u/Frangifer

... ❝British Tars towing the Danish Fleet into Harbour :– the Broadbottom Leviathan trying to swamp Billy's old Boat, & the little Corsican tottering on the Clouds of Ambition.❞

I'd say this counts as propaganda rather than mere political caricature: it's clearly meant to enhearten the British populace regarding the conflict raging @ that time around the ascendancy of Napoleon Buonaparte as major player in European geopolitics.

... adducing a parallel between the situation of the then French Emperor Napoleon Bonaparte & the ancient Babylonian king Nebuchadnezzar - renowned for having had a vision of writing on the wall appear unto him in-connection with the prophetic activity of the Biblical Prophet Daniel ... see The Book of Daniel – The Bible.

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... ie with West @ the top & North to the right.

From

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Wikimedia Commons — Abraham Ortelius, Hiberniae, Britannicae Insulae, Nova Descriptio. Eryn. Irlandt

https://commons.wikimedia.org/wiki/File:Abraham\_Ortelius,\_Hiberniae,\_Britannicae\_Insulae,\_Nova\_Descriptio.\_Eryn.\_Irlandt\_%28FL35475086\_2582119%29.jpg

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I don't know why a 3,840×2,954 image downlodd, because that particular size isn't listed @ that page ... but, one way or another, it did !

A

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12,628×9,715 version is

https://upload.wikimedia.org/wikipedia/commons/7/72/Abraham\_Ortelius%2C\_Hiberniae%2C\_Britannicae\_Insulae%2C\_Nova\_Descriptio.\_Eryn.\_Irlandt\_%28FL35475086\_2582119%29.jpg

¡¡ 173‧31㎆ !!

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is also available ... but that can't be posted here, by reason of the inordinate size of the file.

The second figure originated with the goodly Stefan Felsner, & is actually the point–line dual of the figure @

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my previous post

https://www.reddit.com/r/mathpics/s/wwQ3Rxen5H

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. The rest are of a more technical nature – ancillary to the various reasonings adduced in the treatise the figures are from ...

... which is

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A Pseudoline Counterexample to the Strong Dirac Conjecture

by

Ben D Lund & George B Purdy & Justin W Smith

https://arxiv.org/pdf/1802.08015

¡¡ may download without prompting – PDF document – 139‧37㎅ !!

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. ＡＮＮＯＴＡＴＩＯＮＳ ＲＥＳＰＥＣＴＩＶＥＬＹ

•••••••••••••••••••••••••••••••••••••••••••••••••

Figure 4: The arrangement for j = 1, containing 3(6j + 2) + 1 = 25 pseudolines. Each pseudoline is incident to at most 10 vertices.

Figure 1: The dual of Felsner’s arrangement with 6k + 7 = 31 lines (including the line at infinity) and no line incident to more than 3k + 2 = 14 points of intersection.

Figure 2: A single wedge from Felsner’s arrangement.

Figure 3: The wedge for j = 1, the base case for our induction.

Figure 5: The wedge for j = 2.

... the 'set of lines spanned by the set of points' being the set comprising every distinct infinite line defined by having @least two of the points of the set lying on it.

If the points are in general position - ie no three in a line - then every point is incident to n-1 lines. So this problem is about arranging the points cunningly such that the point with the greatest number of lines incident to it, of all points in the set, has the least number incident to it, over all arrangements of points.

For a good while it was thought that Dirac's conjecture was true with c = 0 , but this infinite family of arrangements of 6k+7 points with none of them incident to more than 3k+2 lines (this instance, the one shown, being the k = 4 instance) proves that c ≥ 1½ .

Note also that the 'plane' in which the configuration is set is the projective plane , as two of the 6k+7 points are points-@-∞ .

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I actually queried this matter a fair-while-back @

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this post

https://www.reddit.com/r/askmath/s/7lJtmS7BxR

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@ r/AskMath

... but I don't know why I didn't put the figure in @ this channel, aswell ... but the recent appearance @ this channel of material about the no-three-inline problem has remounden me of it. At that post, I'm querying how it works, because @first I didn't quite get it ... but once I had got it it started seeming to me that we could actually do-away-with the points-@-∞ & have 6k+5 points with no point incident to more than 3k+1 lines, from which the same lower bound for c would follow ... but, especially considering how I was struggling with it @first, there's a likelihood I've missed something & am in-errour as to that ... & maybe someone here can confirm or refute it.

Also, I'm fairly sure that k needs to be an even № ... but the same caveat applies as just-above anent the reliability of my figuring.

... which carries Stretford Road across Princess Road, & is a nice example of the 'obliquely-set single-arch suspension bridge' trope.

It's actually pure coïncidence that the tree toward the right-hand edge of the frame is leaning in such way as to be parallel with the lower limb of the arch!

😆🤣

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Another Angle On It

https://www.reddit.com/u/Frangifer/s/AxsCW9vlE7)

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(Both photographs are @ that subpost ... so the second one is just the one already here.)

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... ie an arrangement, for each n , of the smallest possible № of points on an n×n grid such that adding a further point will necessarily induce some three in a line.

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By the goodly Robert Israel , from a reference found @

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Online Encyclopedia of Integer Sequences (OEIS) — A277433 Martin Gardner's minimal no-3-in-a-line problem, all slopes version.

https://oeis.org/search?q=A277433&language=english&go=Search

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Robert Israel, Examples for n <= 12 (provably optimal for n <= 10).

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I posted this earlier, & missed-off the last (n=12) one! 🙄

😆🤣

... an instance of which is the way, if we're trying to twist some cords extremely tight - say for an elastic-band-powered toy, or an antient Roman ballista for knocking-in a redoubt by hurling rocks @ it – there'll come a point @which the cords will cease to be nice neat straight muntually-twined helices & suddenly bunch-up into a 'globule', or 'knot'.

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And possibly the simplest instance of it is Michell's instability : if an elastic slender rod be bent-round, unto the two flat ends being upon eachother, to form a torus, & the ends be rotated relative to each other, so that the bent rod gets a twist in it, there'll come a point @ which the ring will convulse out of its plane into, initially, a non-planar lemniscate shape ... & by further twisting we'll have it writhing allover-the-place. It's a nice 'toy model' for more complex instances.

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And the goodly late Augustus Edward Hough Love , in his 1944 book A Treatise on the Mathematical Theory of Elasticity , presents a derivation to the effect that an ideal elastic rod becomes unstsble to small perturbations when the angular coiling density exceeds

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2√(KₛF)/Kₜ

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where Kₛ is the bending moment of the rod ( "s" for "skolition" ᐜ ), Kₜ the twisting moment ( "t" for torsion), & F is the tension applied to the rod.

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(ᐜ This choice stems from the garbagicity of Unicode subscripts. 🙄)

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From

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Numerical solution of a bending-torsion model for elastic rods

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by

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Sören Bartels & Philipp Reiter

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https://link.springer.com/article/10.1007/s00211-020-01156-6

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, @which there's compsibdriabobble disquisition upon this phenomenon, including about Michell's instability.

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The figures are in the order in which they appear in the treatise; & the last (13_ͭ_ͪ) is a montage of screenshots of the annotations excluding that of figure 9 , as I've left the annotation of that one with the figure itself.

... ie a pendulum that has its pivot vertically oscillated @ angular frequency ω that satisfies

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ω > √(2gl)/a

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, where l is the length of the pendulum, & a is the amplitude of the oscillation, & g is Earth's surface gravitational acceleration, & therefore is stable with the point mass _directly above_ the pivot.

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From

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Gereshes — Kapitza’s Pendulum

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https://gereshes.com/2019/02/25/kapitzas-pendulum/

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, @which there's considerable explication of the history & theory of this phenomenon.

Animation (First Item) From

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Emory — Random Close Packing

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https://faculty.college.emory.edu/sites/weeks/lab/rcp/index.html

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Static Image (Second Item) From

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Random-close packing limits for monodisperse and polydisperse hard spheres

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by

Vasili Baranau & Ulrich Tallarek

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https://pubs.rsc.org/en/content/articlehtml/2014/sm/c3sm52959b

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❝

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Fig. 1 Closest jammed configuration at a density φ = 0.662 for a random packing of 10 000 polydisperse spheres. The sphere radii distribution is log-normal and has a standard deviation σ = 0.3. The initial unjammed packing was generated with the force-biased algorithm at a density φ = 0.613

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Apologies for repeated attempts @ posting! ... there seemed to be difficulty with the animation uploading properly. 🙄

From

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Vice — LA Exhibit Traces 25 Years of Artistic Responses to Police Brutality

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https://www.vice.com/en/article/police-brutality-black-lives-matter-art-exhibit/

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I've slipped a second one in from the same page ... but I don't know the date-of-origin of it.

... although they're only 750V DC ones, I'm afraid 🙄 ... but they zig-zag just the same.

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Any train-geek is going to know why they do-so ... but for any non-train-geek reading: it's so that as the vehicle proceeds the point of contact between the wire & the pickup @ the top of the pantograph sweeps to-&-fro across it, greatly evening-out the wear & avoiding the cutting of a groove in the pickup.

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The stretch of track is from Timperley to Navigation Road (– Manchestyre – Hengley-Launde) .

From

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American Rails — Rock Island Railroad: Map, Pictures, Logo, History

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https://www.american-rails.com/crip.html

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. So I gather, the goodly town of

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Tucumcari – New Mexico

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https://youtu.be/gY3t36AF9Nw

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was an important location in that network ... & infact there was a major depot there. See the above-lumken-to wwwebsite that's the source of the image for a detailed disquisition about the network.

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... but (unless, perchance, there was a primitive line to the 'prototown') the movie is in-errour: the railroad was only opened in 1901; & with that event the inauguration of the town is customarily dempt ... but the movie is (I'm fairly sure) setten just-after the Civil War: isn't the goodly late (“I did get off – thank-you” 😄) Lee van Cleef's surly character supposed to have recently been a fairly high-ranking officer in that ghastly war, who's now diverting his well-honed military qualities into bounty-hunting?

Source of Images

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New Atlas — Omar Kardoudi — World's first consumer wing-in-ground effect aircraft takes flight

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https://newatlas.com/aircraft/navee-wig-plane-boat-consumer-wavefly-5x/

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