, because its claims of success are based on false assumptions and outright errors. Any "good" fortune, or "bad" is largely a matter of the throw of the dice. There are no stars which control fortune, just as there are none which determine who is destined to be short, tall, beautiful or ugly. Astrology is based on star configurations two thousand years old, which no longer exist but are treated as if they do! You see, dear reader, there is this very slow process called precession. Anyone who has ever played with a toy gyroscope would have noticed how the axis made a large loop as the gyroscope wobbled.

The Earth acts the same way: like a giant, spinning top. It takes much longer, however, because it is vastly bigger than a toy gyroscope. The whole process for the Earth requires about 25,800 years to go through one complete cycle. Because the polar axis points at different areas of the sky over this time, it means that there is also a continual change in the position of the equinoxes. (These are the points in the sky where the Sun is located on the first day of Spring: March 20 or 21 for the Vernal Equinox, or the first day of autumn: September 22 or 23 for the Autumnal Equinox).

The Vernal Equinox is called the "first point of Aries" by astrologers, since it was originally in that sign two thousand years ago. The problem is that the Vernal Equinox no longer is located in Aries, but in the adjacent constellation/sign: Pisces! In the two thousand years since the Sun signs were devised, the Earth's axis has precessed out of the original alignment, and with it the equinoxes! This means that each of the other eleven so-called Sun signs used in horoscopes is out by the same amount: one entire Sun sign. For consistency, all present day birthdays need to be re-calibrated to the preceding sign. So if your birthday is July 6, like mine, you are no longer a Cancer but a Gemini! If it is October 4, you are no longer a Libra, but the sign just before, Virgo! And so on for each sign of the zodiac, displaced by a full 30 degrees. This means that the entire underpinning of zodiac signs, and their descriptions, is in error! All the personality and temperamental characteristics attributed to "Leo the Lion", for example, really belong to "Cancer the Crab"! This implies that no modern-day horoscope can possibly be correct!

None of the astrologers I've spoken with are overly concerned by this, though they should be for the sake of credibility. They usually say something to the effect that the "real" (original twelve signs) are "forever fixed" as they were two thousand years ago. They insist that my pre-occupation with signs being out of step is not a matter of genuine signs at all, but the constellations (star patterns) associated with them! In their incredible view, the zodiacal constellations are shifted by the Earth's precession, but not the fundamental signs! These signs will remain firmly anchored in the sky, immune to all Earth's motions, forever and ever!

This answer obscures more than it illuminates, and in any case is illogical. What if, for example, the horological school of astrology originated two thousand years later, in our modern era? All the "basic" or "fixed" signs would then be thirty degrees displaced! (Pisces instead of Aries, Gemini instead of Cancer, and so on). Similarly, if horological astrology had been developed two thousand years earlier than it actually was, the signs would be thirty degrees displaced the other way. (Taurus instead of Aries, Leo instead of Cancer, and so on.)

Even if one were generous enough to grant astrologers the benefit of the doubt, other serious problems arise. For example, Greek horological astrology is based on the entirely fallacious concept of a geocentric (Earth-centered) cosmos. In this view (of Ptolemy), the Earth was the center of the solar system and the Sun just another planet going round it! The very brightness of the Sun was like a kind of beacon which, when it appeared in a given sign at the time of birth, disclosed the nature of the person. With the Sun in Leo, one expected a "courageous and fearless" person; with the Sun in Cancer one expected a "reclusive and defensive" person; with the Sun in Taurus one expected a "stubborn or headstrong" person (like a bull). In each case, there is an analogy between the personality traits and the sign's "creature characteristics".

The trouble is that there is no objective basis for supposing any genuine correspondence. The reason is that the stars observed from Earth do not occur in any persistent patterns. Indeed, they are not even all the same distance away! They are purely random, temporary configurations, on to which astrologers have superimposed imaginary images: bull, lion, crab, fish or whatever. Even as I write these words, the stars which make up the zodiacal constellations are speeding in different directions. There really is no such thing as a fixed star, on scales of cosmic time anyway. In about one hundred thousand years, none of the star patterns seen today will even remotely resemble a "Lion", or "Archer" or "Bull" or anything else! I should point out (again) that this would not bother the typical astrologer. Unruffled, he or she would persist in referring to the "basic unchanging signs".

Cunning though they are, astrologers have a harder time justifying their symbolic use of the planets. For example, positive astrological forecasts are the norm when Venus is prominent in one's sign, since that planet is named after the goddess of love. From afar, it also looks like a brilliant jewel set among the other celestial bodies (by virtue of its extreme brightness, third after the Sun and Moon). The truth, astronomically speaking, is that Venus is a hellhole.

However, decades ago, telemetry from the Mariner 4 and Magellan probes, revealed Venus to have surface temperatures of over 750 degrees Fahrenheit and an atmosphere with plenty of sulphuric acid to go with it. If you wanted to locate a single place to call "hell" you couldn't do much better than Venus. The astrologers, however, are concerned exclusively with how it looks in the sky to us, from a tremendous distance. (Though I have at times thought of informing them that Venus' original name was Lucifer!)

Mars is another planet that astrologers invoke in their symbolism. Most often, they attribute a "maleficent influence" to Mars. Its presence is associated with undefined aggression, hostility or some kind of impending confrontation or war. (Naturally, as its color is blood red!) Some astrologers have insisted that those born with Mars in their Sun sign are destined to become warriors, or soldiers. All these associations follow from the color. In truth, the color is not from blood at all but a chemical comprising most of soil of the planet: iron oxide, or rust. As in random star patterns, there is no objective basis for presuming any analogy between human personalities and planets.

It is clear that astrology would collapse under its own weight if the outdated information on which it is based were simply updated, using modern astronomical findings. Astronomers especially wish that this would happen, and the sooner the better. The way they reason is like this: If all those billions of dollars now going into astrology 900-numbers, books and charts were to suddenly stop, it might start reaching some astronomer badly in need of research funding. Projects that are on the verge of being scuttled, for lack of budgetary allocations, could then be saved. The money would have gone towards the acquisition of genuine knowledge about the solar system and universe, instead of supporting bogus "insights" and personal forecasts.

In this regard, another aspect of astrology that has always puzzled me is the lack of any sophisticated instrumentation. For something like two thousand years astrologers have been using the same basic set of Sun signs and charts in arriving at their horoscopes. Because of a lack of instruments, they cannot detect if any new objects are in the zodiacal signs. Indeed, they appear totally oblivious to the prospect of any "hidden" objects there. True, astrologers are now using computers more and more, but these are not observational instruments.

Meanwhile, astronomers have fashioned ever more powerful and sophisticated instruments, many aboard satellites, to reveal a cosmos teeming with dazzling and diverse phenomena from exploding stars to colliding galaxies to black holes. The diversity of the objects discovered is directly related to the diversity of techniques and instruments available.

When asked about this, astrologers are notoriously silent, as if it really doesn't matter. In a memorable debate several years ago, I asked my astrologer counterpart whether astrology reckons the influences of all the colliding galaxies, black holes and x-ray pulsars which have been discovered in the constellations of the zodiac. She said that those objects "belong to astronomy" and: "In any case, the Sun, Moon and planets have far more powerful effects than distant exotic objects". At that point I could not resist referring her to a just-published article by two members of the Committee for the Scientific Investigation of the Paranormal (Kurtz, P. and Fraknoi, A.: 1988, “Belief in the Stars is not a Good Sign”, Skeptical Inquirer.),in which they cited calculations showing the obstetrician at a baby's birth exerts a much more powerful gravitational influence on the newborn than either Sun, Moon or other planet!

"Oh, really?" was her initial response, "I don't think that’s the same as an astrological influence".

Astrological influence? It struck me then and there that she hadn't the foggiest notion of what a gravitational field or force was. She had no conceptual framework for credible (physical) influences that could act: a) between different celestial bodies, and b) between celestial bodies and human bodies. When one thinks about it, however, why should a particular "alignment" of the same planets have any more importance than another? To the astronomer, the difference in gravitational effect is negligible. (The total gravitational force arising from the specious "Jupiter Effect" alignment of planets in 1982 was calculated to produce a tide of only one millimeter on the Sun's surface!)

By contrast, alignments are of pivotal importance to astrologers and can make all the difference in one's fortunes. Certain alignments are "harmonious" and others are "inharmonious". But try to pin down an astrologer about the necessary and sufficient conditions for each. The only response is astrological double talk like "the negativity of a trine is enhanced when Mars is in opposition." What does this mean anyway? In a science like astronomy, precision is demanded by the language, and specific terms have very specific meanings. In astrology, evidently, anything goes!

Most astronomers, understandably, have neither the time nor the patience to peel away all the fictitious layers and pseudo-scientific language to expose astrology for what it is. I've always felt, as an educator, that the effort must nonetheless be made. If scientists are unable to offer an objective critique of astrology, how can they expect students to take them seriously when they insist it is a fraud? How can they expect the thousands of daily horoscope readers to pay more than passing attention to genuine science? How in the world can they expect those who dabble in other astrology-related pastimes, such as numerology or tarot card readings, to sit up and take notice? The answer is they can't if they choose to pretend that pseudo-science doesn't exist.

It would be too easy to ignore the plight of the millions of gullible souls who have been taken in by astrology. The unsympathetic attitude of most professionals is "they deserve what they get for buying into that garbage". But this misses the point, and can be deleterious to all science in the long run. The fact is that a former U.S. President consulted an astrologer, and many businesses and corporations unashamedly employ the services of astrologers to get a "competitive edge". People see this and start to think: "Hey, if these big shots are into it, there must be something to it after all!" Astronomers, like it or not, occupy the final fallback position in the effort to rescue society from this superstitious and pseudo-scientific scam.

Why should a professional astronomer care? Because each dollar diverted into astrology erodes the resource base he depends upon for ultimate support. Each dollar spent on an astrology book or chart is a dollar not spent on genuine knowledge, such as provided by astronomy. The net effect is a further retreat of our nation into backwardness and superstition at a time when it faces stiff competition from more educated foreign competitors. (Who are more likely to market astrology goods and services to us than import it themselves.)

Did I say superstition? Yes, because when the layers of arcane language are peeled away that is essentially what modern astrology is. If one is sufficiently persistent and pressures an astrologer long enough, the truth will emerge. The truth is that the astrological influences of which they write and speak are all psychic in nature. This is why certain alignments are "harmonious" while others are not. It is also why astrologers are not concerned about gravitational forces -they only recognize psychic "forces"!

Astrologers, by employing a clever scientific veneer, entice more folks into taking them seriously. People who would ridicule the idea of going to a psychic or fortune teller have no qualms about consulting an astrologer, since the astrologers' words are cleverly couched in scientific-sounding jargon. The "influence of a spirit" sounds downright medieval and superstitious, but not "The positive influence of the conjunction of Jupiter and Venus in your ascendant third house is highly beneficial".

Why do reasonably intelligent people, many of them college-educated, buy into astrology? There are undoubtedly many cogent reasons. Consider the fact that uncertainty continues to plague many lives: corporate downsizing and wage stagnation continues (at least for middle and working class households), families are being fractured in the time-crunch of the two wage earner necessity, and children are not being parented as a result, with the attendant increases in drug abuse and teen pregnancy. The integrity of the family is thereby eroded and traditional religions no longer seem to have credible answers to these problems. Into this bleak landscape astrology appears with its pat answers and legitimate-sounding jargon, which renders it believable to many.

Most of the people attracted are not ignorant or foolish, or even gullible. In many cases, they are genuinely sincere in seeking meaning or focus in their hectic lives. Astrology not only confers meaning but uniqueness as well. This illusion of unique applicability is a powerful force fuelling the faith in modern astrology. I am always fond of recalling the case of the French psychologist who advertised himself as an astrologer back in the 70's. He received hundreds of requests for his services, in response to which he sent out a single, ambiguous horoscope. The noteworthy point is that over two hundred grateful admirers took the time and trouble to write and thank him for his "accuracy and perceptiveness".

The job for the astronomer, or any scientist, is to show how meaning in life can be restored without the need for astrology. This is not an easy task. Most astrologers offer a simplistic explanation for the troubled times around the globe: it is still the "Age of Pisces" (a sorrowful period), but it will give way to the "Age of Aquarius", when a new era of unparalleled peace and prosperity will be ushered in. With such knowledge in hand, the difficulty in confronting the future is minimized. Living life from day to day is made bearable, where formerly it was intolerable.

By contrast, science imposes a remorseless and stringent demand for objectivity. There is no place for any scheme that bestows meaning on human affairs by codifying celestial events. The scientific view insists it is supreme arrogance to suppose that the entire vast universe is arranged only as a key to understanding the events on a remote, dust-speck of a planet - or worse, the personality traits of one of its billions of temporary inhabitants. This outlook violently contradicts the astrological notion of cosmic objects existing only as a hidden code for interpreting human lives. (Of course, with the advent of Derridian postmodernism, even the objectivity of science has been questioned severely – but that mode of solipsism is a topic for another article!)

How can we recapture the imaginations of those attracted to astrology? One of the most effective techniques I've seen appeared in the first episode of the highly acclaimed video series The Astronomers. In it, the former monk and telescope maker John Dobson is seen traveling around national parks of Oregon and towns in California, setting up his telescopes for the public to view. He does this for free, and treats each passerby to a spectacular scene, whether lunar craters, Jupiter's moons, Saturn's rings or sunspots during the day. The people are, without exception, taken aback and in a state of awe. They are then ready to listen to some of the information which Dobson has to impart. In his own little way, he is undermining the advance of astrology and its cousins by showing the richness of the natural - rather than supernatural, universe.

Most astronomers would not be interested in this sort of approach. Based on my own experience, and the astronomers I've known, they are more inclined to reclusive research or observation. As a rule, they tend not to be extroverted and colorful showmen or carnie barkers with telescopes! The next best thing, however, would be to offer special astronomy courses at a nearby community college. Such courses could be given at night, for no credit, to mainly adult students. This would be an investment of barely an hour or two each week to lecture and answer questions, and the return would be enormous.

If each astronomer could find a way to rekindle people's latent curiosity about the real universe, there would be no need for that curiosity to be squandered in search of a bogus one. Astrology will then die naturally, and I very much doubt anyone will mourn its passing.

The Abuse of Pluto

As much as democracy can be a boon in terms of advancing a common will (or at least majority will), it isn't a model for science to adopt. This was abundantly evident last year in the case of the demotion of Pluto from planetary status.

Worse, I surmised it was only a matter of time before various right wing sites and forums would use Pluto’s demotion as a basis to attack science. In particular, the argument goes: “If science had to backtrack on Pluto, taking away its planethood, why not on evolution and global warming as well? Shouldn’t all science be regarded as ‘tentative’?”

In fact, not.

For one thing the “consensus” achieved on Pluto was accomplished not through objective inquiry and scientific test, but rather a vote at a recent meeting of the International Astronomical Union. As a member of the American Astronomical Society, who totally disagreed with this process, it pains me to write this - but any time a scientific conclusion is reached by vote, it’s time to raise the red flag!

The modern consensus on evolution, meanwhile, was achieved by a body of empirical evidence, including from the fossil record, microbiology and genetics. For example, one of the most powerful lines of evidence has shown that the human chromosome designated '2' was the result of the telomeric fusion of the two ape chromosomes, 2p and 2q. The effect also saw the reduction from 24 chromosome pairs in apes, to 23 pairs in humans. (Source: Yunis and Prakash, 1982, Science, Vol. 215, p. 1525, 'The Origin of Man: A Chromosomal Pictorial Legacy')

Mathematical precision also consolidates scientific discovery in quantitative fields, making them less likely to be overturned. Particle physics is one such example, wherein the original three sub-atomic particles (proton, neutron, electron) have now become a veritable “zoo” numbering in the hundreds – including up, down, top and bottom quarks, W and Z bosons, electron, mu and tau neutrinos and many others too numerous to mention. This panoply of particles didn’t just manifest because the respective particle physicists intended to be “mean” to school children or regular mortals. It emerged out of more refined and detailed experiments that exposed each of the particles.

The same applies to the current consensus on global warming, only arrived at after years of computer analysis of ice cores, changing CO2 concentrations therein, satellite imagery disclosing retreat of polar ice and data from over 1300 temperature sensing bouys in the oceans.

In the end, the error of the planetary astronomers was to cave in to public pressure to avoid “complexity”, e.g. in adding numerous additional planets like Charon, Xena, Ceres etc. This, despite the fact the original IAU definition was perfectly rational in its criteria for a planet.

Alan Stern, executive director of the Space Science & Engineering Division of the Southwest Research Institute – and Principal Investigator for NASA’s New Horizons mission to Pluto- observed that the new planet definition was “sadly flawed, particularly due to the vagueness of the third condition”, e.g. clearing the neighborhood around its orbit – which might also disqualify Earth!)

He added: “A lot of people are going to ignore the (new) definition because it doesn’t make sense.” (Source: Eos, Vol. 87, 29 August, p. 350)

The tragedy of the Pluto vote is that it has not only set back planetary astronomy – showing the preponderance of ego over scientific inquiry – but also public confidence in science. In any case, this retrenchment of scientific position gives no aid or comfort to those who would eschew either evolution or global warming.

Time to get real on global warming

As the Bali Climate Change Conference goes on, there is still far too much quibbling over whether anthropogenic global warming even exists. All of this back and forth is costing valuable time, and causing an incipient problem that might be at least partly corrected, to metastasize to a possibly ungovernable situation - perhaps even a "runaway Greenhouse effect" and transformation of Earth to another Venus within 1ka (one thousand years).

The inherent problem with most skeptics lies in their carelessly invoking “natural warming” cycles – which simply don’t hold up to investigative scrutiny – in terms of the magnitudes of energy input required for the level of warming observed the past century.

Solar physicist John Eddy, who made it is research specialty to study long-term solar variations connected to climate change, noted the period of 12th century warming in his book, ‘The New Solar Physics’, AAAS Selected Symposium, Westview Press, 1979, p. 17.

Eddy noted that this coincided with a period of higher solar activity (i.e. more sunspots) and possibly greater luminosity – on account of the fact that the irradiance is amplified around sunspots owing to redirection of convective heat flow. (Bear in mind the plasma in spots is at lower temperatures, by about 1500C, because of the powerful magnetic fields in them).

During solar cycle 20 – when I also conducted investigations on solar flares and their effects- the then Solar Max satellite used an active cavity radiometer (ACRIM) to measure temperature increases arising from higher activity – especially as generated by more convection at the periphery of large spots. The differential was something on the order of 0.1C at the Sun! Since the radiant energy must now transit 150 million kilometers, and its intensity falls off as the inverse square, one can see this would translate into negligible increases at Earth.

What about longer period increases in solar luminosity associated with its possibly being a variable star – as opposed to sporadic sunspot outbursts?

The maximal magnitude of inherent solar -induced climate variability was probably first highlighted by Sabatino Sofia et al in their paper Solar Constant: Constraints on Possible Variations Derived from Solar Diameter Measurements, in Science, Vol. 204, 1306, 1979. Their estimate was a solar change in irradiance of roughly 0.1 % averaged over each solar cycle. (Irradiance is a measure of the energy per square meter received from the Sun).

Thus – if the solar irradiance effect at Earth (solar constant) is normally about 1360 watts/m^2, this would imply an increase of roughly 1.36 W/m^2.. The problem is that there is no observational evidence to support this in the warming period of the 12th century, or any time in the past century – when global warming spiked to serious levels. (Some like Sofia have argued that even if it had occurred, it would only engender a temp. increase contribution of perhaps one-fourth of one degree, or significantly less than what has been documented.

More recent space-based observations appear to show a variation in solar irradiance of at least 0.15% over the standard 11-year solar cycle. (E.g. Parker, E.N., Nature, Vol. 399, p. 416). However, even with this higher percentage ascribed to solar changes, the heating effect is nowhere near comparable to that induced from man-made global warming. (See, e.g. Martin I. Hoffert et al, in Nature, Vol. 401, p. 764).

As the authors in the latter study point out, the heating component arising from greenhouse gas emissions from 1861-1990 amounted to anywhere from 2.0 to 2.8 watts per square meter. The solar variability component detected over the same period amounted to 0.1 to 0.5 watts per square meter. Thus, even the MAXIMUM solar variability amounted to only a fraction (25%) of the MINIMUM power input from human-induced greenhouse warming!

Most serious climate researchers (such as Gunther Weller – who was at the University of Alaska, Fairbanks when I was there) already believe we're in the first stages of a positive feedback, non-linear effect that is leading right into the maw of the runaway greenhouse. Though most will not own up to it publicly for fear of raising alarm.

The basis has already been described by Sagan and others: Melting of ice caps (already occurring) results in diminished albedo (reflection of solar radiation back into space), and a darker Earth surface - with more IR (infrared or heat energy) absorbed - enhancing global warming. At the same time global warming is accelerated in the oceans, and both El Nino and La Nina are ramped up in the unfolding panorama of global warming (cf. S. George Philander in Eos, March 31, 1998, ‘Who is El Nino?’) . In the meantime, we have new evidence that the melting permafrost (e.g. in Alaska) is releasing 25- 100 times more greenhouse gas as methane and CO2.

As more ice melts from the polar regions, positive feedback proceeds faster. The overall (mean) ocean temperature continues to rise - ultimately becoming too hot for any marine life- and reaching equilibrium temperature somewhere in the next 500 years. All ocean currents, circulation systems will, of course, cease. With atmospheric circulation soon following (as on Venus) , all solar energy going into heating the oceans until their specific heat capacity is reached.


Of course, long before this we will be in a state of emergency. When I was at the Geophysical Institute in Fairbanks, AK I used to chat with some of the Atmospheric scientists there. Almost all agreed that what we may have on our hands will be many times worse than the Luis Alvarez' proposed extinction asteroid impact of 65 million years ago.

People - when they talk blithely and ignorantly of 'adjusting' simply have no idea of how bad it can get. Or that a 'runaway greenhouse' (such as converted Venus to a hellhole) is more than possible. Can they 'adjust' to global mean air temperatures of 60 C (yes, that's Celsius degrees)? I don't think so, especially as all the oceans will have boiled away!Yet, this is our inevitable fate if we don't get a handle on the problem RIGHT NOW. Before the 'runaway' sets in when CO2 starts to be 'out-gassed' from the oceans, and all the trillions of tonnes of carbonate rocks on the planet.

Long before that, of course, a host of exotic diseases will have spread across the world and wrought havoc - and all the fools clamoring for warm temperatures will wish otherwise with the first case of amoebic dysentery or dengue hemorrhaghic fever. Especially as we're already losing efficacy of anti-biotics from over-use(See, e.g. Global Climate and Infectious Disease: The Cholera Paradigm, in Science, Vol. 274, 20 December, 1996, p. 2025.)

And I won't even go into the tech predictions of regularly fried power-grids because so many teeming millions are trying to keep cool.

The late Carl Sagan noted in a CNN interview (with Ted Turner) ca. 1989 (still have it on tape) that the limits for catastrophic climate change- we're talking species exterminating magnitude- are not as high as many think. In fact, he cited the tolerance increment of six degrees Celsius. When global mean temps. veer past that, watch out. Right now - from all data collected- we appear to be approaching a 2 degrees Celsius increase, possibly more, by 2100.


Even if we initiated massive emissions cutbacks today it may be inadequate, since the carbon dioxide deposition time (time for it to remain in the atmosphere and create problems) approaches 100 years, as Sagan (and others) have also noted. Thus, the climate change we're seeing NOW, is really traceable to the conditions of CO2 concentration from ca. 1903!!!Our CO2 inputs, will become recognized ca. 2103. And be added to the cumulative total at THAT time..

Now, while not everyone yet agrees we are in such an emergency – most climate scientists of any repute argue that we cannot afford the luxury of waiting any longer for “absolute proof” to arrive. We must act now to initiate credible policy changes (not merely exchanging "carbon credits") or face dire consequences – and have future generations curse us forever for our inertia and inaction.

Further Speculations on the Nature of UFOs

French astrophysicist, computer scientist and UFO researcher Jacques Vallee first formulated the hypothesis that UFOs represented projections from the human mind-brain. Vallee's thesis rested on the premise that humans are not really aware of what they are doing. This leaves open the possibility that each human is the unwitting 'author' of a UFO event. This is why the conceivable “meta-cosmic” origin of UFOs needs to be understood.

I first began to take this seriously after publishing a 1980 paper on a UFO report submitted to me, or what I then called 'transient optical phenomenon of the atmosphere'. I was enticed, in the wake of this, to look more closely at the total frequency of such sightings around the world. When I discovered there were an average of 10,000 each year, I realized these could not all be 'intelligent craft from other planets'. It was incredible, in my estimation, to have 10,000 (or more) interstellar craft from maybe a thousand other planets, visiting us each and every year!

The problem with the projection hypothesis resides in ascertaining where the underlying energy originates. Clearly, UFOs are not merely 'images' or mass hallucinations, but manifestations of light, heat and even electromagnetic waves (since they also show up on radar screens). The question: How can projections - ostensibly of the human mind-brain, become endowed with substantial physical energy that registers on detectors? In order for this to be possible, the brain must possess the capacity of an enormously powerful transducer.

Consider: the brain is a three pound repository of chemical and electrical energy. The latter, for example, can be measured on a special device known as an EEG (electro-encephalogram) which can pick up the telltale brainwaves. But how can this low-level electric energy be converted into the enormous energy that appears to emanate from typical UFOs? Somehow a dramatic amplification of the low-level energy occurs, coupled with a change in the form of the energy. (For example, from electrical to heat or to light).

Presently, there is only one way for this to occur. It has to do with very tiny structures within the brain’s neurons called “microtubules” and how they function in conjunction with a newly discovered form of matter called bosons. Some have conjectured that a Bose-Einstein condensate forms in brain microtubules when particles called Goldstone Bosons become locked into a single coherent state. In this situation, an amplification of energy occurs in the brain's electrical system. This becomes convertible to other forms. Now, imagine an "alien" brain thousands of years more advanced than our own, and perhaps using 99% of its potential as opposed to barely 2%.

In this respect, it is conceivable that the alien brain's microtubules might act as a kind of electro-magnetic waveguide.. The Goldstone bosons undergo a phase transition (change of state) in the brain.

They then propagate outward from the brain as powerfully amplified electromagnetic (EM) waves, but driven by the nonlocal quantum potential first described by David Bohm. Certain calculations using the Planck length (10^-33 cm) suggest that ample energy exists to explain the apparent energetic UFOs that appear as luminous discs, or leave radiating patches on the ground.

So what exactly what might the UFOs be? Ultimately, pure zero-point energy emanating from a brain that is integrated with the "vehicle" denoted "UFO". That is, energy, but not constrained or limited by the sort of mechanical-chemical constraints and laws to which cosmic energy is naturally subject. This energy, calculated using the energy-time uncertainty relation, is virtual energy. For example, all cosmic energy is constrained to obey the law of conservation of mass-energy. Whatever quantity of energy appears (say in a nuclear fusion or fission reaction) cannot exceed the value of whatever mass difference made it available, multiplied by the speed of light squared. Virtual energy, by contrast, can violate this principle for limited durations of time (at least in the cosmic realm), but leave enduring effects in its wake.

Further, cosmic energy is subject to degradation (entropy increases) since it is used again and again. Not so with zero-point energy. If UFOs are governed by prosaic energy-mass, they would degrade in quality over time. The observed energy density, say as photons contributing to their brightness, would diminish (due to photon wavelengths being lengthened toward longer (low energy) regions of the spectrum. A zero-point or virtual energy source does the opposite, maybe even brightening over time.

In addition, while ordinary mass-energy is interwoven in the 4-dimensional space-time continuum, zero-point energy is hyperdimensional. (See Bohm;s point on this as related to his “implicate order” in his book, ‘Wholeness and The Implicate Order’, p. 191) In this regard, the 4-D space-time continuum arises as a construct from the essential interchangeability of mass and energy through Einstein's famous equation: m = E/c2. This is the core concept of Near-Reality energy. By contrast, hyperdiemsional or “far reality” energy is derived from the quantum potential and the Dirac Ether - and is essentially infinite[1], for any scale size approaching the Planck Length LP. This means that the mass-energy link (defined by the Einstein equation) is not a necessary condition of virtual energy.

The preceding features are important facets of UFO behavior to grasp. One of the most interesting is the apparent deformation of shape that UFOs undergo, including disappearance. Thus, a highly elongated UFO is observed to progressively contract until it becomes a point, then vanishes. In the context of Near-reality predicated on prosaic mass-energy and 4D space-time, this simply does not compute. Certainly, in the same frame of reference, clocks and measuring yards remain constant and not slow down, or shrink.


UFOs clearly are not in our frame of reference, though they appear to be. In fact, UFOs cannot exist in what we call space-time. As emanations of metacosmic or Dirac (zero-point) energy, the UFO shares the hyperdimensionality peculiar to certain D-branes, for example. The specific dimensionality depends on a number of factors, including: the Bose-Einstein condensate which gave rise to it, and the consciousness on which that is contingent.

In a subsequent post I intend to get into these D-branes in much more detail.

A relevant question is: How can a UFO be visible if it is 5- dimensional, say? This is possible through interpenetration of the mutual dimensional planes: applicable to the UFO, and to our normal 4-D cosmos. Below I make use of an analogy that first appeared in the classic book by E.A. Abbott: 'Flatland - A Romance of Many Dimensions'.

Assume you’re a denizen of a two-dimensional world, confined to the plane of that world. How would you be aware of contact with a hyper-dimensional entity? Imagine a 3-D ellipsoidal disk (like a UFO) as an example, which is 'hyperdimensional' compared to a 2-D plane (like a sheet of paper). To enable detection, the disk must progressively intersect the plane. The observer sees a mystifying object suddenly grow to vast size, then contract and finally disappear! (See the diagram below - where the horizontal line is a 1D infinite object, intersected by an elliptical 2D object, which might be the 'cross-section' of the UFO)

(O)
----------- A

----( )----- B

------------- C

In the changing situation A to C, the interpenetration is such that the dimensions of the intruder appear to contract or shrink. But this is only a consequence of its departure from our own (linear, in this case) dimensionality. This is precisely analogous to the case of UFOs regularly observed to 'shrink' or contract.

The phenomenal velocities of many UFOs are also explainable using zero-point, hyperdimensional energy. In many UFO sightings, a common feature I call 'skipping' manifests. The UFO appears to be at one location, say over a building, and in the next instant far displaced, without having traversed the intervening distance. It appears to have simply jumped or 'skipped' the intervening space from A to B. This sort of quantized displacement is another example of hyperdimensional influence, since no normal macroscopic object exhibits quantum jumps in transport. However, a nonlocal and hyperdimensional agent can, because it possesses an inherent ability to 'fold space', e.g. a D- brane-space..

Again, I reiterate that I am not discounting that a small proportion of UFOs could be genuine, nuts and bolts extra-terrestrial artifacts. But this percentage is so low that not more than 2 or 3 sightings per decade (out of 100,000 plus), fall in this category, if that many.

Humans and Carrying Capacity - Why care?

“Carrying capacity” and whether humanity can attain it in terms of its population on Earth, may determine if the species survives or becomes extinct. The concept itself is not that difficult a concept to grok – and makes perfect sense when one thinks about it.

For example, a spacecraft with 3 astronauts sent to the Moon has enough food & oxygen supply for a five day round trip. For those specific conditions, ‘3’ is the carrying capacity of the craft. ‘Three” is the maximal population that the craft can sustain given its food, water, fuel supply.

Double the number of astronauts and you must double the resource supply for the craft, or you have to reduce the time of the journey by a factor two. There is NO other way! (And bear in mind for a spacecraft redundancy of resources is built in just in case of an emergency - e.g. a micro-meteorite puncturing the shell of the craft and causing oxygen loss).

In many ways, Earth is like a self-contained “space craft”. It is isolated in space, not infinite in extent, and its resources (fuel, water, food supply) are also finite and set for a certain maximum consumption – unless the inhabiting species radically alter.

In determining this, one must bear in mind that a planetary population’s impact extends vastly beyond immediate space, living habitat. Recently some studies (e.g. Millennium Ecosystem Assessment Project) have translated impact into hectares consumed over a lifetime. The avg. – when all humans are factored in – amounts to nearly 2.3 hectares per person reckoning in crops and meats-cattle etc. consumed over a life not to mention water, other resources. No high population density scheme that doesn’t factor acreage impact in can be taken seriously.

Americans consume an exorbitant amount in planetary hectares – around 10 hc per American – which is exactly why the planet can’t afford “two US of As”. (Not that China is paying much attention as they ramp up their economy!) It also means the distribution is that much less for other nations, peoples. In general then – the base relationship defining carrying capacity for Earth is:

Carrying capacity = (usable land-water resource base providing water + food + fuel) / (individual food, fuel + water requirement)

Now, if the numerator is ~ 11.4 x 10^9 hectares of usable aggregate equivalent land-water resource base and if 6 hectares is the ideal "mean individual requirement" over a lifetime (e.g. meet all basic needs and have a few private luxuries) , that means:

CC = (11.4 x 10^9 hectares) / 6 hectares/person ~ 2 billion

Obviously, this can be increased if the numerator can be increased or the denominator (each individual's ecological footprint) decreased. The problem is how to achieve it? (Especially if the total population continues to increase at 2-3% per year)

As we know, all three land base contributors to the numerator (land base for food supply, land base for water supply, land base for fuel supply) are decreasing. By 2025, for example, non-contaminated potable water hectares is expected to be roughly halved.

This means, according to ‘The State of the World’ (2000) report notes (and at present rates of declining watersheds, and increased population) the number of people living in water-stressed countries will rise from 470 million to 3 billion by 2025. This is more than a sixfold increase. Add in projected new climate change data and the expanding drought, desertification regions - and likely effects (see. eg. recent issues of Eos) and the stressed populations increase nine or tenfold.

Similarly, arable land supply hectares are diminishing owing to drought from climate change and encroaching development on previous farmland (here in Colorado, for example, farmers have sold nearly one-fourth of their lands to developers). True, people could make do with less – especially Americans- but WILL THEY? Short of major political pressure there is no reason at all to become optimistic that Americans will change their piggish, planet greedy ways on their own. (E.g. the U.S. with 5% of the world's population consumes more than 25% of its oil)

Technology is great BUT: 1) Technology means nada if the political will to put the money where gums are flapping doesn't exist. One could as well be talking about zero point energy as savior. 2) Technology is not an energy source but energy consumer and often an energy use accelerator as Matt Savinar has noted in his excellent online text, ‘Life After the Oil Crash’. .

Thus again, technology and even maximal brain power to develop it means nada if the energy sources aren't there for its expeditious use and distribution. To put it another way, Prof. Wizardo may design a super-duper automatic crop creator to use in deserts. However, if the energy E isn't there to: a) run the damn thing (say 10^7 Joules per month), and the water W (hectares) isn't available to irrigate the result, his device amounts to a waste of time, effort and more resources. And Prof. Z could have an IQ high enough to qualify got the 'Triple Nine' society and it still wouldn't matter. Without the resources to enable his invention, it's dead on arrival.

Isaac Asimov in one of his early essays (in his anthology ‘The Stars in Their Courses’) on population and carrying capacity, put it bluntly:

“Humans have a simple choice: they can either decrease their numbers on their own to live within the Earth’s carrying capacity or they can let nature do it for them”.

With the emergence of antibiotic resistant bacteria such as c. diff. and MRSA in recent years, it more and more appears the latter is what is transpiring.

Of G-O-D and God-concepts

With an election year approaching, religion has once more reared its ubiquitous head, as we behold candidates competing to establish (for the media’s sound bite benefit) which party is the more “godly” and righteous. Or, which candidate is the truer Christian.

The main problem with religion in this country, especially the version peddled by the Christian Right - is the lack of temperance and the prevalence of "god talk". They forget that the mere existence of the noun does not mean there is any existent behind it - or (even assuming there is) that it is in any way understandable by a finite human mind.The tendency to posit a superior, supernatural being to account for physical existence is nearly universal in every human culture. Indeed, every major sacred book - whether Bible or Talmud, or Koran, attributes a divine origin to the universe.

The problem inheres in taking this literally, and then extrapolating to an entity 'God' - that allegedly is involved in the cosmos.On a more Materialist level, of course, the entity isn't required. According to the energy-time uncertainty relations, a quantum fluctuation can provide an immense quantity of energy if the corresponding time in which it occurs is exceedingly small.

This means the universe could have conceivably brought itself into existence by a (random) quantum fluctuation. This process, called "quantum bootstrapping" is now the basic answer provided by physics in regard to "what came before" the Big Bang.

I have been asked after some lectures I've given: "Well, be that as it may, where's the harm in at least acknowledging the possibility of a divine or supernatural origin for the Big Bang - admitting that while the detailed evidence may not be available, the possibility cannot be dismissed?"

This is a fair question, but misses the point. As a limited human enterprise, in terms of funds and resources, science is compelled to make judicious (and painful!) selections of problems to investigate. Indeed, the problems posed by the natural world often tax the resources of science beyond its capability to extract practical solutions. Now, add to this the (pseudo-) problems of a realm that no one can be certain exists, and it becomes clear why scientists are averse to venturing beyond their domains. At least this is the attitude of most natural scientists, competing for scarce funding.

At another level, science excludes all hypotheses which are judged impractical in terms of confirmation, or empirical test. God is such a hypothesis. What determines whether a hypothesis belongs in the excluded category? For one thing, whether the proposed entity can be defined in terms recognizable by science. For scientists, definitions are "operational" - which means they are framed in terms of other (familiar) scientific concepts: energy, fields, mass, volume or whatever.

However, if a concept falls within the purview of Godel's Incompleteness theorems, then it isn't addressable by science. 'God' is such a concept, since there aren't enough testable axioms or tenets to prove it - or even to identify the necessary criteria for adequacy of operation (which these days more often than not passes for what we call "proof"). Thus, "deity" is unprovable by any system of axioms that can be conjured up by the finite human brain or collection of brains.

Put another way, the typical human brain can make 'x' statements about "God" - but these will always be at least (N- x) short of encapsulating the concept in fullness and adequacy. The gap between the statements that can be given and must be given is usually referred to as the "undecidable propositions".

Now, some aver or say that just as they may have a "faith" in deity or "God", I have "faith" the Sun will rise each morning. This is not quite the same thing. In my case it isn't so much any "faith" (in terms of the definition or acceptance of sights unseen) but rather pre-supposition predicated on a host of predictable past behaviors! This is always the case when when experience repeatedly validates that the probability of controversion of the underlying physical laws is null.

For example, I can “pre-suppose” that each and every time I get up in the morning I will not suddenly levitate upwards, with the law of gravity demolished. I can pre-suppose, based on my years and years of observed experience, a law of gravity exists - even if I may never have seen it crafted verbally or demonstrated scientifically. Similarly, I can “pre-suppose” that the Sun will rise every morning, and set each evening, albeit not at the same time. But the rising and setting themselves are indicative of larger dynamical laws to do with the Earth’s rotation.

In addition, I can “pre-suppose” that as I’m writing this, my computer keyboard will not suddenly de-constitute into individual atoms, molecules. One may not “know” the exact laws that apply (in the case for atomic physics) but one’s recurring experience validates it to high confidence levels.Even if a scientific or research hypothesis may include some open or meta-statements (evidently leaving the door open for undecidable propositions) there are nevertheless empirical checks and tests that can close the system parameters.

Consider this relatively simple example from celestial mechanics:

the magnitude h, of the angular momentum vector is:

h = r x r’ = r x (dr/ dt) =

(y z’ - z y’)
(z x’ - x z’) = (C1 C2 C3)
(xy' - yz')


so (r x r’) = (C1/ h, C2/ h, C3/h)

and inserting variables one finds:

C1/ h = sin U sin (i)

C2/ h = - cos U sin (i)

C3/h = cos(i)

where U is the longitude of the ascending node, and (i) is the inclination of orbit

Now since (i) is known (23.5 deg) and therefore cos(i) can be determined, then sin(i) can be as well.

Also h can be determined, since: h = C3 / cos(i) = (GMm a (1 – e^2)^1/2

where all the constants are known (a = semi-major axis of orbit, e = eccentricity of orbit)

We also know: h = (C1^ 2 + C2^2 + C3^2)^1/2 = [C^2]^ 1/2

and we can take:

(C1/ h)/ (C2/ h) = sin U sin(i)/ [- cosU sin(i)]or C1/ C2 = - tan U

We basically already know, from the above (and using some basic algebra), that:

k (const.) = (C1^2 + C2^2)^1/2

Also, U = W - w, where W can be obtained from a table based on observations, and w can be obtained using a Fourier expansion of the mean anomaly M, e.g.

w = M + (2e – e^3/ 4) sin M + 5 e^^2/4 sin 2M + ... etc.

Once U is known, C1 and C2 and C3 are known, and there are no circles or open ends

Similar limit provisions and closed operational explication do not apply to "G-O-D". At another level, it is of interest to explore how deity varies as a concept between religions. This discloses there can't be one uniform human perception for the concept. To fix ideas, the typical western Christian regards his or her deity as a personal God, while the typical Hindu regards his or her deity (Brahmin) as impersonal.

Even if all other things are equal, how can there be such a vast gap in human perceptions? The fact of such a perceptual chasm must mean either: a) God does not exist - at least as specified by either group, or b) God exists, but no human mind is capable of grasping even the most elemental conception accurately.

Philosopher Joseph Campbell once observed:

"God" is an ambiguous word in our language because it appears to refer to something that is known. But the transcendent is unknowable and unknown. God is transcendent, finally, of anything like the name of "God". God is beyond names and forms.

Thus, it seems the more one uses the term "God" the less he or she is really committed to it, or respects the underlying concept. The more the noun is bandied about, in other words, the less

power is derived from it, and the less it impresses one's fellows.

Based on Campbell’s explication that the noun G-O-D omits vastly more than it embodies, it seems to make sense that the implicit use of god-concepts (as opposed to asserted absolutes), reinforces a judicious attitude of cautious forbearance.

The implicit relativism acts as a restraint, hopefully backing the believer away from a militant stance of absolutism. Ideally, this should dispose him or her to be more tolerant: tolerant toward unbelievers, and tolerant toward those of different religions. Far from conceding to evil, this necessary acceptance of relativity offers an escape from evil. It is an admission of intellectual humility. An admission that human brains are too limited in capacity and function to access the fundamental answers to life - or to have an exclusive grasp of the "one, true God", somehow denied to all those of other faiths.

The use of the term god-concept also recognizes implicitly (by acknowledgment of a finite intelligence confronting an assumed "infinite" entity) that the question of the existence of God is certainly unanswerable. (If it were not, then any believer ought to be able to set out the necessary and sufficient conditions for his God to exist.)

In this sense, it is important to be able to distinguish the nearly universal allegiance to god-concepts from the separate issue of the factual existence of a deity. In other words, the widespread use and appeal of god-concepts does not necessarily mean that there is a genuine correspondent in reality, supernatural or otherwise. Acknowledging the god-concepts at the center of all God-talk instead tempers the absolutist presumption that merely saying a word establishes the underlying (assumed) reality.

An Excursion into Algebraic Homology

Algebraic homology is a branch of topology that is used to analyze higher-dimensional structures. This is accomplished by first converting them into flat, two-dimensional configurations, then assigning algebraic symbols to each 'dimension' (chain). Let's consider a relatively simple example: the basic torus pattern shown above:

If one were now to fold over the left and right sides so they join, ABA-left to ABA-right, s/he would be well on the way to re-forming the torus. Taping the two sides together, for example, would form a cylinder or straight tube. To complete the process, one simply joins the oppositely situated circles, ADA-top to ADA-bottom.

In the sketch, four basic regions are shown - defined as separate homology spaces. Each is of a different ‘equivalence class’ as well. Starting with the lower left corner ‘space’ and going around clockwise, these can be written:

BCDA
CBAD

ADCB
DABC

with each one representing an individual rectangular region within the whole. In a comprehensive approach, arrows are used to define consistent directions, and either numbers or Greek letters are assigned to the box sides. This is for ease of identification of the particular equivalence classes.

For example, arrows assigned to segments AB and BA on both sides of the shape shown above are made to point in the same direction, say top to bottom. The same direction implies two sides have to blend together when connected. A similar consideration applies to the bottom ends (AD + DA) when joined. So that the arrow from A to D on top would match an arrow direction from A to D on the bottom.

Thus, for the ‘top’ side of the torus:

A ---->-----D ----->------ A

and, for the ‘bottom’ side:

A ---->-----D -----> ------ A

One could go one step further, as I indicated, and assign Greek letters to the different segments. For example:


ALPHA: A ---->-----D -----> ------ A

BETA: A ---->-----D -----> ------ A


We now have a one-dimensional homology space (H1) denoted by:

H1 = (ALPHA + BETA)

The same applies to the complementary homology space (H1') that runs vertically so as to join the left and right sides, which we might denote by:

H1' = (DELTA + GAMMA)

These are not just homological spaces but cycles - that are themselves not boundaries. For example, one large cycle would be made by going around the outermost ‘space’ in a clockwise sense, astarting from the ‘A’ in the upper left corner. We would have:

(A-D-A) -> (A-B-A) ->[-(A-D-A)] ->[-(A-B-A)]

where the -(minus) signs precede the last two terms and help to distinguish their direction from opposite the ‘positive’ space- defined above. This could also be written in a shorthand form:

H1 + (H1') - H1 - (H1')

It can clearly be see that the ‘boundary’ vanishes, since both pairs of sides (H1, H1') cancel out (having opposite signs for opposite directions). This can, of course, be written to include the ‘space’ elements:

[(ALPHA + BETA)] + [(GAMMA + DELTA)]- [(ALPHA + BETA)] - [(GAMMA + DELTA)]

whence we clearly see mutually cancelling space elements

Note here that 1-cycles in a triangulated space can be generated by closed curves of the space formed by the edges of the triangulation. One can thereby form the factor group:

H1 = Z1/ B1

which amounts (roughly) to counting the closed curves that appear in the space (which are not there simply by virtue of being the boundary of a 2-dimensional segment)

Re-posing the factor group: dim H1 = dim Z1 - dim B1

where dim Z1 = [b + 1 - n] for any connected complex

and: b = branches, n = nodes

For the torus:

b = 4 and n = 4

so: dim Z1 = 4 + 1 - 4 = 1 = dim H1

and dim B1 = dim Z1 - dim H1 = 1 - 1 = 0

Well, what does all this gain us? Where might we be going? The beauty of this branch of math is that higher dimensionality can be represented with simpler, lower dimensional configurations.

And, as I showed above, the three-dimensional torus can be represented in two -dimensions.

Future posts will apply this to other, more ineteresting areas.

Travails of the Milankovitch hypothesis

The grandiose name “Milankovitch theory” purports to account for the recurrence of the ice ages. In fact, it is more correct to call it the “Milankovitch hypothesis” especially as few astronomers that I know accept it. On that note, I would like to discuss the basis for general non-acceptance of Milankovitch’s theory into conventional astrodynamics and astrometry.. (The former is what has been called “celestial mechanics” in the past, while the latter focuses on methods of position updating for celestial objects)

Start with its contention that the obliquity of the ecliptic (inclination of Earth to its orbital axis) varies from 21 to 24 degrees over a 41,000 period in a process called ‘nutation’. This is certainly a magnitude in excess of a half degree (1800”) on either side of its current 23.5 deg.

Astronomers-astrometrists recognize no such period or differential of axial tilt. The following is from the book, Astronomy- Principles and Practice by A.E. Roy and D. Clarke, 1978, Adam Hilger Books, p. 118:

“Because of the nutational wobble in the Earth’s axis of rotation, the obliquity of the ecliptic (KP in Fig. 10.32) varies about its mean value. The magnitude on either side is about 9.”2.”

For the benefit of non-astronomers, the magnitude cited (9.”2) isn’t even one hundredth of a degree! Indeed it is nearly a factor 4 LESS than a hundredth of a degree! (which translates to 36”- there are 3600” = 1 degree))

Going now to Eichhorn and Mueller’s standard text in astrometry and geodesy- p. 69, “astronomic nutation’:


”The main term of astronomic nutation is produced by the non-coincidence of the Moons’ orbit with the ecliptic in conjunction with the retrograde moton of the lunar nodes. This results in a periodic change in the obliquity of the ecliptic termed nutation in obliquity, denoted by d x

The astronomic nutation, from now on called simply ‘nutation’ is not to be confused with the true nutation appearing as a force-free precession (Eulerian motion) of the Earth’s rotation axis about its principal moment of inertia axis, which is part of the polar motion described in 4.13.

The first six terms of the expression for nutation in obliquity are:


delta eta =

(9.”2100 + 0.”00091t) cos Z - (0.”0904 - 0.”0004t) cos 2Z – (0.”0024 cos (2w _m + Z) + 0.”0002 cos (2w_ s – Z) + 0.”0002cos 2 ( w_ m + Z) + (0.”5522 – 0.”00029) cos 2L_ s


where t denotes the time interval measured from 1900 January 0.5 d ET in Julian centuries (1 JC = 36525 mean solar days), Z is the longitude of the mean ascending node of the lunar orbit on the ecliptic measured from the mean equinox of date, w_m is the ‘argument’ of the point where the Moon is nearest the Earth (i.e. from the lunar perigee), w s is the mean longitude of the solar perigee measured from the mean equinox of date, and L_ s is the geometric mean longitude of the Sun measured from the mean equinox of date. The terms are illustrated in Fig. 4.6


Most interesting in the above – which I merely give for the sake of completeness- is that even jacking up the value of t by 41,000 yrs. (e.g. 410 JC) doesn’t appreciably alter the magnitude from seconds of arc – very small seconds of arc (e.g. about 8.”85 with Z = 160 deg and counting only the first order term)


After giving all this a lot of thought – here is what I think, and why safi and I are at loggerheads:

The Milankovitch theory’s first problem is that it was developed by a Serbian civil engineer (& later meteorologist) who acquired the avocation of astronomy – as a personal abiding interest. This is admirable, but puts him basically in the same class as Immanuel Velikovsky – who was a Russian psychologist – trying to do astronomy. Since astronomers wouldn’t let Velikovksy into the club, it was doubtful they’d let Milankovitch. (Though to be sure, his theory had much more to commend it than Velikovksy’s gobbledegook)


The second problem is that his theory acquired its cachet and bulk of earnest support from outside the astronomical community and inside the geological research community that also spawned Wegner and Croll. The astronomical community cannot claim it as uniquely their own, so like the solar physics community earlier – in regard to space physics incursions into their ‘territory’ in the mid- 20th century – has left it “still born” or a step child. This is perhaps why it appears in no standard astrometric texts or astrodynamic texts. It is the bastard offspring of a non-astronomer and doesn’t follow at all from any standard celestial mechanics principles, equations or theories.

This marginalization can also be rationally justified (within the astronomically- affected community) on the basis that the Milankovitch theory hasn’t yet been adequately tested (e.g. beyond finding geological correlations - which of course is not causation) to prove itself worthy to compare to rigorous astronomical theories – say like the one for the lunar libration. All we have are assorted correlations, which most in the astronomical community can still attribute to flukes or haphazard research.

For example, when conditions are favorable for an ice age in the northern hemisphere, they’re not favorable for one in the southern hemisphere. How could the Milankovitch Cycles cause a global change in climate then? Also, Milankovitch cycles can only account for a temperature difference of 1° to 2°. How is it possible then that sediment records show temperature differences of 7° to 10°? The 100,000 yr cycle is dominant in the record, yet it has the weakest astronomical effect; moreover, in the record, it doesn’t always occur at 100,000 years - ranges from 80,000 to 125,000. How can these variances be explained? Until they are – most astronomers won’t embrace the theory.


Adding to that, in the paper ‘A Causality Problem for Milankovitch”, Daniel B. Karner and Richard A. Muller from the Dept. of Physics, University of California, note an earlier paper by W. Broeker (1992) ‘Upset for Milankovitch Theory’ – in which he discussed a troublesome new measurement. That is, oxygen isotope data from a cave in Nevada called ‘Devils’ Hole’ appeared to show that the timing of the penultimate termination of the ice ages- called ‘Termination II’ – was incompatible with the standard Milankovitch theory (cf. Winograd et al, Science, Vol. 258, p. 255; Ludwig et al, Science, Vol. 258, p. 284)

The data indicated a shift in (delta 16) to interglacial values that was essentially complete by 135 thousand years ago (ka). But at this time, the Northern Hemisphere summer insolation had not yet warmed to the point at which it should have triggered anything extraordinary, let alone a glacial termination. The termination event appeared to precede its cause.

Though the Milankovitchites attempted valiantly to rebut this, as the authors noted, the “causality” problem remained and it was really all an (initially) skeptical community needed to keep the Milankovitch theory from being cemented into standard celestial mechanics. Too many loose ends!

Worse, the Devils Hole data had not been the first to indicate a problem. As far back as 1974,Bloom et al. (Quatr. Research, Vol. 4, p.185) had suggested that sea level had reached a high point, from melting glacial waters, by as early as 142 ka. Their work was based on U-Th ages of coral terraces from the Huon Peninsula in Papua New Guinea. These results were not used when Imbrie et al. (‘Milankovitch and Climate – Part I’, Doredrecht Reidel)) created the SPECMAP template, the most widely used model for explaining how insolation could drive ice age cycles. Instead, Imbrie et al. set the termination at 127 ±6 ka, based on radiometric dates from Barbados corals by Mesolella et al. (J. Geology, Vol. 77, p. 250) and Shackleton and Matthews (‘Nature’, Vol. 6, p. 445)

All of the above provides just enough ‘ammunition’ to those already skeptical – to justify their resistance to Milankovitch theory and to preventing supporters from nudging it into text books (like evolutionists seek to prevent ID’ers)

This is reinforced by the fact that most astronomers’ prevailing skepticism is fuelled by the lack of a precise dynamic time scale, which would make it possible to test the match between the supposed cycles recorded in ocean sediments and the Milankovitch cycles calculated on the basis of the Earth's orbit in standard celestial mechanics. Until this is done, the Milankovitch theory will rightly not be regarded as a part of legitimate celestial mechanics – but rather a marginal or fringe spinoff.

While the cycles with periods near 100,000 years, 41,000 years, and 23,000 years, based on sediment data are intriguing – they don’t get an astronomer’s blood boiling. They merely show circumstantial evidence for the claim. Ultimately, the claim has to be tested and verified in space – to get an astronomer to invest credulity.

Theoretically, at least, there is more than enough Earth orbital data right now to be able to make solid predictions. The trick is to be able to make testable predictions, and then, meld those into a coherent theory of exactly how the orbital forcing occurs and what it does.

A first start would be using sophisticated numerical simulations – piping in the orbital (a, e, i, pi etc.) elements and their perturbations claimed by the Milankovitch crew. Thus, use standard equations of celestial mechanics (e.g. Kepler’s equation, n(t – T) = E – e sin E) and show that the assumed changes actually occur in space. Show that 100,000 yrs. from now Earth will be in such and such predicted position (according to the Milankovitch theory), and ditto for the 41,000 year scale, and so on. Capture these graphics, then publish them.

Interestingly, this was exactly the method used by space physicists to make their case for accepting dynamo processes as applicable to solar flares! Years of work finally paid off, when the most elaborate numerical simulations could no longer be disputed.

When all that happens, the Milankovitch theory may finally be embraced by the key segment of the astronomical community for which it would matter most: astrometry and celestial mechanics. This might be like the space physics community (at least part of it!) was finally embraced by the solar physics community after yrs. of squabbles..

One can, of course, look on these turf battles and territorial defense mechanisms as “immature” and unbefitting science – but scientists are human too. They want to protect what they have, what they have fought for – and not surrender turf without a good fight. If the Milankovitch lot are up for that, their "theory" may finally find its place in standard celestial mechanics text books – maybe in the next fifty yrs. Maybe earlier.

We will see.

A Perspective on the UFO

I've long since concluded that the "UFO" is not a subject that astronomers can avoid. Over the years, at nearly every public lecture I've given, at least one curious questioner has asked: "Do you believe in UFO's?" or "Have you ever seen a UFO"?

Such questions deserve to be respected and not dismissed out of hand. Knee-jerk rejection or debunking is contrary to what the scientific temperament is all about. My own strategy has been to first clarify the definition of UFO, while challenging the assumption that it is a synonym for "spaceship from another world". It means exactly what the acronym implies: Unidentified Flying Object. As such, UFOs are an observed and recorded fact, as much as the Sun rising every morning. The existence of UFOs is a matter independent of "belief".

What remains open to question is the particular interpretation of the UFO. Here is where the astronomer can provide useful insight by introducing his or her audience to the notion of scientific hypotheses. Several are applicable to UFOs, which I will explore in detail later. They include: 1- a misidentified known object (planet, airplane, or balloon); 2- an unknown natural-meteorological phenomenon; 3- a psychological phenomenon and 4- a genuine craft from another planetary civilization. Note that depending on viewpoint, an astronomer may or may not regard (1) as a valid hypotheses. I do, because further research can be used to verify it.

From all the available UFO statistics, hypothesis (1) has most often been demonstrated by sheer attrition of reported sightings. These statistics show that about 95 per cent of initially reported UFOs turn out to be "IFOs" or identified flying objects. I used to have enormous difficulty coming to grips with this statistic, but I don't any more. The fact is that the majority of sporadic skywatchers really don't know what is up there. Many cannot distinguish a star from a planet, or an extremely bright planet or star from a man-made device. I cannot begin to recount here all the times an excited caller has phoned me to breathlessly report...... Venus! Incredibly, when I correct them they retain an attitude of disbelief that any "natural" object could be that bright! Surely it must be an unnatural object! An alien craft!

As long as the night sky remains unknown to the vast majority of people, misidentifications will be the rule for "UFOs". On the other hand, I'm intrigued by the 5 per cent of reports which persist as unidentified after close scrutiny: what I call the "signal" (as distinct from the 95% "noise".) These reports cannot be solved after years of comprehensive investigations. They are what I call "the genuine UFO reports" (I use the term "UFO reports" rather than UFOs, since the existence of the reports is not in question, though their interpretation is open). These reports, in all probability, will fall into one of the other hypotheses: (2), (3) or (4). However, as we shall see, it is also possible to arrive at other hypotheses which are not so neat and tidy.

The difficulty in formulating UFO hypotheses, and interpreting UFO reports, is a first-hand experience for me. Not only have I investigated other people's reports and published the results[1] but I have seen a "UFO"myself. The incident occurred in the summer of 1962 while at the opening of a shopping center in Carol City, Florida. While awaiting the start of festivities I happened to look up at the night sky, being the amateur astronomer that I was. Amazingly, I witnessed a brilliant orange disc, at least the same diameter as a full Moon, moving rapidly from north to south. It hovered for two to three seconds above the crowd at the shopping center and I detected the odd "Oooh" or "Aaah" from random spectators. Thus, I knew I was not having a simple hallucination (at least not by myself!)

The most ironic and notable thing to me was the complete absence of sound. No whirring, like one would expect from a helicopter's propeller blades, or engine noise. The object - if "object" it was - appeared to be a light source rather than just reflecting light from elsewhere. After about three seconds it took off due south at what I estimated to be an incredible speed. As a seasoned sky observer, even at the age of 16, I was able to quickly eliminate all known man-made or natural objects from consideration. The exceptional luminous and dynamical behavior allowed this. Nevertheless, to this day I am not prepared to pinpoint a specific hypothesis in any dogmatic sense, though up until recently I have gravitated toward (4).

Why? From my knowledge of physics, it is the only one I could fit into a conceptual framework. The key is the fact that the UFO, certainly for the brief time I observed it, exhibited a remarkable degree of intelligent control. Weather phenomena, like ball lightning, simply do not behave in this fashion. They tend to be governed by stochastic or random forces. And, on a probability scale of 0 to 1 (with 1 certainty) I would put the existence of a hitherto "unknown" meteorological phenomenon at about 1 in ten million, or 0.0000001.

There may be rare weather phenomena capable of precisely imitating
the dynamics of my UFO, but I wouldn't bet on it! Ball lightning, which is a known phenomenon, comes closest - but what I saw was not ball lightning by any stretch of imagination. (For one thing there were absolutely no clouds visible at the time, and ball lightning does not move at the speed this moved).

What about hypothesis (3), a psychological effect? This cannot be ruled out absolutely, but it is rendered highly improbable by there being a dozen or so other witnesses. If it was psychological, then it was a shared hallucination, which stretches credibility in yet another direction. That leaves hypothesis (4), or does it? At face value it would seem to be the most reasonable thing to accept that what I observed was an intelligently controlled spacecraft from another world. Somehow, however, it seems to me too facile an explanation.

Forget about my observation for the moment, and consider the vast number of UFO sightings each year. That 5 per cent of permanent unknowns translates, on average, into about ten thousand sightings each year. Surely, there cannot be ten thousand different craft visiting us each year.[2]

Even allowing for repeat sightings, the variety of shapes and sizes would suggest a figure of at least one thousand. One thousand craft from another world - or from a thousand different worlds? No way! I simply can't accept that any race of supposedly intelligent beings would regard the human species and its humdrum little world as that important - and consume so much energy in such an extended endeavor!

Energy, even for very intelligent aliens, must be extremely resource-intensive. And though highly advanced, I can't see how these hypothetical beings could bypass fundamental physical laws such as entropy[3]. To imagine that any civilization can afford to squander vast energy on thousands of yearly visitations to a backwater world borders on the ridiculous.
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[1] See: Transient Optical Phenomena of the Atmosphere - A Case Study, in The Journal of the Royal Astronomical Society of Canada, Vol. 74, No. 3, June 1980.

[2] Jacques Vallee, in his book: Revelations: Alien Contact and Human Deception (Ballantine Books, 1991, Appendix) presents worldwide data and statistics which extrapolate to 14 million sightings over a thirty-year period. This figure,of course, renders the extraterrestrial hypothesis even more improbable. Is Earth really the "grand central station" of the Milky Way? I somehow doubt it.

[3] The entropy law, or 2nd law of thermodynamics, states that for every useful conversion of energy/fuel there will be a large amount of waste energy (e.g. heat) accompanying it. A corollary is that as the frequency of useful energy conversion grows, energy efficiency must go down and the amount of waste energy must increase.

What is "brane space"?

This blog is named "brane space". This is not an error or misspelling. Branes are surface representations of cosmic strings, and include p-branes, and D-branes. The latter are most useful for blog description purposes since they comprise a five-dimensional hyper-surface that propagates in time, thereby sweeping out a six-dimensional space-time. (The 'D' in D-brane refers to "Dirichlet boundary conditions" but we won't get into that right now!)

Anyway, the name for this blog implies multi-dimensional (or multi-faceted) content in time. I will blog on aspects of science (especially astronomy, astrophysics, quantum mechanics etc.) as well as on pseudo-science (astrology, Velikovskianism etc.), philosophy, religion, mathematics and even economics. Most of the time the blogging will simply reflect ruminations on current events, developments or scientific discoveries, but sometimes the writing will embody current worries......whether of economic malaise and dollar decline, or global warming, or perhaps when the next major asteroid may strike Earth.

Occasionally, solutions may be offered, but these ought to be regarded as an aspect of blogging and not based on expert opinion - at least most of the time!

My main goal is to provide captivating content that enlightens and maybe entertains as well. Comments are welcomed, and - as with my other website- I tend to keep them open and uncensored unless I have cause to do otherwise.

Enjoy!