Showing posts with label Neil Turok. Show all posts
Showing posts with label Neil Turok. Show all posts

Tuesday, December 07, 2010

Cyclic model

Physical cosmology
WMAP 2010.png
Universe · Big Bang
Age of the universe
Timeline of the Big Bang
Ultimate fate of the universe
A cyclic model is any of several cosmological models in which the universe follows infinite, self-sustaining cycles. For example, the oscillating universe theory briefly considered by Albert Einstein in 1930 theorized a universe following an eternal series of oscillations, each beginning with a big bang and ending with a big crunch; in the interim, the universe would expand for a period of time before the gravitational attraction of matter causes it to collapse back in and undergo a bounce.



In the 1930s, theoretical physicists, most notably Albert Einstein, considered the possibility of a cyclic model for the universe as an (everlasting) alternative to the model of an expanding universe. However, work by Richard C. Tolman in 1934 showed that these early attempts failed because of the entropy problem that, in statistical mechanics, entropy only increases because of the Second law of thermodynamics.[1] This implies that successive cycles grow longer and larger. Extrapolating back in time, cycles before the present one become shorter and smaller culminating again in a Big Bang and thus not replacing it. This puzzling situation remained for many decades until the early 21st century when the recently discovered dark energy component provided new hope for a consistent cyclic cosmology.[2]

One new cyclic model is a brane cosmology model of the creation of the universe, derived from the earlier ekpyrotic model. It was proposed in 2001 by Paul Steinhardt of Princeton University and Neil Turok of Cambridge University. The theory describes a universe exploding into existence not just once, but repeatedly over time.[3][4] The theory could potentially explain why a mysterious repulsive form of energy known as the "cosmological constant", and which is accelerating the expansion of the universe, is several orders of magnitude smaller than predicted by the standard Big Bang model.

A different cyclic model relying on the notion of phantom energy was proposed in 2007 by Lauris Baum and Paul Frampton of the University of North Carolina at Chapel Hill.[5]

The Steinhardt–Turok model

In this cyclic model, two parallel orbifold planes or M-branes collide periodically in a higher dimensional space.[6] The visible four-dimensional universe lies on one of these branes. The collisions correspond to a reversal from contraction to expansion, or a big crunch followed immediately by a big bang. The matter and radiation we see today were generated during the most recent collision in a pattern dictated by quantum fluctuations created before the branes. Eventually, the universe reached the state we observe today, before beginning to contract again many billions of years in the future. Dark energy corresponds to a force between the branes, and serves the crucial role of solving the monopole, horizon, and flatness problems. Moreover the cycles can continue indefinitely into the past and the future, and the solution is an attractor, so it can provide a complete history of the universe.
As Richard C. Tolman showed, the earlier cyclic model failed because the universe would undergo inevitable thermodynamic heat death.[1] However, the newer cyclic model evades this by having a net expansion each cycle, preventing entropy from building up. However, there are major problems with the model. Foremost among them is that colliding branes are not understood by string theorists, and nobody knows if the scale invariant spectrum will be destroyed by the big crunch. Moreover, like cosmic inflation, while the general character of the forces (in the ekpyrotic scenario, a force between branes) required to create the vacuum fluctuations is known, there is no candidate from particle physics. [7]

The Baum–Frampton model

This more recent cyclic model of 2007 makes a different technical assumption concerning the equation of state of the dark energy which relates pressure and density through a parameter w.[5][8] It assumes w < -1 (a condition called phantom energy) throughout a cycle, including at present. (By contrast, Steinhardt-Turok assume w is never less than -1.) In the Baum-Frampton model, a septillionth (or less) of a second before the would-be Big Rip, a turnaround occurs and only one causal patch is retained as our universe. The generic patch contains no quark, lepton or force carrier; only dark energy - and its entropy thereby vanishes. The adiabatic process of contraction of this much smaller universe takes place with constant vanishing entropy and with no matter including no black holes which disintegrated before turnaround. The idea that the universe "comes back empty" is a central new idea of this cyclic model, and avoids many difficulties confronting matter in a contracting phase such as excessive structure formation, proliferation and expansion of black holes, as well as going through phase transitions such as those of QCD and electroweak symmetry restoration. Any of these would tend strongly to produce an unwanted premature bounce, simply to avoid violation of the second law of thermodynamics. The surprising w < -1 condition may be logically inevitable in a truly infinitely cyclic cosmology because of the entropy problem. Nevertheless, many technical back up calculations are necessary to confirm consistency of the approach. Although the model borrows ideas from string theory, it is not necessarily committed to strings, or to higher dimensions, yet such speculative devices may provide the most expeditious methods to investigate the internal consistency. The value of w in the Baum-Frampton model can be made arbitrarily close to, but must be less than, -1.


  1. ^ a b R.C. Tolman (1987) [1934]. Relativity, Thermodynamics, and Cosmology. New York: Dover. LCCN 34032023-{{{3}}}. ISBN 0486653838. 
  2. ^ P.H. Frampton (2006). "On Cyclic Universes". arΧiv:astro-ph/0612243 [astro-ph]. 
  3. ^ P.J. Steinhardt, N. Turok (2001). "Cosmic Evolution in a Cyclic Universe". arΧiv:hep-th/0111098 [hep-th]. 
  4. ^ P.J. Steinhardt, N. Turok (2001). "A Cyclic Model of the Universe". arΧiv:hep-th/0111030 [hep-th]. 
  5. ^ a b L. Baum, P.H. Frampton (2007). "Entropy of Contracting Universe in Cyclic Cosmology". arΧiv:hep-th/0703162 [hep-th]. 
  6. ^ P.J. Steinhardt, N. Turok (2004). "The Cyclic Model Simplified". arΧiv:astro-ph/0404480 [astro-ph]. 
  7. ^ P. Woit (2006). Not Even Wrong. London: Random House. ISBN 97800994488644. 
  8. ^ L. Baum and P.H. Frampton (2007). "Turnaround in Cyclic Cosmology". Physical Review Letters 98 (7): 071301. doi:10.1103/PhysRevLett.98.071301. arXiv:hep-th/0610213. PMID 17359014. 

See also

Further reading

  • P.J. Steinhardt, N. Turok (2007). Endless Universe. New York: Doubleday. ISBN 9780385509640. 
  • R.C. Tolman (1987) [1934]. Relativity, Thermodynamics, and Cosmology. New York: Dover. LCCN 34032023-{{{3}}}. ISBN 0486653838. 
  • L. Baum and P.H. Frampton (2007). "Turnaround in Cyclic Cosmology". Physical Review Letters 98 (7): 071301. doi:10.1103/PhysRevLett.98.071301. arXiv:hep-th/0610213. PMID 17359014. 
  • R. H. Dicke, P. J. E. Peebles, P. G. Roll and D. T. Wilkinson, "Cosmic Black-Body Radiation," Astrophysical Journal 142 (1965), 414. This paper discussed the oscillatory universe as one of the main cosmological possibilities of the time.
  • S. W. Hawking and G. F. R. Ellis, The large-scale structure of space-time (Cambridge, 1973).

External links

Thursday, June 29, 2006

Early Universe Formation

An Energy of Empty Space?

Einstein was the first person to realize that empty space is not nothingness. Space has amazing properties, many of which are just beginning to be understood. The first property of space that Einstein discovered is that more space can actually come into existence. Einstein's gravity theory makes a second prediction: "empty space" can have its own energy. This energy would not be diluted as space expands, because it is a property of space itself; as more space came into existence, more of this energy-of-space would come into existence as well. As a result, this form of energy would cause the universe to expand faster and faster as time passes. Unfortunately, no one understands why space should contain the observed amount of energy and not, say, much more or much less.

I had been doing some reading and some thoughts came to mind about the measures one may use to see how our universe is doing. While it is really early here for any great revelation :) it did seem that issues could arise in my mind, if we used the "distance" to measure what exactly the universe is doing.

A Determinism at Planck Scale?

I'll tell you why in a second and then leave for now, as I have to continue with finishing the "foundation" with my son. Getting ready for backfilling tomorrow.

Andrey Kravtsov's computer modelling comes to mind, and how I was percieving early universe modelling in terms of a supersymmetrical state of existance. Holding this very idea in terms of this whole universe, it seemed to me, that the very "dynamical situation" and rise from such motivations, would have revealled principles as inherent in how "GR" would arise from this beginning. If the 5d consideration ha dbeen reduced to the 4 spacertime coordinated frame of reference, then what use any supersymmetrical state, or the motivation for such universe expressions?

Scientists have detected a flash of light from across the Galaxy so powerful that it bounced off the Moon and lit up the Earth's upper atmosphere. The flash was brighter than anything ever detected from beyond our Solar System and lasted over a tenth of a second. NASA and European satellites and many radio telescopes detected the flash and its aftermath on December 27, 2004. Two science teams report about this event at a special press event today at NASA headquarters

So there are two issues here that in my mind which make measurement extremely difficult. Two events within each other, that reveal something acute about the closeness of the beginnings in the universe, as very closely mappped to what exists now in our views revealled in GRB events

It was further complicated in my mind by two more issues that hold reference to these high energy events releases, that layout the schematics drawings, that the new WMAP indication holds in regards to analogistical sounds, revealled as the underpinnings of movement within this same universe.

So what about the WMAP and it's current reveallings?

If such equillibrium states are recognized as they are in placing detectors to position. Wouldn't this also reveal an opportune time for how we see this information, and provide for quick travel?

How did these "holes" create a problem for me?

If energy from these events found the "fastest route," then what would any lensing have looked like, effected by the very influences that the photon's travelled held, unduly holding to a fifth dimensional view?

The universe may of then looked like a swiss cheese? :)

Within conventional big bang cosmology, it has proven to be very difficult to understand why today's cosmological constant is so small. In this paper, we show that a cyclic model of the universe can naturally incorporate a dynamical mechanism that automatically relaxes the value of the cosmological constant, including contributions to the vacuum density at all energy scales. Because the relaxation time grows exponentially as the vacuum density decreases, nearly every volume of space spends an overwhelming majority of the time at the stage when the cosmological constant is small and positive, as observed today.

Link for article above here. Paul Steinhardt's homepage here.

If gravity and light are joined in the fifth dimension, what would this mean?

Wednesday, February 15, 2006

Big Bang:One Man's Change of Heart

Thanks Paul

One definitely needs some perspective around this and how such information is given. I refer here for consideration, about perspective, and how it can be exploited for further consideration on what is emitted, and what manifests in weak gravitational field measure, as neutrino effects(quantum gravity).

Microperspective and methods of examination, raise the issue fo cerenkov radiation and what it tells us about such interactive phases?

Here in refractive consideration, ICECUBE, paints a different picture of what began somewhere else in cosmological high energy collisions. "Neutrinos and strangelets" are part of the developing scenario with which the universe has consequences, if held to the initial conditons of our universe. You had to know where to look for these.

"Nothing" in stated form was and always is "nothing" which would have not allowed any further discussion. "Zero" in our conversation is a much different kind of thinking. I understood that as well. "Zero" would have been the equivalent to "i" in the Dirac's matrices?

Physics at this high energy scale describes the universe as it existed during the first moments of the Big Bang. These high energy scales are completely beyond the range which can be created in the particle accelerators we currently have (or will have in the foreseeable future.) Most of the physical theories that we use to understand the universe that we live in also break down at the Planck scale. However, string theory shows unique promise in being able to describe the physics of the Planck scale and the Big Bang.

I wanted to add this post, and to centralize some references that were found that helped form my perspective on "nothing." What! I guess I'm done?:)

Seriously, this had to be confronted, and who better then from our layman perspectve, then the admission of a leaders in science, who can change theirs mind after some thinking?

Cosmological Constant SeeSaw in Quantum CosmologyMichael McGuigan

Lubos shares his perspective on linked section of titled paper above.

One interpretation of the coupling of Wheeler-DeWitt functions is that it originates from topology changing effects. Topology change seems to be inevitable in quantum gravity. To treat topology change properly is a very complicated calculation using today’s mathematical tools.

I wanted to add these links here for consideration, as well what link given by Paul for consideration in regards to Penrose, the figure of the man's change of heart that ighlight's this post. In Phase transitions the comments have been quite enlightening.

Before the Big Bang BBC News, with Stephen Sackur
Sir Roger Penrose has developed a new theory on what happened before the Big Bang.

These pages were created by Jack "Turtle" Wong, Spring 1999

  • First of all, how do we think the universe began?

  • The Big Bang theory.

  • Resolving the inadequacies of the big bang theory.

  • The Hawking-Turok Instanton theory: Stephen Hawking's

  • The Hawking-Turok Instanton theory: Neil Turok's ideas.

  • The Hawking-Turok Instanton theory: the result of merging
    two interesting theories.

  • Is the search over?

  • Bibliography / Further Reading

  • See Also:

  • Cycle of Birth, Life, and Death-Origin, Indentity, and Destiny by Gabriele Veneziano

  • Ekpyroptic and cyclical models
  • Saturday, December 03, 2005

    General Relativity

    I took GR because I thought Neil Turok was dreeeamy.

    Well I dunno? He certainly got me thinking about brane world collisions, along with steinhardt, that’s for sure. We are most certainly dealing with a cosmological placement here with General relativity, but has been extended, as we look at string/M theoretical successes.

    You had to make "certain assumptions I know" in order to get here in the picture, and you had to have some inkling of what gravitational waves were and how they were transmitted.

    Completed 720 degree rotations, as "tidbits" of the process which are given to us from a cosmological standpoint.

    So what is transmitted in the bulk in terms of "gravitational lensing" has some relation, to what we see in the picture above. Look at the placement of the gravitons in bulk perspective and how they are concentrated on and around the brane.

    So it is not without reason that we see bulk perspective as a extension and not scientifically up to the challenege because Peter Woit say so?

    Modifications to General Relativity

    So "six weeks" we should have known something by now with respect to below statements? Jo-Anne, of cosmic varaince selected this answer next to the Pioneer Anomalie.

    Eric Adelberger on Aug 12th, 2005 at 2:37 pm
    Please don’t get too excited yet about rumors concerning the Eot-Wash test of the 1/r^2 law. We can exclude gravitational strength (|alpha|=1) Yukawa violations of the 1/r^2 law for lambda>80 microns at 95% confidence. It is true that we are seeing an anomaly at shorter length scales but we have to show first that the anomaly is not some experimental artifact. Then, if it holds up, we have to check if the anomaly is due to new fundamental physics or to some subtle electromagnetic effect that penetrates our conducting shield. We are now checking for experimental artifacts by making a small change to our apparatus that causes a big change in the Newtonian signal but should have essentially no effect on a short-range anomaly. Then we will replace our molybdenum detector ring with an aluminum one. This will reduce any signal from interactions coupled to mass, but will have little effect on subtle electromagnetic backgrounds. These experiments are tricky and measure very small forces. It takes time to get them right. We will not be able to say anything definite about the anomaly for several months at least.

    As stated maybe this "anomalie" might be significant and for scientists it is necessary such a quirk of nature be seen and understood. I relayed Einstein's early youth and the compass for a more introspective feature that such anomalies present.

    The Eotwash Group is a sign of relief, for the speculative signs attributed from other scientists, made this topic of extra-dimensions unbearable and unfit for the general outlay for scientists who did not understand this themselves.

    Deviations from Newton's law seen?

    So what does Lubos have to say about this in his column?

    Lubos Motl:
    The most careful and respected experimental group in its field which resides at University of Washington - Eric Adelberger et al. - seems to have detected deviations from Newton's gravitational law at distances slightly below 100 microns at the "4 sigma" confidence level. Because they are so careful and the implied assertion would be revolutionary (or, alternatively, looking spectacularly dumb), they intend to increase the effect to "8 sigma" or so and construct different and complementary experiments to test the same effect which could take a year or two (or more...) before the paper is published. You know, there are many things such as the van der Waals forces and other, possibly unexpected, condensed-matter related effects that become important at the multi-micron scales and should be separated from the rest.

    On Relativity again

    According to General Relativity, the key qualities of strong sources of gravitational waves are that they be non-spherical, dynamic (i.e. change their behavior with time), and possess large amounts of mass moving at high velocities. So prime suspects should exhibit one or more of the following characteristics.

  • 1. Spinning

  • 2. Mass tranfer

  • 3. Collpase

  • 4. Explosion

  • 5. Collision

  • As to “online resources” for General Relativity, is there one preference if you do not have access to the Hartle book or the other?

    Lecture Notes on General Relativity, by Sean Carroll

    These lectures represent an introductory graduate course in general relativity, both its foundations and applications. They are a lightly edited version of notes I handed out while teaching Physics 8.962, the graduate course in GR at MIT, during the Spring of 1996. Although they are appropriately called \lecture notes”, the level of detail is fairly high, either including all necessary steps or leaving gaps that can readily be filled in by the reader. Nevertheless, there are various ways in which these notes differ from a textbook; most importantly, they are not organized into short sections that can be approached in various orders, but are meant to be gone through from start to finish. A special effort has been made to maintain a conversational tone, in an attempt to go slightly beyond the bare results themselves and into the context in which they belong

    Or a link to this one for a historical look?

    The Special and General Theory

    Saturday, December 18, 2004

    Warped Field Creates Lensing

    The statement of this post, is distilled from the collaboration of some of the images to follow.

    In cosmic string developement there are these three points to consider.

  • 1. Cosmological expansion

  • 2. Intercommuting and Loop Production

  • 3. Radiation

  • I am always looking for this imagery that helps define further what gravitational lensing might have signified in our perception of these distances in space. How the cosmic string might have exemplified itself in some determination, as we find Lubos has done in the calculation of the mass and size of this early event. This image to follow explains all three developemental points.

    Bashing Branes by Gabriele Veneziano
    String theory suggests that the big bang was not the origin of the universe but simply the outcome of a preexisting state

    The pre–big bang and ekpyrotic scenarios share some common features. Both begin with a large, cold, nearly empty universe, and both share the difficult (and unresolved) problem of making the transition between the pre- and the post-bang phase. Mathematically, the main difference between the scenarios is the behavior of the dilaton field. In the pre–big bang, the dilaton begins with a low value--so that the forces of nature are weak--and steadily gains strength. The opposite is true for the ekpyrotic scenario, in which the collision occurs when forces are at their weakest.

    The developers of the ekpyrotic theory initially hoped that the weakness of the forces would allow the bounce to be analyzed more easily, but they were still confronted with a difficult high-curvature situation, so the jury is out on whether the scenario truly avoids a singularity. Also, the ekpyrotic scenario must entail very special conditions to solve the usual cosmological puzzles. For instance, the about-to-collide branes must have been almost exactly parallel to one another, or else the collision could not have given rise to a sufficiently homogeneous bang. The cyclic version may be able to take care of this problem, because successive collisions would allow the branes to straighten themselves.

    The most strongest image that brought this together for me was in understanding what Neil Turok and Paul Steinhardt developed for us. It was watching the animation of the colliding branes that I saw the issue clarify itself. But before this image deeply helped, I saw the issue clearly in another way as well.

    The processes of intercommuting and loop production.

    It was very important from a matter distinction, to understand the clumping mechanism that reveals itself, after this resulting images of the galaxy formation recedes in the colliding brane scenrio viewing. If such clumping is to take place, we needed a way in which to interpret this.

    Branes Reform Big Bang By Atalie Young

    Friday, December 03, 2004

    Quantum Microstates: Gas Molecules in the Presence of a Gravitational Field

    Andy Strominger:
    This was a field theory that lived on a circle, which means it has one spatial dimension and one time dimension. We derived the fact that the quantum states of the black hole could be represented as the quantum states of this one-plus-one dimensional quantum field theory, and then we counted the states of this theory and found they exactly agreed with the Bekenstein-Hawking entropy.

    I do not know of many who could not have concluded that microstates would have been something of an issue, as one recognizes this focus towards cosmological considerations. One aspect of Einstein’s general relativity, helped us recognize the value of gravitation that is extremely strong in situations where energy values are climbing. We had to look for these conditions and work them out?

    Strominger: That was the problem we had to solve. In order to count microstates, you need a microscopic theory. Boltzmann had one–the theory of molecules. We needed a microscopic theory for black holes that had to have three characteristics: One, it had to include quantum mechanics. Two, it obviously had to include gravity, because black holes are the quintessential gravitational objects. And three, it had to be a theory in which we would be able to do the hard computations of strong interactions. I say strong interactions because the forces inside a black hole are large, and whenever you have a system in which forces are large it becomes hard to do a calculation.

    The old version of string theory, pre-1995, had these first two features. It includes quantum mechanics and gravity, but the kinds of things we could calculate were pretty limited. All of a sudden in 1995, we learned how to calculate things when the interactions are strong. Suddenly we understood a lot about the theory. And so figuring out how to compute the entropy of black holes became a really obvious challenge. I, for one, felt it was incumbent upon the theory to give us a solution to the problem of computing the entropy, or it wasn't the right theory. Of course we were all gratified that it did.

    If we did not have some way in which to move our considerations to the energy states that existed in the beginning of this universe what other measures would you use? How would you explain a cyclical model that Neil Turok and Steinhardt talked about and created for us?

    Is this a predictive feature of our universe that had to have some probablity of expression and mathematically, if one wanted some framework, why not throw all things to the wind and say, Pascal's triangle will do?:)

    The animation shows schematically the behavior of the gas molecules in the presence of a gravitational field. We can see in this figure that the concentration of molecules at the bottom of the vessel is higher than the one at the top of the vessel, and that the molecules being pushed upwards fall again under the action of the gravitational field.

    One had to have some beginning with which to understand what could have emerged from such energy configurations. If such energies are concentrated and found to bring us to the supersymmetrical values assigned on that brane, then how would cooling functions of the CMB have figured a direct result would be expressive of those same events? Was there no way to measure chaoticness. Maybe it was all Fool’s Gold?:)

    Monday, November 29, 2004

    Cycle of Birth, Life, and Death-Origin, Indentity, and Destiny by Gabriele Veneziano

    Was the big bang really the beginning of time? Or did the universe exist before then? Such a question seemed almost blasphemous only a decade ago. Most cosmologists insisted that it simply made no sense - that to contemplate a time before the big bang was like asking for directions to a place north of the North Pole. But developments in theoretical physics, especially the rise of string theory, have changed their perspective. The pre-bang universe has become the latest frontier of cosmology.

    The new willingness to consider what might have happened before the bang is the latest swing of an intellectual pendulum that has rocked back and forth for millennia. In one form or another, the issue of the ultimate beginning has engaged philosophers and theologians in nearly every culture. It is entwined with a grand set of concerns, one famously encapsulated in an 1897 painting by Paul Gauguin: D'ou venons-nous? Que sommes-nous? Ou allons-nous? "Where do we come from? What are we? Where are we going?" The piece depicts the cycle of birth, life and death - origin, identity and destiny for each individual - and these personal concerns connect directly to cosmic ones. We can trace our lineage back through the generations, back through our animal ancestors, to early forms of life and protolife, to the elements synthesized in the primordial universe, to the amorphous energy deposited in space before that. Does our family tree extend forever backward? Or do its roots terminate? Is the cosmos as impermanent as we are

    One had to know at a deeper level how we might have engaged the cyclical universe? Could bubble nucleation fall in line with the ideas about the origins of this universe and find itself too part of this creative scenario?

    Colliding branes had to have some recognition in the world that we would have considered, bubble nucleation, as manifesting itself over and over again within the confines of our own universe now? Would this have made it likely that such manifestions really go to the source of what could have begun, has always been and wil continue to evolve, in this cycle of birth death and being reborn?

    Neil Turok

    The Myth of the Beginning of Time

    The new willingness to consider what might have happened before the big bang is the latest swing of an intellectual pendulum that has rocked back and forth for millenia. In one form or another, the issue of the ultimate beginning has engaged philosophers and theologians in nearly every culture. It is entwined witha grand set of concerns, one famosly encapsulated in a 1897 painting by Paul Gauguin: D'ou venons? Que sommes-nous? Ou allons-nous? Scientific America, The Time before Time, May 2004

    Sister Wendy's American Masterpieces":

    "This is Gauguin's ultimate masterpiece - if all the Gauguins in the world, except one, were to be evaporated (perish the thought!), this would be the one to preserve. He claimed that he did not think of the long title until the work was finished, but he is known to have been creative with the truth. The picture is so superbly organized into three "scoops" - a circle to right and to left, and a great oval in the center - that I cannot but believe he had his questions in mind from the start. I am often tempted to forget that these are questions, and to think that he is suggesting answers, but there are no answers here; there are three fundamental questions, posed visually.

    "On the right (Where do we come from?), we see the baby, and three young women - those who are closest to that eternal mystery. In the center, Gauguin meditates on what we are. Here are two women, talking about destiny (or so he described them), a man looking puzzled and half-aggressive, and in the middle, a youth plucking the fruit of experience. This has nothing to do, I feel sure, with the Garden of Eden; it is humanity's innocent and natural desire to live and to search for more life. A child eats the fruit, overlooked by the remote presence of an idol - emblem of our need for the spiritual. There are women (one mysteriously curled up into a shell), and there are animals with whom we share the world: a goat, a cat, and kittens. In the final section (Where are we going?), a beautiful young woman broods, and an old woman prepares to die. Her pallor and gray hair tell us so, but the message is underscored by the presence of a strange white bird. I once described it as "a mutated puffin," and I do not think I can do better. It is Gauguin's symbol of the afterlife, of the unknown (just as the dog, on the far right, is his symbol of himself).

    "All this is set in a paradise of tropical beauty: the Tahiti of sunlight, freedom, and color that Gauguin left everything to find. A little river runs through the woods, and behind it is a great slash of brilliant blue sea, with the misty mountains of another island rising beyond Gauguin wanted to make it absolutely clear that this picture was his testament. He seems to have concocted a story that, being ill and unappreciated (that part was true enough), he determined on suicide - the great refusal. He wrote to a friend, describing his journey into the mountains with arsenic. Then he found himself still alive, and returned to paint more masterworks. It is sad that so great an artist felt he needed to manufacture a ploy to get people to appreciate his work. I wish he could see us now, looking with awe at this supreme painting.