March 2011 - mathfun@mailman.xmission.com

Re: [math-fun] High school calculus?
by Dan Asimov 11 Mar '11

11 Mar '11

Anna Johnston asks: << . . . The question is: Why is calculus still the perceived linear end point to HS math and what is the best way to change perceptions and curriculum? >> In my view, different students have different needs, and if at all possible a school should have the flexibility to cater to them. For some students who have finished the normal H.S. curriculum, calculus is just what they need. But if we're talking about students who are likely to take just one math course beyond the normal H.S. curriculum (whether this course is in H.S. or college), I would say that by far the most important subject for most of them is *statistics*. Discrete math is important for those who will go on to program computers, especially if they will need to go beyond the most basic level. But usually such discrete math is made part of a computer science curriculum, so need not be part of a math curriculum. (It may be artificial to separate math from computer science, but that's the way things are these days.) --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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[math-fun] Affine manifolds
by Dan Asimov 11 Mar '11

11 Mar '11

Speaking of affine, there's a famous conjecture in differential geometry (by Shiing-Shen Chern): Define an *affine manifold* to be one having an atlas all of whose transition functions are affine. For example, R^n / x ~ 2x is an affine manifold that's topologically the product of spheres S^1 x S^(n-1). CONJECTURE: The Euler characteristic of any compact affine manifold is equal to 0. This has remained unresolved for over 50 years. --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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Re: [math-fun] Affine manifolds
by Dan Asimov 11 Mar '11

11 Mar '11

Correction. This example: << For example, R^n / x ~ 2x is an affine manifold that's topologically the product of spheres S^1 x S^(n-1). >> should have excluded the origin: << For example, R^n - {0} / x ~ 2x is an affine manifold that's topologically the product of spheres S^1 x S^(n-1). >> And just to be clear, an affine function in the previous post means (possibly a restriction of) any map f: R^n -> R^n of the form f(x) = Ax + b where b is in R^n, and A: R^n -> R^n is an invertible linear map. --Dan << Speaking of affine, there's a famous conjecture in differential geometry (by Shiing-Shen Chern): Define an *affine manifold* to be one having an atlas all of whose transition functions are affine. For example, R^n / x ~ 2x is an affine manifold that's topologically the product of spheres S^1 x S^(n-1). CONJECTURE: The Euler characteristic of any compact affine manifold is equal to 0. This has remained unresolved for over 50 years. >> ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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Re: [math-fun] A rhinoceros' pancreas
by Dan Asimov 11 Mar '11

11 Mar '11

I wish that references to (e.g.) Cederberg contained enough context to know what it is that was said, which might enable making sense of comments like the following: << However the first interpretation seems inconsistent with Cederberg, who defines her "affine" group to fix an "absolute" quadric (whatever that precisely means) . . . >> ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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[math-fun] Fun math I've been doing lately
by Mike Stay 11 Mar '11

11 Mar '11

Brice Due made a 2048x2048 Life pattern that simulates the Life rules. All the machinery is in a thin layer around the edge; the center is an empty field that gets flooded with LWSS's when it's turned on. Since the size of the pattern is a power of 2, Golly can memoize its internal states efficiently and can run it as fast as a single pixel. There's a "metafier" script in Golly that takes any pattern and constructs a grid of metapixels running that pattern. I took part of the metapixel itself, a stream of LWSSs, and then made a small video that zooms out and speeds up, only to return to the image at the beginning of the video: http://www.youtube.com/watch?v=QtJ77qsLrpw I then did a version of Escher's "Print Gallery" http://media.cleveland.com/ent_impact_arts/photo/ax192-7f8a-9jpg-b7073d105f… but using the LWSSs: http://reperiendi.wordpress.com/2011/03/10/print-gallery-metapixel/ Here are some alternate versions involving doubled or reversed spirals: http://reperiendi.wordpress.com/2011/03/10/more-views-of-the-metapixel/ I made a stereographic projection of the 16-cell and 24-cell polytopes out of hula hoops: http://www.youtube.com/watch?v=-0rFECk-uRg http://reperiendi.wordpress.com/2011/03/01/hooped-up-24-cell/ My brother and I made this image from the famous "Powers of 10" video: http://reperiendi.wordpress.com/2011/03/10/powers-of-10/ Can you figure out how? -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

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Re: [math-fun] A rhinoceros' pancreas
by Dan Asimov 10 Mar '11

10 Mar '11

<< Google in its infinite wisdom won't let me see anything at either of those links. Perhaps it will be different for others. >> In may case it let me see so much that it didn't help. --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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Re: [math-fun] A rhinoceros' pancreas
by Dan Asimov 09 Mar '11

09 Mar '11

<< Just an observation - why quote Wikipedia when Affine Space, Functions etc. are defined at mathworld.wolfram.com >> There seems to be an assumption here that MathWorld is somehow a superior reference, at least in this case, to Wikipedia. I know of no grounds for this assumption. --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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Re: [math-fun] Factorials in non-hypergeometric sums
by James Cloos 09 Mar '11

09 Mar '11

> 'sum(1/(x^k + k!), k, 0, inf); That looked intriguing. Looks to have poles at -1 and -6^(1/3) and -120^(1/5); the limit from below at those is -inf and the limit from above is +inf. To 53 bits (the precision of a 64 bit float), it about ten to fifteen terms is enough to converge. Would finding all of the poles help to determine a closed form, if there is one? There does seem to be a pattern to those poles. I didn't look at values in C, just R.... -JimC -- James Cloos <cloos(a)jhcloos.com> OpenPGP: 1024D/ED7DAEA6

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Re: [math-fun] A rhinoceros' pancreas
by Gareth McCaughan 09 Mar '11

09 Mar '11

On Tuesday 08 March 2011 06:00:56 Fred Lunnon wrote (to just me rather than math-fun, but that was an error): [me:] > > A(Cx+d)+b = (AC)x + (Ad+b). In what sense are these transformations > > not closed under composition? [Fred, whole message kept for the sake of context:] > This (on the face of it) perfectly reasonable objection made me realise > how accustomed I have become to thinking about these matters in terms > of the symmetries acting on the space, to the extent that it did not > occur to me that there might be more than one way to define "composition" > in this context. > > It also points up very nicely one of the major problems concealed in > the matrix times vector representation of geometric transformations: > that it breaks down as soon as you want to transform anything else > besides a point, such as another isometry. > > In 2-space say, the translation by T of a rotation R around the origin > equals rotation about the translated centre, represented by T^{-1} R T > --- which now cannot expressed in the form A x + b for A in GL(2). 1. Rotation about a nonzero point *is* of the form x -> Ax+b. 2. The definition of an affine transformation does not require that A be in GL(n). What am I missing here? > The situation deteriorates further in 3-space and above, where even > the axial coline is not canonically representable. What do you mean by "not canonically representable", and why is it a problem that it isn't, and why is that a problem *with the notion of affine transformation*? > The interpretation that symmetries act on other symmetries becomes > inevitable when the geometry is defined in a Kleinian fashion: > for instance, it seems clear to me that the author of the discussion at > http://en.wikipedia.org/wiki/Affine_geometry > is also (implicitly) taking this view. I wonder too if some of the > difficulty I have experienced with the use of the term "affine" stems > from the same dichotomy. > > Incidentally, for anyone (like most practitioners of computer graphics) > acustomed to thinking of subspaces (particularly points) as fundamental, > but transformations as a derivative concept, to reverse the bias demands a > substantial mental realignment. I can still recall the shocking discovery > that planes, lines, points could all be represented perfectly well > as if they were instead the reflections, half-turns etc. leaving them > pointwise fixed --- a fundamental feature of the geometric algebra > approach. -- g

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[math-fun] Factorials in non-hypergeometric sums
by quad 09 Mar '11

09 Mar '11

I saw this sum somewhere as a question as to whether it has a closed form. I am pretty sure it doesn't have any elementary closed form, but I'd think that this isn't a whole new class of functions (with respect to special functions). (%i1) 'sum(1/(x^k + k!), k, 0, inf); inf ==== \ 1 (%o1) > ------- / k ==== x + k! k = 0 for some real x (we might even just consider positive integral x). Maybe even just (%i2) 'sum(1/(1 + k!), k, 0, inf); inf ==== \ 1 (%o2) > ------ / k! + 1 ==== k = 0 Can we write this in any other useful forms? (cue the Gosperfunctionology) -Robert

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