Wikipedia:Reference desk/Archives/Mathematics/2013 December 11

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December 11

Lawson's Klein bottle

I've looked casually for a parametrization of Lawson's Klein bottle in S3, with no luck. Is this it?

w,x,y,z = cosθ cosφ, cosθ sinφ, sinθ cosφ, sinθ sinφ

The circles at φ=0 and φ=π, or at θ=0 and θ=π, coincide but in opposite senses. —Tamfang (talk) 00:21, 11 December 2013 (UTC)

That just gives the torus. To see this, draw the lattice in the universal covering space R^2. It is true that the mapping you gave is a double cover of the surface by the standard torus (with symmetry $(\theta ,\phi )\mapsto (\theta +\pi ,\phi +\pi )$, but this is an orientation preserving symmetry. (The circles at θ = 0, π have the same orientation, just starting at different points.) If you rewrite your parameterization in terms of the variables $\theta +\phi ,\theta -\phi$ you should be able to see directly that it's the torus. Sławomir Biały (talk) 17:09, 11 December 2013 (UTC)

D'oh! —Tamfang (talk) 19:29, 13 December 2013 (UTC)

When is $\int {\sqrt[{m}]{P_{n}(x)}}\ dx$ Expressible in Terms of Elementary Functions ?

When is the antiderivative of the m^th root of a polynomial that meets the following criteria expressible in terms of elementary functions ?

$P_{n}(x)=\sum _{k=0}^{n>2}a_{k}x^{k}\quad ,\quad {\begin{cases}a_{n}\neq 0\\a_{0}\neq 0\\m\in \mathbb {N} \smallsetminus \{0,1\}\end{cases}}$ — 86.125.207.160 (talk) 08:05, 11 December 2013 (UTC)

See Elliptic integral which only deals with the square root case and a polynomial of degree 4 or less, for cube roots etc the situation is even less tractable. Dmcq (talk) 10:44, 11 December 2013 (UTC)

In general, the Risch algorithm can provide a conclusive test (although I don't know exactly why).--Jasper Deng (talk) 21:55, 14 December 2013 (UTC)

Figuring out some shapes for sewing together.

I'm trying to sew a needle book and I'm stuck on what shapes to cut out. I'm using pieces of an old pair of jeans and some of the sides will not be straight. I can mirror the first piece easily enough but the part I can't figure out is the middle or spine of the book. I want the spine to be rectangular with opposite sides parallel. In this picture you can see the piece I have used as a template for cutting the copy and the spine which I'd like attached to the copy so I can then sew them together. Is this even possible? — Preceding unsigned comment added by 78.148.110.243 (talk) 23:19, 11 December 2013 (UTC)

We seem to lack a needle book article, but apparently it's a "book" made of fabric instead of paper, to showcase various fabrics, stitches, etc., and sometimes to store basic sewing tools. Here are instructions to create one: [1] (although they specify felt, which I don't find to be very durable).

As for the shapes, you would either need one flat side of approximately the same length, on each piece, if you intend to sew the spine together, or you can make a fold-over book, where each fabric piece is two "pages". In the later case you may want some type of hard cover, to give it shape, since this isn't provided by the spine.

Also you want each page to be roughly the same shape and size, but some variation is fine. If you have one very long sample, which extends beyond the cover, you may have to make that a fold-out page. You could use Velcro to hold the fold-out page in place when folded up. For small, irregular shaped fabric pieces, you should sew them onto a backing fabric, which will fit better as pages in the book.

BTW, I'm not sure I understood your actual question, which seemed to be about cutting multiple pieces of fabric so the sum of the profiles is rectangle. Do I understand that correctly ? If so, then yes, this is possible. The simplest solution seems to be large rectangular covers to give the book it's shape. Assuming you never intend to wash this book, corrugated cardboard covered in fabric might be the simplest way to make the covers. If you want something stain resistant, you could put vinyl on the outside.StuRat (talk) 00:14, 12 December 2013 (UTC)