[math-fun] A First Look at "Snow Crystals"
1. The substantial work of Kenneth Libbrecht "Snow Crystals" [1] was released in October 2019, and recently popularized by Quanta [2]. Once again, the Quanta article buys in to egocentric optimism, and fails to produce a critical book review (my opinion: what they should be doing here). Yet this article does discuss relatively original and timely research. Also, awesome video header [3]! 2. The book is very long at 500+ pages, but if nothing else at least read the first chapter for history from Kepler to Ukichiro Nakaya. Nakaya, a contemporary of Tomonaga, was the first scientist to chart a temperature-dependent phase diagram for snow crystals, see figures 1.24, 1.25, 1.12 (copied on quanta as "A Wintry Mix"). 3. Figures 3.3 and 3.4 explain discrepancy in adsorption; however, they also contradict the phase diagram figure 1.12. The author even admits that this formation paradox remains incompletely resolved, so still an open question of science. Chapter 3-4 contains in-depth science worth more serious critical reading. 3b. Quanta figure "Growing Snowflakes" is extremely confusing, and excessively over-simplifed. The use of "face" and "edge" is flat out _Wrong_. This is not a good summary of chapters 3-4. 4. Chapter 5 "A Progression of Snow Crystal Models" is an exciting prospect; though, somewhat of a disappointment in execution, especially considering diligent history of Chapter 1. 4b. The "Packard Snowflake" *is not* the earliest relevant C.A. model of dendrite growth. Unfortunately Packard 1986 [4] does not cite Ulam 1962 [5]; though, OEIS does, see [6-8]. Ulam mentions that Holladay already knew of 2^n extinctions as early as 1960 [8]! 4c. A151723 includes a picture [9] that shows the structure of a sectored plate, and features the same 2^n extinctions. This picture is drawn over the half-hexagon tiling. From a mathematical perspective, half-hex is a better analog to 2^n C.A. than the Koch Snowflake. 4d. A similar comparison between Chair tiling and another growth C.A. is available at [10]. After this, it is possible to show that Code 686 C.A. in fact *builds* the chair tiling[11]. It follows in some sense that 686 ~ trilobite and crab. (Did C.G.S. know this?) 4e. We do not know if a variation of the C.A. from A151723 can be used to build the Socolar-Taylor tiling, or any of its variants [12]. This is a worthwhile open question. 5. Even before Holladay & Ulam, L. Pauling began to develop the "ice-type model" in 1935 [13]. Why isn't this cited? 5b. Pursuant to 4b. to 4e. a perspective combining matching-rule tiling theory and cellular automata *is not being presented* in Libbrecht's "Progression of snow crystal models". Does the author have a reason for such an exclusion? 5c. Hypothetical: In a matching-rule tiling theory for describing snow crystals, obviously the binding energies would change with thermodynamic conditions, but also, the matching-rules could change over temperature, pressure, vapour saturation, etc. 6. Chapters 6-9 are out of my price range, but describe incredible experiments. Thankfully lots of pictures and videos are available for purpose of data analysis. 7. Chapter 10, WOO-HOO! Libbrecht speaks: "Keeping an eye out for interesting crystals is a fine pursuit whenever you happen to be outside during a light snowfall. You could be riding the chair lift at your local ski area, taking a stroll through the winter woods, or just waiting in your car somewhere. If the snow is falling all around you, why not have a look from time to time to see what you can find?" 7b. Couldn't agree more, great advice Kenneth! Here are some specimens I collected at about 9800 feet elevation in the Rocky Mountains on Christmas morning: https://0x0.st/zDFW.jpg 7c. The photographs were rushed, but do show plates, sectored plates, stellar dendrites, and simple stars. From the existence of these structures, I think we can infer temperature of formation in the magic range around -15c, and decently high vapour saturation. 7d. Similar structures persisted at least 1000 feet above the height of base camp; though, I did not collect photo evidence. 8. HATE WARNING: The website "snowflakevictory.com" is not actually a website about physical snowflakes. It is a political site for the impeached Donald Trump, and uses the word "snowflake" as derogatory slang [14]. 8b. I supposed that domain name registration is solely a question of money ($$$) at this point. However, if decisions were being made by more fair criteria, I would think that snowflakevictory.com would simply redirect to Kenneth Libbrecht's awesome website: http://www.snowcrystals.com/ Any thoughts on 4e., 5c., or snowflake calculations in general? Anyone else getting decent snowflake pictures this season? Happy new-flakes, --Brad [1] https://arxiv.org/abs/1910.06389 [2] https://www.quantamagazine.org/toward-a-grand-unified-theory-of-snowflakes-2... [3] https://d2r55xnwy6nx47.cloudfront.net/uploads/2019/12/SnowStructure_2880x162... [4] http://www.scipress.org/e-library/sof/pdf/0095.PDF [5] https://oeis.org/A002858/a002858.pdf [6] https://oeis.org/A002858 [7] https://oeis.org/A151723 [8] https://oeis.org/A322662 [9] https://oeis.org/A151723/a151723.png [10] https://oeis.org/A147562/a147562_1.png [11] https://demonstrations.wolfram.com/Code686BuildsTheChairTiling/ [12] https://demonstrations.wolfram.com/LimitPeriodicTilings/ [13] https://en.wikipedia.org/wiki/Ice-type_model [14] https://en.wikipedia.org/wiki/Snowflake_(slang)
* Brad Klee <bradklee@gmail.com> [Dec 30. 2019 13:04]:
[...] Libbrecht's awesome website:
Nice indeed.
Any thoughts on 4e., 5c., or snowflake calculations in general? Anyone else getting decent snowflake pictures this season?
Sort-of-ish: All the following files are in https://jjj.de/tmp-math-fun/ The file 3464-rotational-x02-prototiles-plus-B3--hr.pdf is the image of a snowflake I found. To see what this is about, look at the above together with these two: 3464-rotational-x02-prototiles-plus-A6--hr.pdf 3464-rotational-x02-prototiles-plus-b3--hr.pdf Oh, here is another one: 3464-degenerate-curve-B-const-star-hr.pdf BIG pdf image, size = 5 MB, so use a decent viewer. Best regards, jj
Happy new-flakes,
--Brad
[...]
Hmm... A chiral snowflake would be impossible in roughly homogeneous, isotropic growth conditions, but in a different ambient field, maybe it would be possible! One idea is to connect the base of an e-needle to a slow moving electric motor, and have e-needle + hexagonal plate rotate during formation. Or install fans to give a chiral velocity field for water vapor. Either technique seems likely to produce chiral symmetry breaking in formed crystals. Another possibility is to grow the snow crystal around the center of a circular magnetic field. However, I'm not sure if the Lorentz force law would cancel out anisotropy on average? Are there filaments of low enough resistance to generate a B-field without getting too hot to prevent dendrite growth? --Brad On Mon, Dec 30, 2019 at 6:38 AM Joerg Arndt <arndt@jjj.de> wrote:
All the following files are in https://jjj.de/tmp-math-fun/ 3464-degenerate-curve-B-const-star-hr.pdf BIG pdf image, size = 5 MB, so use a decent viewer.
* Brad Klee <bradklee@gmail.com> [Dec 31. 2019 13:44]:
Hmm... A chiral snowflake would be impossible in roughly homogeneous, isotropic growth conditions, but in a different ambient field, maybe it would be possible!
Note that this flake tiles the plane. This does not seem to be the case with real snowflakes. That other family of three (prototiles) only tile the plane if they work together. These were carefully morphed from a family of two (a triangle and a hexagon), in the process the triangle was split into the two roughly triangular things shown. Best regards, jj
[...]
Joerg, we must have different definitions. To my sight, snowflakes seem quite capable of tiling space over a wide range of elevations from plains to alpine. These tilings are non-compact, but opaque and reflective nonetheless. Let's not get distracted from the physical question of whether or not a broken-symmetry "rotoflake" could have fractal spiral arms. It's a possibility, but perhaps not the most likely. Another possibility is redemption of Gosper's "flowsnake". Obviously it is not a natural snow flake, but have we been too dense to consider that it could describe the shape of an *unnatural* snowflake? Here are a few more alternatives, made from a generalization of the Holladay-Ulam snowflake (see Mma code at the end): https://0x0.st/zkpa.png The pinwheels on the bottom right seem to be the most likely outcome of an experiment with a rotating hexagonal seed. Assume T = -15C, i.e. thin snow crystal. Rotation will produce a +/- pressure differential on the left/right halves of each prism facet. Pressure directly affects saturation, so the differential will alternatively encourage and suppress dendrite growth. If you cut, say, the left half off every axial dendrite, essentially you get a pinwheel. Could the experiment be done with Plate on a Pedestal?
http://www.snowcrystals.com/designer/designer.html
I don't think it would be infeasible to motorize rotation of the sapphire substrate, but then alignment becomes an issue. Alignment would also be an issue with rotating e-needles. The other issue would be additional heat from electronics. If an adequate solution could be found, I would say go for it! For one, it would lead to new and more unique works of art. More practically, natural snowflakes are not made sitting still, so experiments should develop motion control to probe the affect of rotational motion on formation. Happy 2020, --Brad ==========Mma Code========== MoveStar = {Cos[Pi/3 #], Sin[Pi/3 #]} & /@ Range[6]; ONRep[inds_] := ON[or_, x_] :> ( ON[or + MoveStar[[Mod[x + #, 6] /. {0 -> 6}]], Mod[x + #, 6] /. {0 -> 6}] & /@ inds) DelDup[ONs_] := Flatten[Select[ Cases[ONs, ON[#, _]] & /@ Union[ONs[[All, 1]]], Length[#] == 1 &]] Iterate[data_, inds1_, inds2_] := With[ {it1 = DelDup[Flatten[data /. ONRep[inds1]]]}, Union@Join[DelDup[Flatten[Join[data, it1] /. ONRep[inds2]]], it1, data]] DepictRepCol = ON[x_, y_] :> { Blend[{Darker@Blue, Darker@Cyan}, y/6], EdgeForm[Thick], Polygon[x + 1/Sqrt[3] {Sin[Pi/3 #], Cos[Pi/3 #]} & /@ Range[6]]}; AXIOM = MapIndexed[ON[#1, #2[[1]]] &, MoveStar[[1 ;; 6]]]; (* Example: *) Graphics[Nest[ Iterate[#, {-1, 0, 1}, {-2, -1, 1, 2}] &, AXIOM, 15] /. DepictRepCol] On Tue, Dec 31, 2019 at 6:57 AM Joerg Arndt <arndt@jjj.de> wrote:
Note that this flake tiles the plane. This does not seem to be the case with real snowflakes.
participants (2)
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Brad Klee -
Joerg Arndt