November 2011 - mathfun@mailman.xmission.com

[math-fun] weird 2F1s
by Bill Gosper 04 Nov '11

04 Nov '11

Rich pointed me to ( http://arxiv.org/abs/1110.6284 , 37kb) whose abstract ends: "An interesting by-product of our analysis is the evaluation $$_2F_1(1/24,7/24,5/6, -\frac{2^{10}\cdot3^3\cdot5}{11^4})=\sqrt6 \sqrt[6]{\frac{11}{5^5}} $$ and other similar identities." Note the argument > 9.4. Empirically, the two 2F1s produced by the 1/z transformation satisfy Hypergeometric2F1[1/24, 5/24, 3/ 4, -(14641/138240)] == ((45 + 26 Sqrt[3])^(1/3) Gamma[7/12] Gamma[3/4])/ (2^(1/6) 3^(1/24) 5^(19/24) Sqrt[\[Pi]] Gamma[5/6]) and Hypergeometric2F1[7/24, 11/24, 5/4, -(14641/138240)] == (2 3^(17/24) (2 (-45 + 26 Sqrt[3]))^(1/3) Sqrt[\[Pi]] Gamma[1/12])/ (11 5^(13/24) Gamma[3/4] Gamma[5/6]) which together imply the "denesting" -(3 (-45 + 26 Sqrt[3]))^(1/3) + (3 (45 + 26 Sqrt[3]))^(1/3) == 6 --rwg

1 1

Re: [math-fun] Tetrahedral altitudes
by Henry Baker 03 Nov '11

03 Nov '11

[This question is probably aimed at Clark Kimberling.] I liked the little table at the beginning of this paper about which triangle thingy's generalized to 3-space and n-space. Is there a more encyclopedic version of this table, together with all the proofs? One of the most common questions by (insightful) high school geometry students is "what happens to this thingy in 3-space?". Also, the computation of such objects in higher spaces becomes more & more involved (as the computer graphics community knows all too well). Hopefully, such an encyclopedia would include some of these computational formulae. It would be nice to know where to point. At 09:14 PM 11/2/2011, Fred lunnon wrote: > See > http://www.geometrie.tuwien.ac.at/havlicek/pub/hoehen.pdf >Hans Havlicek, Gunter WeiÃ >"Altitudes of a Tetrahedron and Traceless Quadratic Forms" >Tech. Univ. Vienna (no date?)

2 1

[math-fun] Tetrahedral altitudes
by Fred lunnon 03 Nov '11

03 Nov '11

Richard's paper mentions mentions that the 4 altitude lines of a tetrahedron, through vertices and perpendicular to faces, do not necessarily concur in a common point. A long time ago I came across a charmingly antiquated tome: Nathan Altshiller-Court "Modern pure solid geometry" (1935) which I seem to remember stated the theorem that the altitudes always lie on a quadric; though his notation --- undefined --- for this property was something confusingly obscure. There were several other theorems of a similar nature. Can anybody confirm this? Has any further work in this direction taken place more recently? WFL On 10/29/11, Richard Guy <rkg(a)cpsc.ucalgary.ca> wrote: > ... I've just submitted a paper, ``Five-point > circles, the 76-point sphere, and the Pavillet tetrahedron'' > to the Monthly. The theme is that triangle geometry is not > dead. I would attach a copy if attachments were allowed; I > will entertain individual requests. ...

3 11

Re: [math-fun] function "image" v. function "range"
by Dan Asimov 01 Nov '11

01 Nov '11

An arbitrary function's image can't in general be determined, so a "range" (by which I mean here codomain) must be specified -- just as the domain must be -- as part of the definition of the concept of a function. It is unfortunate that authors of calculus books got "range" and "image" so confused that new terminology (codomain) had to be invented. (Though maybe it's good terminology from the point of view of category theory.) It took a few centuries to hammer out the correct definition of a function: A subset f of the cartesian product XxY of the domain X and codomain Y such that for each x in X there is exactly one y in Y such that (x,y) belongs to f. This should not be forgotten or tampered with. --Dan "Things are seldom what they seem." --W.S. Gilbert

2 1

Re: [math-fun] function "image" v. function "range"
by Henry Baker 01 Nov '11

01 Nov '11

"image" v. "range" suddenly got a lot more important after it was found in the last century that the image set of a function might not be recursive. "Range"/"codomain" is important in computer science, where one would like to specify for a function an output data type which is capable of holding any value in the function's image; due to undecidability issues, it is good enough to specify a convenient superset of the image. At 01:01 PM 11/1/2011, Schroeppel, Richard wrote: > If you can't be sure about what's in the Image, >it's still useful to know some superset. >Perhaps the mistake is to speak of "the" range. > >Rich > >-----Original Message----- >From: math-fun-bounces(a)mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Henry Baker >Sent: Tuesday, November 01, 2011 1:45 PM >To: math-fun(a)mailman.xmission.com >Subject: [math-fun] function "image" v. function "range" > >I noticed that the Khan Academy still uses the term "range" instead of "image" in order to describe the smallest (in terms of subsets) range for a function. > >The "image" of f(x)=x*x (where the domain of f(x) is R) is {x in R|x>=0}. The "range" of f(x) is any superset of the image, e.g., R itself. > >http://en.wikipedia.org/wiki/Image_%28mathematics%29 > >http://en.wikipedia.org/wiki/Range_%28mathematics%29 > >I assume that the Khan Academy teaches this usage for "range" because the standardized tests require it. > >However, I thought that this usage of "range" was hopelessly obsolete, even 50 years ago. > >Is this continuing usage of "range" instead of "image" an artifact of school budgets that still can't replace 50-year-old textbooks, or what?

1 0

[math-fun] function "image" v. function "range"
by Henry Baker 01 Nov '11

01 Nov '11

I noticed that the Khan Academy still uses the term "range" instead of "image" in order to describe the smallest (in terms of subsets) range for a function. The "image" of f(x)=x*x (where the domain of f(x) is R) is {x in R|x>=0}. The "range" of f(x) is any superset of the image, e.g., R itself. http://en.wikipedia.org/wiki/Image_%28mathematics%29 http://en.wikipedia.org/wiki/Range_%28mathematics%29 I assume that the Khan Academy teaches this usage for "range" because the standardized tests require it. However, I thought that this usage of "range" was hopelessly obsolete, even 50 years ago. Is this continuing usage of "range" instead of "image" an artifact of school budgets that still can't replace 50-year-old textbooks, or what?

3 3

[math-fun] An alternative way to count the rationals?
by Joshua Zucker 01 Nov '11

01 Nov '11

Well, the Calkin-Wilf tree is still the best, of course. But given the grid, really I suppose I should say the ordered pairs of positive integers than the rationals, the usual way to traverse it is by diagonals, that leads to a mapping like (n,k) -> (n+k choose 2) + k - 1, or something along those lines. At least, that's what I always seem to see in Cantor and infinity talks and books out there. Today I had a 6th grader suggest that we should show that it's a countable set with the mapping (n,k) -> (2k-1) * 2^(n-1). Probably this is old hat to people who have thought about this sort of thing, but I had never thought of it that way before. It seems much more "natural" because usually when I'm telling people about this stuff, we first use the mapping n -> n+1 to show we can add 1, then (1,k) -> 2k-1 and (2,k) -> 2k to show that we can double things, and then we need to find a whole new solution for "infinity squared". But we can just iterate the same trick we used for "2 times infinity" instead! First apply it to rows 1 and 2, then 2 and 3, then 3 and 4, and so on. [She doesn't know any algebra, so that's the way she explained it, and I showed the formula in terms of a power of 2 times an odd factor.] Another nice thing about her method is that it's quicker and easier to show that it's 1-1 because the inverse function is so simple to explain. Is there a relevant OEIS entry? Ah, yes, http://oeis.org/A135764 . There it is. Does it need a comment about this being a 1-1 mapping from ordered pairs of positive integers to the positive integers, or is that so obvious that it goes without saying? --Joshua Zucker

2 1