"Excluded volume" between close-packed spheres

 "Excluded volume" between close-packed spheres  

I remember reading that the 'theoretical' limit on volume density for close-packed uniform rigid spheres was circa 0.74, (so that in a bucket of marbles the volume of trapped air is 26% of the bucket volume.). Indeed, the Wikipedia article on "Sphere Packing" tells us that Johannes Kepler thought that the densest possible packing would be 0.74048 (i.e. π/(3√2)), a result confirmed by Carl Friedrich Gauss 2 centuries later. Further, that Thomas Callister Hales claimed a "proof" of Kepler's conjecture which a referee found 99% convincing. 

 I thought I ought to be able to calculate that myself, now that I have retired and have the leisure time needed for such an non-essential occupation.

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First I considered laying pennies on a flat surface, and was delighted to find that exactly six discs can be fitted round a central disc such that all are touching (Fig. 1).

Next, I considered a football (radius r) closely fitted inside a cubic container (side 2r; Fig. 2). 

The volume of the ball (V_b) is V_b = 4 π r³/3; that of the container (V_c) is V_c = (2r)³ = 8 r³. So the volume density (V_d) of the container filled by the ball is: 

    V_d = V_b/V_c  =  (4 π r³/3)/(8 r³)

∴ V_d = 4 π / 24

∴ V_d = π/6 = 0.5235987756 (when rounded up). 

I concluded that, if many spheres were aligned (touching) in straight Euclidian lines mutually perpendicular in three dimensions, the density would still be 0.5235987756 (and the excluded volume would be 47.64%, or nearly so). This array can be seen as a series of x/y planes stacked one on top of the other in the vertical (z) dimension. Each sphere touches 6 neighbouring balls; 4 in the same plane, plus 1 above, and 1 below. But they can clearly be packed more tightly. 

So I constructed a single plane as follows. First I laid a file of spheres in the x dimension. Then a second file of spheres, but those of the second file do not coincide with those of the first file, but are offset to the right by r, while those of the third file are offset to the left by r (and thus coincide with the first file).  Clearly the centre of a spheres in the second file is equidistant from two spheres in the first file. Indeed, they form an equilateral triangle, side 2r (Fig. 3). 

Drawing a perpendicular (length m in units of r) from the apex to the base, we can use Pythagoras's theorem to calculate m (See Fig. 4 below).

    m² + r² = (2r)² 

∴ m² = (4-1)r²

∴ m = 1.73205081 x r

It is clear that a single layer of this array is more compact than the first array by the factor m/2 (=0.8669254); or more "volume dense" by the factor 2/m (= 1.154700536759). It would be the same if we stacked many layers of the 60º-offset array, such that each sphere touches 6 neighbours in the plane, with one above and one below.

∴ V_d of Loose 3d array = 0.5235987756

∴ V_d of semi-compact 60º-offset array =0.5235987756/0.8669254 = 0.6045997881

But there is more compacting we could do.

Looking down on our single layer of the 60º-offset array (i.e. the marbles in Fig. 3 above), we see that there is a dip formed by 3 contiguous spheres where a sphere of the next layer should lie, touching all three spheres in the lower layer. We can see that there will form a triangular pyramid where every straight line between the apices is of length 2r; indeed a 'regular tetrahedron'. But how high (in the z dimension) is the upper sphere above the 3 lower spheres?

The apex of a vertical equilateral triangle drops a perpendicular of length m to its base, as we calculated above:

m² + r² = (2r)² 

∴ m² = (4-1)r²

∴ m = 1.73205081 x r

But this equilateral triangle is leaning inwards to form one face of the pyramid. The perpendicular line (length p), from the apex to the point on the base equidistant from the three sphere-centres (point e), can be found in the same way; but this was (for me) the most difficult step. I kept getting the wrong answer. It was only after  3½ days that I solved the puzzle. The right triangle we have to solve has hypotenuse 1.73205081 (length in units of r), while the base is not r but Tan(30º) = 0.577350 (in units of r). 

To see this look at the equilateral triangle that forms the base of the pyramid. There is a point (a) midway along one side (so r from either end). A line can be drawn across the triangle to the opposite point. That can be repeated for the other sides. These lines intersect at the point we called e, cutting each line into a shorter and a longer portion. The shorter portion is Tan(30º) while the whole line is Tan(60º). By Pythagoras's theorem:

p² + (Tan(30º))² = (1.73205081)² 

∴ p² =  (1.73205081)² - (0.577350)²

∴ p = 1.63293936 x r  

If our layers are extended infinitely in the x and y dimensions this compaction in the z dimension will affect the overall volume density of the array:

∴ V_d of compact array = 0.6045997881 x 2/1.63293936 = 0.74048049

Q.E.D.

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Apple versus Google

 Apple versus Google

There is said to be a ‘cold war’ between Apple and Facebook (New Statesman 16-22 July 2021). There may also be a rivalry between Apple and Google. 

I have spent three days trying to get to the bottom of this puzzling problem. My ageing MacBookPro is getting slower and slower. On Monday this week I noticed that it was (in its own words) “Downloading 98,000 emails”. Which is quite daft. And after watching the first 1,000 come down, it was clear that the machine would be out of commission for the whole day. I did not want any of those emails. I do not mind Google storing all my mail (going back 15 years) on its capacious servers; if it wants to do that. But I keep on my MacBook the emails I want to keep, and delete the 2/3 that I do not want.

I found that unplugging the power inlet stopped the nonsense. Then I tried ‘googling’ the problem and found considerable discussion from the period 2017-2019. Thus, an article by Elizabeth Jones from 2019 is titled “Mail App Always Downloading on Mac? How-To Fix” (https://appletoolbox.com/mac-mail-app-always-downloading-fix/). She recognised the problem, and tried to meet it, suggesting:

• Relaunch the Mail app while holding the Shift key on your keyboard;
• Change your mail account’s setting for storing the Drafts Mailbox Behaviors to “On My Mac”;
• Take your Mail Account offline temporarily and then take them back online
• Remove the Mail Account and then add it back;
• Try rebuilding and reindexing your Mailbox.

Five remedies immediately aroused my suspicions. Why not cut to the chase and use the right fix? But I tried the first 3, setting all my email addresses to “store drafts in a ‘drafts’ folder on the MacBook; with no immediate benefit. (The last two seemed excessively (in medical parlance) "invasive".)

I surfed further, till I came to the suggestion of emptying the Draft Emails box, (easily done but useless), then the deeper suggestion of going to the G-mail site on a web browser and emptying the Draft Emails box there. Now, that was interesting. For there were 2,100 draft emails visible on the browser web-mail site, but none when I used AppleMail’s IMAP approach.

Well, after breakfast, I started to delete these 2,100 drafts. But they only emerged singly, or at best 8 at a time. This was going to be tedious. A draft appeared called “North Downs way”, was selected, deleted, but then up it popped, again; like the Hydra’s head. I seemed not to be gaining on the monster, till I noticed that the number of remaining drafts did reduce each time; 2,000, 1,999, 1998 etc.. Then I twigged! While writing an email, the Apple Mailer records a draft every 20 seconds or so, all under the same name as in the subject line. There may be 20, or 80 drafts of a single email. 

By mid-afternoon I had got rid of 1,100 of my unwanted drafts, but 1,000 remained. 

Some people like “threading” emails into “conversations”, so that if you reply this week to mine of last week they are stored together. I do not. I like my emails to appear in date order, and I had found and unticked the “Please thread my emails” option on AppleMail. With a stroke of genius, I thought to go to the web-mail “settings” and re-tick the threading option there. Now, like ‘Jack-the-Giant-Killer’ each swipe got rid of 50 or 100 drafts all dangling on that silly ’thread’’. In two more minutes the job was done. I had freed up 1 GByte of Google storage space.

But I was now in open country, on my own and ahead of all the boffins and commentators. No one had suggested that the problem was to do with ’threading’ . Could it be that Apple defaults to no-threading, Google to threading? Or did Apple misunderstand Google’s signal for threading? Or, (more sinisterly) did Google deliberately change the signal without telling Apple? They both emerge from the affair a little “muddy”.

I am still bothered by a slow machine. And am highly scornful of the way Apple has messed around with its iCloud (giving me 2, 4, and sometimes 8 copies of each photo). And of its aloof indifference to customers’ problems. And at the way perfectly good hardware becomes obsolete, and needs replacing. 

I hope this train of thought might be useful to some others. 

☆  ☆  ☆  ☆  ☆

Tom Nairn: Scotland's leading political theorist?

Tom Nairn has been called “Scotland's greatest thinker”, and “Britain’s leading political theorist”, and “by far Scotland’s pre-eminent political intellectual”, all of which sound over-the-top. However, there is no denying that he can claim to be the most prescient prophet of the disintegration of Britain, on the basis of his 1977 book “The Break-Up of Britain”. But who is he, and what has he contributed to political thought in the last 70 years.

Rory Scunthorne’s article in the New Statesman (30 Jul - 19 Aug 2021; from which I take most of my quotes), piqued my interest. But its tedious length and its disjointed stream of inscrutable quotations left me confused and frustrated. So I went to the web. Here below I offer a synthesis; and a conclusion.

CV

    ▪    Born in 1932 in Freuchie, (a small town in Fife, Scotland), where his father was a local head teacher. Nairn attended Edinburgh College of Art, and Edinburgh University where he graduated MA in philosophy in 1956.
    ▪    In 1957, with a British Council scholarship, Nairn enrolled in the Scuola Normale Superiore in Pisa, where he studied politics and encountered the evolving communism of Gramsci, and the strategy of the “long haul”.
    ▪    From 1962, with Perry Anderson in the New Left Review (NLR), he developed a thesis (the "Nairn-Anderson thesis") to explain why Britain did not follow other European nations in their rejection of established religion, and monarchy.
    ▪    He taught at the University of Birmingham (1965-6) and elsewhere.
    ▪    In 1968, Nairn was fired from his teaching job in Hornsey College of Arts, for participation in  a lengthy utopian “sit-in” involving both students and staff. He was clearly shunned by British academic institutions for decades.
    ▪    From 1972–76, with help from a NLR colleague (Anthony Barnett), Nairn was employed in the Transnational Institute, Amsterdam; a non-profit think-tank largely funded by the Dutch Government.
    ▪    He spent 1994-5 at the Central European University (Austria-Hungary) with the sociologist Ernest Gellner, who had argued that Nationalism had helped the development of industrialization.
    ▪    In 1995, he set up and ran (1995-1999) a Masters course on “Nationalism” at the University of Edinburgh .
    ▪    In 2001-2010 he was invited to take up an “Innovation Professorship in Nationalism and Cultural Diversity” at the Royal Melbourne Institute of Technology, Australia,
    ▪    Returning to the UK he became a fellow at the Institute for Advanced Study of Durham University (2009).

Personality

He was said to be an excellent cook by a flat-mate in 1970 in Edinburgh. Said also to be “utterly single-minded” yet “resigned”; even “optimistic”. Also “reserved”, and lacking in even the British level of sociability; fiery in writing, but shy in person. He enjoyed Italy, and became proficient in Italian, but also spent time in Amsterdam, Paris and Vienna.

Thinking

Several ‘periods’ can be discerned, and with each an influencer or colleague.
[1] The Italian period and the romance of communism, immersed in Gramsci.
[2] The Perry Anderson period (1962-1965), and the New Left Review. The Nairn-Anderson thesis was that the British state was archaic. The early revolution of 1642-1660 established a consolidated ‘pre-modern’ political structure in England by 1688.  After the Union of 1707 this was inherited by Scotland.
[3] Political period. Nairn was pro-European, and therefore impatient with the UK’s Labour Party which was insular. He joined the Scottish Labour Party (1976) to advocate devolution in a European context (c.f. the 'auld alliance'). His book The Break-Up of Britain (1977, revised 1982) predicted by 45 years the present state of the British union that is ‘Great Britain and Northern Ireland’.
[4] With his coinage “UKania”, Nairn ridiculed the Ruritanian elements that survive in Britain. His anti-monarchical views were concentrated in his book The Enchanted Glass (1988).
[5]  ‘Nationalism’ period. With Ernest Gellner in Vienna, Nairn developed an analysis of Nationalism that extended to post-colonial countries and incorporated the role of myths and artefacts in the creation of national consciousness. He rode this wave in Edinburgh and Melbourne, and is still active in Durham.
[6] All his life Nairn has been a prolific writer. In all, he wrote 14 major books and numerous articles in the New Left Review and the London Review of Books, and elsewhere.

Conclusion

I get the impression of a shy intellectual who benefits from collaboration, but who takes up and develops a thesis with great focus and tenacity. A product of his time and place. So: Scottish; unselfish, indeed actively anti-selfish; anti-privilege, anti-London, anti-Conservative, anti-royalist, pro-Europe; not above using some of the ‘tricks’ of nationalism to further his objectives. Is he Scotland’s pre-eminent political thinker? He maybe the winner, but of an barely contested prize.

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