[math-fun] Re: Melbourne, city of Math
A nice cube is the Atomium. http://mathworld.wolfram.com/Cube.html
I propose that someone build a tetrahedron with vertices at 33.87 S, 151.20 E Sydney, Australia 24.80 S, 70.00 W northern Chile, S of Antofagasta 66.69 N, 151.70 W central Alaska, near Bettles Field 3.35 N, 39.92 E northeast Kenya (Thanks to Fred Helenius for identifying this "land tetrahedron": see http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/geographicals .) I picture something like the Atomium, with struts that disappear into the earth. Presumably the struts would terminate a few feet below the ground, although the viewer would be invited to imagine them as uninterrupted, several-thousand-mile-long beams. Who would pay for this? Well, if the struts are scaled-up version of the Zome construction system, the Zome guys might view the project as free advertising. Alternatively, the tetrahedron's latent It's-a-small-world-after-all, Building-bridges-between-nations political message might attract some potential funders of this conceptual art project. (To underscore the one-world theme, one could add video cams that let the viewers at each site watch what's happening at the other sites, along with a globe that shows the four vertices and the signal-paths that join them.) Or, maybe Sydney would fund the whole project, as a way of besting Melbourne? :-) Jim Propp
Jim Propp wrote:
I propose that someone build a tetrahedron with vertices at
33.87 S, 151.20 E Sydney, Australia 24.80 S, 70.00 W northern Chile, S of Antofagasta 66.69 N, 151.70 W central Alaska, near Bettles Field 3.35 N, 39.92 E northeast Kenya
(Another tetrahedron has vertices near Auckland, NZ; Cape Town, South Africa; Tegucigalpa, Honduras; and Irkutsk, Russia.) I think the four "tetrahedron vertex" sculptures sounds like an outstanding art project. If none of Jim's suggestions work, surely there's someone who made far too much money in the dot-com boom, who's just sitting around waiting to fund something like this! --Michael -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.
James Propp wrote:
...I propose that someone build a tetrahedron...
Michigan artist David Barr designed his "Four Corners Project" in 1976. It is an Earth-sized regular tetrahedron that spans the planet, with just the tips of its four corners protruding. These visible portions are four-inch tetrahedra, which protrude from the globe at Easter Island, Greenland, New Guinea, and the Kalahari Desert. Barr traveled to these locations and was able to permanently install the four aligned marble tetrahedra between 1981 and 1985. For details, see: Sandra L. Arlinghaus and John D. Nystuen, Mathematical Geography and Global Art: the Mathematics of David Barr's "Four Corners Project" Institute of Mathematical Geography monograph series 1986 http://www-personal.umich.edu/%7Ecopyrght/image/monog01/fulltext.pdf George http://www.georgehart.com
Many years ago i saw an abstract in the Oberwolfach Vortragsbuch entitled "Squares in Lake Michigan", which for a long time I thought proved theorems such as "any Jordan curve - or distorted circle in the plane - contains 4 points which are at the vertices of a square". But I never saw anything more about this, and began to doubt my memory. Just now I found the following on MathSciNet, so maybe it was not a dream: MR0474041 (57 #13698) Fourneau, René; Leytem, Charles Sur l'existence de $n$-losanges réguliers inscrits dans un corps compact convexe. (French) Comment. Math. Univ. Carolinae 19 (1978), no. 1, 151--164. This paper provides a solution for the generalized problem of "squares in Lake Michigan" for some classes of convex sets. The authors show that, in $R^d$, one can inscribe a regular crosspolytope (the generalization of the square and of the regular octahedron) in every compact convex body with an axis of revolution, and in every centrally symmetric compact convex body. In both cases, the uniqueness of such an inscribed regular cross polytope is investigated. NJAS
On 9/4/06, N. J. A. Sloane <njas@research.att.com> wrote:
Many years ago i saw an abstract in the Oberwolfach Vortragsbuch entitled "Squares in Lake Michigan", which for a long time I thought proved theorems such as "any Jordan curve - or distorted circle in the plane - contains 4 points which are at the vertices of a square". But I never saw anything more about this, and began to doubt my memory. Just now I found the following on MathSciNet, so maybe it was not a dream:
Hi Neil and all, I was just reading Peter Winkler's book _Mathematical Puzzles_, and in it I saw the equivalent of "squares in lake michigan" as an unsolved puzzle. He says there are proofs that sufficiently smooth curves always contain a square, but no general proof that every Jordan curve contains a square. --Joshua Zucker
On Tue, 5 Sep 2006, Joshua Zucker wrote:
On 9/4/06, N. J. A. Sloane <njas@research.att.com> wrote:
Many years ago i saw an abstract in the Oberwolfach Vortragsbuch entitled "Squares in Lake Michigan", which for a long time I thought proved theorems such as "any Jordan curve - or distorted circle in the plane - contains 4 points which are at the vertices of a square". But I never saw anything more about this, and began to doubt my memory. Just now I found the following on MathSciNet, so maybe it was not a dream:
Hi Neil and all, I was just reading Peter Winkler's book _Mathematical Puzzles_, and in it I saw the equivalent of "squares in lake michigan" as an unsolved puzzle. He says there are proofs that sufficiently smooth curves always contain a square, but no general proof that every Jordan curve contains a square.
I thought I saw an article in the popular press within the last year or two in which a mathematician noticed his table wobbling, and found that by rotating it he could get all 4 legs stable. So he investigated it and wound up proving that it's possible to do so for any surface. I'm not turning anything up on google, though. -J
Quoting Jason Holt <jason@lunkwill.org>:
I thought I saw an article in the popular press within the last year or two in which a mathematician noticed his table wobbling, and found that by rotating it he could get all 4 legs stable. So he investigated it and wound up proving that it's possible to do so for any surface. I'm not turning anything up on google, though.
Apparently old problems neither die nor fade away (saying from back then). This "theorem" was running around math students back in the early fifties, as I recall. Doubtful that it was new even then. Had something to do with the theorem of the mean, I think. - hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
participants (7)
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George W. Hart -
James Propp -
Jason Holt -
Joshua Zucker -
mcintosh＠servidor.unam.mx -
Michael Kleber -
N. J. A. Sloane