|Snowflakes are crystals of solid ice that arise out of a gaseous vapor -- all it takes is a "cooling wave" that drives the change. As water vapor condenses into a snowflake, possible shapes turn into an actual shape.|
|The images show two stages of growth of a single "snow crystal" produced in the laboratory of Cal Tech physicist Kenneth G. Libbrecht, a leading researcher. (link to his "snowcrystals.com" website.) The span of the larger crystal is 1.2 mm.|
"What Synchronizes the Growth of the Six Arms?" "Nothing. The six arms of a snow crystal all grow independently, as described in the previous section. But since they grow under the same randomly changing conditions, all six end up with similar shapes. "If you think this is hard to swallow, let me assure you that the vast majority of snow crystals are not very symmetrical. Don't be fooled by the pictures -- irregular crystals (see the Guide to Snowflakes (link)) are by far the most common type. If you don't believe me, just take a look for yourself next time it snows. Near-perfect, symmetrical snow crystals are fun to look at, but they are not common."Of course, Libbrecht grew the snowflakes shown above in his laboratory under fixed, ideal conditions. Regardless of the conditions, I am not persuaded by Libbrecht's argument. It is "too hard to swallow." The symmetry of a snowflake extends across distances that span an enormous number of molecules. There are many variations visible in the six arms of the snowflakes in the images, but the symmetry stands above accumulated variations. There are an enormous number of possible snowflake arms but the six actual arms follow a single form and a single growth pattern. In a review article on research in the field (link), Libbrecht describes the mechanical model of snowflakes as "a reasonably accurate caricature." An especially troublesome area for explanation involves "subtleties of the surface" that lead him to conclude: "For now at least, we are left with the unsettling fact that we still cannot explain, even at a qualitative level, some of the most basic characteristics of snowflakes." On his website Libbrecht links to a paper on "scientific aspects of snow crystal growth." To my critical eye, the paper demonstrates the failures of the Mechanical Cosmology. Libbrechts' claims of successful modeling are completely eroded and undermined by the limitations and qualifications he also provides. In its engagement with snowflakes, the Mechanical Cosmology fails to deal satisfactorily with a very simple phase change that yields a rich class of variants and that is examined under ideal conditions. If the Mechanical Cosmology fails to meet the challenges of snowflakes, its general and grand claims are subject to doubt. In the Introduction to the paper at 861, Libbrecht claims to identify the mechanical causes of "Complexity and symmetry" that arises during the growth of the snowflake:
"As the nascent snow crystal grows, faceting will often create a simple hexagonal prism morphology. Diffusion limits the growth as the crystal becomes larger, and eventually this causes branches to form. Because ice is so sensitive to the local environment, it frequently happens that an abrupt motion of some kind will cause all six corners of a simple plate-like crystal to sprout arms at the same time."The second statement of causation is undermined by its qualifiers, namely, "it frequently happens" and "an abrupt motion of some kind." "Because ice is so sensitive to the local environment," is dubious as a statement of causation high sensitivity would seem to lead to a high degree of variance in the shape rather than symmetrical growth. Why do "all six corners ... sprout arms at the same time" and not at different times -- since timing is what is at issue -- or why not five corners sprouting, with one not sprouting, or four sprouting, etc. An abrupt motion as a "cause" of sprouting fails to make the causal connection between the two, at least as far as I can imagine. In sum, I do not find any explanatory meaning in Libbrecht's second statement of causation. The first statement of causation ("Diffusion limits the growth...") is based on imagery that is set forth several times in Libbrecht's paper, so it does not suffer from the meaninglessness of the second statement of causation. However, tracing the imagery through the paper reveals that it, too, is beset by qualifiers, dubious extensions and assertions that fail to make the connection they claim to make. At the start of the trace is the following (at 864):
"Snow crystal growth is typically dominated by attachment kinetics in combination with ..particle diffusion...and heat diffusion ...particle and heat diffusion are well understood at a fundamental level, so in principle we can compute how they limit crystal growth. In practice, of course, this may be difficult owing to the complex geometry of the solidification front. Attachment kinetics, by contrast, is very much not understood in detail..."Please observe how the "causation" statement quoted from p. 861 mentions only "diffusion," which is "well understood at a fundamental level" but neglects to mention the dominant influence, the "attachment kinetics," which "by contrast, is very much not understood." One possible "cause" of widespread symmetrical growth is not mentioned by Libbrecht, namely, that the form is "fixed" at the time the snowflake starts to grow. Such a "cause" would say that the synchronized placement and growth of secondary branches was "programmed." Libbrecht's imagery is entirely of assembly on the fly, with aspects of the assembly depending only on the momentary status of the surface that is growing.
"Attachment kinetics ultimately derives from the molecular dynamics present at a crystal surface, and the dynamics is intimately coupled to the molecular structure of the surface. In the case of ice, numerous studies have found that the surface structure is both complex and quite temperature dependent near the melting point...we do not yet understand the ice surface structure very well, nor its impact on growth, and thus we still cannot explain the snow crystal morpology diagram." (p. 871-872.)Moreover: "Surface melting has a profound effect on the surface structure of ice, and attachment kinetics undoubtedly depends on surface structure. ... Unfortunately, at present we do now know, even qualitatively, how surface melting in general affects the attachment kinetics governing crystal growth." (p. 874.) Of course, Libbrecht is a foremost scientist specializing in experimental techniques involving snowflakes and growth of ice crystals. Regardless of his theoretical commitments, he brings many resources of skill and knowledge to the investigative endeavor. He also states: "In general, there exists a large gap between crystal growth theory and the phenomenology of growing real crystals. This is especially true with a material like ice, for which even the equilibrium surface structure is not well known." (p. 875.) Libbrecht's explanations in terms of the Mechanical Cosmology are not persuasive. I do not believe that processes limited to the molecular scale are sufficient to account for the shape of a snowflake. On the contrary, I believe that some influence is operating across the whole thing. It appears to me that Libbrecht is rationalizing his belief in the Mechanical Cosmology rather than reasoning from physical principles. Every theory is most clearly expressed under ideal conditions. It is conceivable that the ideal conditions of Libbrecht's laboratory are the best way to express a previously-unknown collective influence in the growth of snowflakes, equally with being the best way to express the Mechanical Cosmology. The fact that ideal conditions produce long-range order does not weigh in favor of any theory of how that order comes into being. In his paper, Libbrecht forthrightly acknowledges that, at least presently, there are limits to scientific knowledge about snowflakes:
"Many aspects of snow crystal growth are well understood at a quantitative level. For example, we know the crystal structure of ice, the interactions between water molecules, the ice phase diagram, and much of phase transitions in general. Other pieces of the snow crystal puzzle, like diffusion-limited growth and the equilibrium structure of the ice surface, are fairly well understood, at least in a qualitative sense. And then there are some rather basic aspects of this phenomenon, like the snow crystal morphology diagram, that are not yet understood even at a qualitative level.Libbrecht also states:
"A better picture of the dynamics of the ice crystal surface during growth may also shed light on some of the many remaining mysteries surrounding the dynamics of the different solid and liquid states of water." (p. 857.) "...it is possible in principle to model the growth of an ice crystal as a function of time ...This turns out to be an exceedingly tricky task in practice, even for simple faceted crystals, because the attachment kinetics are such a strong function of surface orientation..." (p. 869.) "Systematic errors in ice crystal growth measurements. At least one thing has become abundantly clear over several decades of ice crystal growth experiments—there are a great many troublesome systematic effects that can influence ice crystal growth and thwart one’s attempts to make quantitative measurements under well-controlled conditions." (p. 876.)
"What does this tell us about snowflakes? For a start, it tells us that snowflakes ought to be comprehensible. If we can hope to discover the shape of the universe, then we ought to be able to handle a snowflake. Conversely, it tells us that whatever explanation we give for the shape of a snowflake, it won't be the last word. The best we can do is to tell convincing stories about snowflakes. Good enough to let us see that they make sense, good enough to suggest interesting experiments that actually work, good enough to make snowflakes to order in the laboratory. Beyond that, humans cannot and should not aspire. The Truth (with a capital 'T') must lie forever outside the realms of our understanding. (if indeed there is such a thing as Truth, which I doubt). What we can aspire to is truths, with a small 't', that is to say, scientific stories that work in their own limited realm - and work surprisingly well given the simplicity of their ingredients."In Fearful Symmetry: Is God a Geometer (1992), Stewart and co-author Martin Golubitsky introduce "symmetry" as a basis for the more advanced concept of "symmetry-breaking." Symmetry-breaking occurs in a variety of physical systems, including symmetry-breaking in both space (at the origin of the Universe) and time (there is an excellent presentation on the different modes of walking and running in animals, called "gaits," especially gaits of four-legged creatures like horses). "Symmetry-breaking" is a chief feature of Critical Point principles, to be realized as Shimmering Sensitivity in Quad Nets. Stewart/Golubitsky also refer to "critical" quantities, like a "critical speed," and to "universals." See pages 115-116. Early studies of the Critical Point by the great Russian physicist, Lev Landau (1908-1968) were the source of the concept of symmetry-breaking. (See C. Domb The Critical Point: a historical introduction to the modern theory of critical phenomena (1996) at 17-18.) For Stewart and Golubitsky, the climax of development occurs when the Geometer God meets up with the Dicing Deity. (Page 251 et. seq.) The Dicing Deity comes from another Stewart book, Does God Play Dice: The New Mathematics of Chaos (1989), which emphasizes chance events and how, from a mathematical perspective, chance events can have major consequences. Together, the Geometer God and the Dicing Deity provide a "Theory of Everything." (Fearful Symmetry at 252 et. seq.) For me, the Geometer God and Dicing Deity are unsatisfactory replacements for traditional religion. Stewart acknowledges that the "Theory of Everything" is imperfect. In particular, he identifies difficulties in modeling metallic crystals made up of repeated units that are arranged in "lattices" but that also contain imperfections, called "dislocations."
"Mathematically, both lattices and dislocations are solutions to the equations that govern the state of a large quantity of atoms: why select one solution as being superior to another? Both lattice and dislocation are concepts imposed by our perception of reality, rather than being inherent features of the way reality itself operates." (P. 259.)Stewart's approach suggests discrepancies between mathematical perceptions of reality and a yet-to-be-defined "way reality itself operates." I suggest that there is a gap between the reach of our intelligence and "reality itself," a gap in which discrepancies prevent our mathematical skills from reaching a satisfactory solution. As a practical matter, I suggest that the gap is a consequence of the limited repertoire of our intelligence: our minds have only so many tricks that work in only so many ways and they don't suffice to cover everything. As a theoretical mnatter, such a gap is generated by what I call the "blind spot in our intelligence." (See "A place for freedom: in the blind spot of the mind" in Researches in Personal Freedom, Introduction, "An Objective Kind of Freedom," § 4, available in the Archive on Embodiment of Freedom.) A blind spot would account for the fact that there are mysteries beyond the the reach of our knowledge, including the mystery of snowflakes.
"Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; 'modern' physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of small particles of matter will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber..."2. David Ruelle, Chance and Chaos (1991) at 122-124:
"One puzzling phenomenon is the boiling of water, and the freezing of water is no less mysterious. If we take a liter of water and lower the temperature, it is not unreasonable that it should become more and more viscous. We may guess that at low enough temperature it will be so viscous, so stiff, as to appear quite solid. This guess about the solidification of water is wrong. As we cool water we see that at a certain temperature it changes to ice in a completely abrupt manner. Similarly, if we heat water it will boil at a certain temperature, i.e., it will undergo a discontinuous change from liquid to water vapor. The freezing and boiling of water are familiar examples of phase transitions. These phenomena are in fact so familiar that we may miss the fact that they are strange indeed, and require an explanation. ... So here is a problem for theoretical physicists: prove that as you raise or lower the temperature of water you have phase transitions to water vapor or ice. Now that's a tall order! We are far from having such a proof. In fact, there is not a single atom or molecule for which we can mathematically prove that it should crystallize at low temperatures. These problems are just too hard for us." (Emphasis in original.)
Shimmering Silences in Beautiful Music -- presentation and development of timing device models through applications to auditory experience
Timing Devices: the original presentation of the timing devices model. The message is that brains are not computers.
Quad Nets: general designs for brain-like devices; timing devices are a special class of devices within the more general class.
Embodiment of Freedom: integrated models - based on Quad Nets - of brains and experience, physics and psychology.
Testimony of Freedom: re-states prior models and extends the inquiry into social and spiritual matters.