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skip to content menu and widgets about me about this blog contact me video efficient networking service recent posts the 0.7% puff pastry my longest shortest shadow faces in the snow categories art (1) calculations (8) making (8) programming (1) science (11) uncategorised (4) tag cloud arduino art chemistry environment ergonomics estimation geometry making maths mountains paraboloids physics polyhedra programming seasons thought experiment vision search for: archives march 2018 january 2018 december 2017 november 2017 october 2017 july 2017 may 2017 april 2017 january 2017 september 2016 august 2016 july 2016 may 2016 april 2016 february 2016 january 2016 november 2015 meta log in entries rss comments rss wordpress.org the 0.7% no, this post isn’t about wealth inequality. it’s about daylight inequality. today is the day of the spring equinox. for the past three months, the days have gradually been getting longer, and from tomorrow, the sun finally starts to spend more time above the horizon than below it*. it’s also the day when everyone in the world enjoys a day of approximately equal duration. but from tomorrow onwards until the autumnal equinox in september, the further north you are, the longer your days will be, in the sense that the sun will spend more time above the horizon. i wondered where edinburgh, where i live, fits into this scale of day lengths. for the next six months, we’ll have longer days than anyone living south of us. what fraction of the world’s population is that? i estimate that, over the coming summer, we’ll have longer days than roughly 99.3% of the world’s population. i was very surprised at how large this number is. and delighted too: it somehow seems to make up for the seemingly endless dark dreich dampness of the scottish winter. the calculation rather than counting the number of people who live south of edinburgh, it’s easier to count the much smaller number who live further north. there are a few countries that are wholly north of edinburgh, namely finland, norway, latvia, estonia, iceland, the faroes, and greenland. there are some countries that are partially north of edinburgh: sweden, denmark, and lithuania. and then there is russia, which spans a vast range of latitudes, but which has relatively few cities north of edinburgh, of which the largest and/or most well known are st petersburg, nishny novgorod, perm, yekaterinburg, tomsk, archangelsk, and murmansk (though i counted a few more). there’s one state of the usa: alaska. finally we have aberdeen, inverness, dundee and perth, the main centres of population further north in scotland. to roughly compensate for the fact that these counts don’t cover minor centres of population, and that the northern part of moscow probably overlaps somewhat with edinburgh, i included the whole of sweden, denmark and lithuania in the sum, despite the fact that the countries have major towns that are south of edinburgh. when i added it all up, it came to just under 44 million people living north of edinburgh, against a world population at the time (i actually did the sums a few years ago) of 6.8 billion. expressed as a percentage, 99.35% of the world’s population live south of edinburgh, which i’ll round to 99.3% to avoid overstating my case. *actually, the sun appears to be above the horizon even at times when, geometrically speaking, it is slightly below it. this is because the light rays are refracted by the atmosphere and so travel in slight curves rather than straight lines. posted on 20th march 2018 categories calculations tags seasons leave a comment on the 0.7% puff pastry i was rather taken by the video above, which i first saw on core77 . i started wondering how many times you have to put the roll of silicone material through the machine to get satisfactory mixing of the two colours of material. the people in the video consider the job done after four passes. what does that mean in terms of the thickness of the red and white layers within the material? the roll is a rather complicated object, so i worked with an idealised version of the real process, where the sheet emerging from the rollers isn’t rolled up, but cut into several pieces which are stacked up before being passed through the rollers again. i came up with the following: after only 2 passes, the layers in the slab are too thin to see with the naked eye. and by some margin, too: there are over 600 of them and they’re only a fortieth of a millimetre thick. if you made a perpendicular cut through the slab, it wouldn’t appear to have red and white layers in it. after only 4 passes, a standard compound microscope operating in visible light wouldn’t be able to resolve the layers in the slab. after only 6 passes, the layers would be thinner than the width of the molecules of the silicone material. at this stage the concept of red and white layers no longer makes sense. these results will only apply to material near the centre of the roll. it’s easy to see from the video that material near the edges is not mixed so well. the calculation from the video, it looks like there are about 9 turns in the roll. each time the roll is flattened by the rollers, those 9 turns are converted into 18 layers. the resulting sheet is rolled up and passed through the rollers again, multiplying the number of layers by 18, and so on. this doesn’t work at the sides of the roll. we’ll ignore that complication, and work with a flat analogue of the actual situation. we’ll assume that we start with two long rectangular flat sheets of material, a white one and a red one, laid on top of each other. we’ll cut this assembly into 18 identical pieces, and make a stack of them; this stack will have 36 layers. we now flatten this stack in the rollers, cut it into 18 pieces, stack them up (giving us 648 layers), and repeat. on emerging from the roller, the sheet appears, by eye, about 1.5 cm thick. we’ll assume that we start with two layers of half this thickness. the table below shows the number of layers and the thickness of each layer after 0, 1, 2, 3… passes through the rollers. number of passes number of layers layer thickness (m) 0 2 7.50 × 10 -3 1 36 4.17 × 10 -4 2 648 2.31 × 10 -5 3 11 664 1.29 × 10 -6 4 209 952 7.14 × 10 -8 5 3 779 136 3.97 × 10 -9 6 68 024 448 2.21 × 10 -10 we can identify various milestones, as follows: limit of visual acuity. a person with clinically normal vision can resolve detail that subtends roughly 1 minute of arc at the eye. at a viewing distance of 30 cm, this corresponds to about 0.1 mm (10 -4 m). the layers of material are much thinner than this after only 2 passes. if you made a perpendicular cut through the slab of material, after two passes you wouldn’t be able to see the layered structure. (this might not be true if the cut was oblique.) limit of standard light microscopy. a compound microscope working in visible light can resolve detail down to about 200 nm (2 × 10 -7 m). the layers become thinner than this after only 4 passes. single-molecule layers. the question here is the number of passes needed before the layers are less than a molecule thick (at which point the idea of layers fails). the difficulty is that molecules of silicones are long chains, and these chains are almost certainly bent, so their size is ill-defined. this part of the calculation will be hugely approximate. we’ll be as pessimistic as possible, assuming that the molecules are roughly straight and that they lie parallel to the layers in the slab of material. polydimethylsiloxane a common silicone material is polydimethylsiloxane or pdms. this consists of a silicon-oxygen backbone with methyl groups attached. the lengths of carbon-silicon and carbon-hydrogen bonds are 1.86 × 10 -10 m and 1.09 × 10 -10 m respectively. so the width of the molecule is going to be, very, very approximately, of the order of 4 × 10 -10 m. the layers are thinner than this after only 6 passes. posted on 17th january 2018 17th january 2018 categories calculations , sc

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http://bencraven.org.uk/tag/ergonomics/
http://bencraven.org.uk/2017/01/14/a-shape-designed-by-nature/
http://bencraven.org.uk/tag/environment/
http://bencraven.org.uk/2017/05/22/the-kinetic-energy-of-a-drifting-tectonic-plate/#respond
http://bencraven.org.uk/#content
http://bencraven.org.uk/tag/art/
http://bencraven.org.uk/2018/03/20/the-0-7/
http://bencraven.org.uk/tag/geometry/
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http://bencraven.org.uk/2016/07/
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http://bencraven.org.uk/category/science/
http://bencraven.org.uk/2016/02/21/irregular-but-not-random/
http://bencraven.org.uk/2017/12/
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edwinpickstone.co.uk
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