What if the second map lands on its side?
Do you have some intuitive explanation of Borsuk-Ulam? From wikipedia:
The case n = 1: n=1 can be illustrated by saying that there always exist a pair of opposite points on the Earth's equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously.
Not that I doubt the result, but it's not al all obvious to me why this should be true.
2: Pick two opposite points on that circle. They have temperatures X and Y and we have their difference X-Y. Rotate both points along the circle (maintaining the fact that they're opposite) until they've switched place. Now the difference is Y-X. Somewhere between X-Y and Y-X the difference had to go through zero, and that's it.
2: Take an arbitrary point on the equator, P, and its temperature, T(P) which is a continuous function of position. Let -P be the antipode of P. If T(P) = T(-P), we're done. Otherwise, form the function f(P) = T(P) - T(-P), which will also be a continuous function of position. f(P) = -f(-P). So if we rotate P through 180 degrees to -P, by the intermediate value theorem, there must be some intermediate point Q for which f(Q) = 0, as the function passes from positive to negative values (or vice versa). That means that T(Q) = T(-Q).
For the crumpled map of the US, I think you are looking for the Brouwer fixed-point theorem (see the Illustrations section), which seems to be related to Borusk-Ulam. The country map needs to be "without holes", so presumably it only applies to the lower 48, and only if the crumpled map lies completely within the original country boundaries. Being on its side (#1) wouldn't be a problem as long as it fits within the original boundary - all the points in a vertical line would map to the point directly below them.
Have the British voters noticed that trade agreements with the United States now have bigger problems than our filthy chickens?
The fun thing is that for a N-dimensional sphere (counting Earth as a 2-dimensional sphere; not sure that that's the usual terminology) you can do this for N variables at once. On the equator ("1-D sphere") you can find opposing points with the same temperature; on the whole Earth you can find opposing points with the same temperature and pressure, or pressure and humidity, or elevation and windspeed (but not all of them at once).
Informally, since you can do 1 variable on any great circle you pick on the sphere, there's actually a continuous loop around the Earth where any one variable is the same at every set of opposing points in the loop; there's a loop for every variable; and any two such loops must intersect at at least one (pair of opposing) points.
Have the British voters noticed that trade agreements with the United States now have bigger problems than our filthy chickens?
I'm pretty sure the Earth had three dimensions. Some of my best friends live there.
I checked Wikipedia and it still does.
9: Yeah, but the surface of the Earth is 2 dimensional (you can specify any point on it by (latitude, longitude)). So it's considered a 2-sphere. If you include the interior, then you get to three dimensions.
9: I didn't know you had friends in low places.
For the crumpled map of the US, I think you are looking for the Brouwer fixed-point theorem (see the Illustrations section), which seems to be related to Borusk-Ulam.
Oh, you're right. I didn't think it through.
12: Apparently I can't, since the Earth is a smooth sphere.
Ulam had a random career. Besides the hydrogen bomb and Borsuk-Ulam, he also contributed to set theory (measurable cardinals). He also invented Monte Carlo simulation (which is weird to think that someone had to invent).
Monte Carlo itself was more socially constructed than invented.
I'm reassured that the article in question is from the LRB - going from the contents of the rest of the particular site which mirrors it.
That last paragraph is really not supported by the rest of the article, is it?
No. Doesn't say a goddamn thing about maps.
No. Assuming it's all true it does show narrowly that GB essentially secured a major ongoing subsidy for its nuclear complex, and on the face of it it's reasonable to assume that this also foreclosed co-operation with the French. It doesn't show the broader significance of all that. But that's worth thinking about. So think, British people. Or French, if you're lurking.
There is a bit of a jump from "Anglo-French collaboration on weaponeering" to "common European nuclear defence policy" which probably needs examining.
Also, subsidy is probably the wrong word because it implies a cash transfer. The US is sharing knowledge which it has anyway, and which otherwise the UK would have to discover itself, at great cost. But the act of sharing knowledge doesn't cost the US anything.
And you could even argue that it's belated compliance with the wartime agreement on nuclear tech that Truman decided to break unilaterally in 1946.
"If you crumple up a map of the United States and drop it on another map of the United States, some geographical point will land on itself."
Is this also true if you drop a map of the United States on the ground (assuming you are in the United States)?
23: Fine. The British nuclear program secured substantial cost savings.
24: The effect was the same either way.
22: Yes. I think the alt-hist is still interesting well short of that threshold, even short of the Anglo-French threshold. If GB financed its nukes independently the cost presumably would have come from other defense spending: where, with what consequences, and what effect on British self-image? Earlier withdrawal west of Suez? Earlier drastic navy cuts? Earlier independence for a bunch of colonies? Disorderly withdrawal from Malaya?
The French option is more nebulous because much would have depended on how it was spun. Britons might have ended up as ignorant of the (Franco-) British nuclear deterrent as of the (Americo-) British (though I suspect CDG wouldn't have allowed that). And in any case I don't know how important the indepent(ish) nuclear deterrent actually is. (Or more precisely, how much play of it was made during the formative years of the current Brexiteer core.)
And there's more than weapons. The French weapons program was tied up with power generation: what if weapons co-operation had produced similar power generation (the French certainly would have tried). And what might that have done to anti-nuclear politics? To British coal and unions? On the outside, what if the French nuclear power project was successfully folded into the European project?
And in any case I don't know how important the indepent(ish) nuclear deterrent actually is. (Or more precisely, how much play of it was made during the formative years of the current Brexiteer core.)
For as long as I can remember, and considerably before, the Tories have used CND/unilateral disarmament as a stick to beat Labour with. How effective that has been has varied over the years, I suspect.
And was/is France held as an example of fitlhy softheaded naive leftism over the same period?
Let me be the first to suggest that a nuclear winter night counteract global warming. But it needs to be a smaller one than U.S. vs Russia or China. France vs North Korea?
I'm probably not the first to suggest it.
Nukes are way down since START. US-Russia would be fine.
And was/is France held as an example of fitlhy softheaded naive leftism over the same period?
Pretty much, though not so much on military/foreign policy. The right pointed (still does) to France's periodic civil unrest as a sign of the evils of unionisation, and the shortness of the French working week was always bandied about.
Somebody parked a car outside the FBI's Pittsburgh office with "Russia plants microradio devices in ppls ears & controls then using a sound similar to a dog whistle. Subliminal hypnosis" painted on the side. Personally, I think they're just paying Trump in money, but I understand the fear.