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April 30

partition unit interval into countably many identical pieces

This problem, proposed by Hugo Steinhaus, was solved by John von Neumann in 1928. I read about it in Freeman Dyson's article about stuff von Neumann had worked on.([1] p. 156 col. 2) The problem is to partition the real unit interval into a countably infinite collection of subsets so that each one is a rigid translation of any of the rest.

Dyson's article didn't describe von Neumann's solution completely but it gave a few hints (mainly that the axiom of choice is involved). From then it doesn't sound hard: use AC to well-order the points in the interval, then deal them into countably many "buckets" by transfinite recursion keeping the rigid translation invariant. The procedure is start (say) with b_i={1/i} for i=1,2,3... and find the difference set D={1/i - 1/j | j >= i} (so 0 is in the difference set). Then as you peel off each real from the ordering, see if there's a number already in one of the buckets that differs from the new one by an amount in D. If yes, put the new number into the right bucket for that distance. Otherwise, put it in an arbitrary bucket (maybe the first one). I think this can be formalized straightforwardly.

Of course that simplicity is from a perspective of 80 years later, while von Neumann had to invent all the machinery from nothing. Still, is there some subtlety to this problem that I'm missing? I'm not sure of my solution. Thanks. 173.228.123.166 (talk) 08:25, 30 April 2018 (UTC)[reply]

This is usually called the Vitali set, and our article doesn't mention von Neumann, but says it was described by Giuseppe Vitali in 1905. I'm not quite sure what your question is, and I don't quite follow your construction, though that might be because I haven't read it carefully enough. It looks like the idea is at least similar to the standard one, though. --Trovatore (talk) 08:50, 30 April 2018 (UTC)[reply]

Thanks! My question is basically whether my solution is any good. It's possible I misunderstood the problem, or made an invalid step, or otherwise goofed up in some dumb way. But now I should find out whether the Vitali set is different from what von Neumann did. His paper (per Dyson) is Die Zerlegung eines Intervalles in abzählbar viele kongruente Teilmengen, Fund. Math 11, 230–238 (1928) so I might try to find it. 173.228.123.166 (talk) 10:00, 30 April 2018 (UTC)[reply]

It might be that I misunderstood the problem. The Vitali-set stuff I pointed you to is about translations with wraparound. That is, you partition the unit interval into a countably infinite collection of pieces, such that each piece is a translate of each of the others on the unit circle.

I wouldn't have thought it was possible to do it without wraparound, but I haven't been able to think of a contradiction, so maybe it is. Your argument may be good. I haven't thought it through thoroughly yet. --Trovatore (talk) 19:06, 30 April 2018 (UTC)[reply]

Ok, yes I looked at the Vitali set article and it's different from this. Maybe to some extent I'm asking a history-of-math question. If my solution above is right, then the problem is trivial with today's "technology", yet it still took a giant like von Neumann to solve it in 1928. Von Neumann's 1926 PhD thesis was about the N part of NBG set theory and he also invented our current notion of ordinals as sets, and I guess transfinite induction as well. If the 1928 paper was where that happened, it would have been a major advance, but Dyson described it as an almost-throwaway. Maybe it was a case of inventing an important part of set theory, and then separately as an aside, recognizing an existing open problem to use it on. 173.228.123.166 (talk) 00:48, 1 May 2018 (UTC)[reply]

Fundamental theorem of normal axonometry

Our article Carl Friedrich Gauss currently asserts that "Gauss stated and proved the fundamental theorem of normal axonometry." But what is that theorem? ◄ Sebastian 13:51, 30 April 2018 (UTC) Oops, the claim is actually in List of things named after Carl Friedrich Gauss. ◄ Sebastian 13:53, 30 April 2018 (UTC)[reply]

Web search finds some descriptions like here. The theorem seems to say that if you project a cube to a plane with one vertex at the origin, then treating the plane as the complex numbers a+bi, then the three adjacent vertices of the cube will be at complex points A,B,C where

A^{2}+B^{2}+C^{2}=0

. 173.228.123.166 (talk) 16:27, 30 April 2018 (UTC)[reply]

The link provided by 173.228.123.166 does not contain a description and therefore does not answer the question. It is interesting, though, in that it asserts that Gaus stated that theorem (but without proof) in a book published in 1876. Incidentally, I found the wording for The fundamental theorem of oblique axonometry: “Any three non collinear segments in a plane with the same endpoint can be considered as the oblique parallel projection of a tripod.”[2]. ◄ Sebastian 09:57, 2 May 2018 (UTC)[reply]