n-Cylinders
 

Am I right in assuming that any (n+1)-cylinder (including slanted) can be made by taking a n-Sphere and extending it nondegeneratively along a line?

For example, using a circle defined like this:
x^2+y^2=1
Let z=0 to 5
The length of the cylinder is 5, with a diameter of 1.
ax+by+cz=k where k is a constant and cz<>0 allows for the creation of 'slanted' cylinders as well.

Is there a counter-example for higher dimensions?


Also, where else is this sequence seen?
1(1), 2(2), 3(4), 4(8), 5(16), 6(32)... (n+1)(2^n)
I found it for the m-plane cross sections of a n-sphere. Let 0<m<n+1 with both variables in the set of positive integers. It turns out that you can make a binary array (implemented as a truth table). It's also possible to get something along the lines of sum(n, (n-1)[(n-1)+(n-2)+...+(1)], ..., n) which of course has larger terms in the center.

[QUOTE=nibble4bits;115738]Am I right in assuming that any (n+1)-cylinder (including slanted) can be made by taking a n-Sphere and extending it nondegeneratively along a line?

For example, using a circle defined like this:
x^2+y^2=1
Let z=0 to 5
The length of the cylinder is 5, with a diameter of 1.
ax+by+cz=k where k is a constant and cz<>0 allows for the creation of 'slanted' cylinders as well.

Is there a counter-example for higher dimensions?[/QUOTE]
Well, I suppose it depends on how you would define a cylinder in extended dimensions.

Perhaps the extension of a 3-d cylinder into the 4th dimension can just as easily be called an '4-cylinder' as what you've described here. But your description seems just as good a definition as any other.

Drew

a cylinder is defined as [tex]S^1 \times [0,1][/tex] (similarly, a torus is [tex]S^1 \times S^1[/tex])

so define an n-cylinder to be [tex]S^n \times [0,1][/tex]. What exactly is there to counter with an example?

You can also define an infinite cylinder as [tex]\mathbb{R}^2[/tex] modded out by a translation (i.e. take an infinite strip of constant width and glue the edges together), and similarly, an (infinite) n-cylinder would be [tex]\mathbb{R}^n[/tex] modded out by a translation.

Two very interesting answers. The most obvious method is to just translate the circle perpendicular to a linear equation in an arbitary number of dimensions... hmm thanks you two, this'll leave me a lot of paper to write on. :)

An n-sphere translated along an (n+1)-dimensional line is probably a little more convoluted than most people would want to deal with, however it is as legitimate as any other possible extension. I seem to remember this being a problem when trying to generalize 3D extensions of 2D shapes such as polygons -> prisms. There's an actual branching structure much like what is seen when programming trees. A square can make a cube, but there's also monclinic and triclinic polygons based on quadlaterals(rectangle/rhombus/etc.). I read a book that discusses 4D+ shapes like hypercubes and simplexes, but notes that there's very few shapes like dodecagons because of the angles involved.

Where's a good book on that [tex]S^1 \times S^1[/tex] system of describing the topology? I'd love to sit down with one that has lots of examples to work out using my computer, a pen, and a piece of paper. I seem to remember it being in reference to 'multiplying' one shape by another. For example, a triangle extended along a line can make a prism or if the angle is changed, a slanted prism or polygon(one triangle's axis parallel to line).