"Ends" of the cubinder?

Discussion of shapes with curves and holes in various dimensions.

"Ends" of the cubinder?

Postby Keiji » Sun Dec 19, 2010 5:04 pm

Well this is a question that could have been answered a decade ago :D

I was looking into the toratopes again, and came to the Toraspherinder page. Now I know I originally wrote this, but it says "It can also be formed by taking an uncapped cubinder and connecting its ends in a loop."

Well... now that I think about this... how can you connect the ends of a cubinder? Unlike a cylinder or a spherinder, a cubinder doesn't really have two "ends"! Should I be connecting diagonally opposite circles, or orthogonally opposite cylinders? Both? Is this entire concept impossible?

Working very loosely with an extension of the abstract polytope concept to curved shapes, I've come up with an analogy that says a toracubinder should have four tubular faces in addition to its sole tubular cell, but this is probably completely wrong...
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Re: "Ends" of the cubinder?

Postby PWrong » Mon Dec 20, 2010 1:42 am

Rather than connecting ends, what you're doing is like taking a square and gluing the sides to make a sphere. The cubinder is square # circle, so the toraspherinder is sphere # circle. A better way to describe it is a sphere with a circle at every point. The plane of each circle is given by the vector coming out of the sphere, and a vector perpendicular to the sphere.

Working very loosely with an extension of the abstract polytope concept to curved shapes, I've come up with an analogy that says a toracubinder should have four tubular faces in addition to its sole tubular cell, but this is probably completely wrong...


I don't think it should have any faces or corners or anything like that. Any closed toratope is completely smooth with just an inside and an outside. The cubinder however should have four solid cylindrical faces and another face.
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Re: "Ends" of the cubinder?

Postby quickfur » Tue Jan 25, 2011 7:30 pm

The cubinder has 4 ends, all of which are shaped like a 3D cylinder. Maybe "ends" is the wrong word for it, since they are connected to each other in a square loop. :) Wrapped around them is a square torus, which is the actual curved surface that the cubinder can roll on. (Remember that a 3D cylinder in 4D is flat and cannot roll at all. Well, it can roll the same way a flat disk in 3D can roll, I suppose, or the same way a cylinder's caps can roll if you set them off vertically, but they will eventually fall flat.)
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