Re: [math-fun] Zonohedral surfaces
Yes (I presume this means in R^3). To get on with all faces rectangles, drill a hole straight through a rectangular solid and look at its boundary. Now apply a random linear mapping and all the rectangles will probably turn into parallelograms. —Dan Jim Propp wrote: ----- George Hart’s Celebration of Mind talk (just concluded) makes me wonder: is there a non-self-intersecting torus surface made of parallelograms? -----
I was just overlooking the obvious! Jim On Thu, Oct 22, 2020 at 5:15 PM Dan Asimov <dasimov@earthlink.net> wrote:
Yes (I presume this means in R^3).
To get on with all faces rectangles, drill a hole straight through a rectangular solid and look at its boundary.
Now apply a random linear mapping and all the rectangles will probably turn into parallelograms.
—Dan
Jim Propp wrote: ----- George Hart’s Celebration of Mind talk (just concluded) makes me wonder: is there a non-self-intersecting torus surface made of parallelograms? -----
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All, The talk Jim had referred to is now online here in a more completed form. Polyhedra fans might enjoy it: https://www.youtube.com/watch?v=_PSdVX02Vbs The question which I thought might have been intended (during the CoM discussion, or initially by Jim) was whether one could make a *homogeneous* zonohedral surface in 3D which is topologically toroidal without self-intersection. By homogeneous, I mean following the algorithm described in the video from a given initial state. (One with all vertices of degree four would be homogeneous, but that isn't a necessary condition; homogeneous examples might have vertices of any order, e.g., polar zonohedra.) One can take a product of two polygons to create a toroidal surface composed of parallelograms and having all vertices of degree four, but it will be self-intersecting. And one can approximate any continuous surface by joining homogeneous patches, e.g., gluing together small cubes. (The video shows how to join larger homogeneous patches seamlessly.) But I would conjecture that a homogeneous zonohedral surface can only be toroidal if it is self-intersecting. George http://georgehart.com On 10/22/2020 5:33 PM, James Propp wrote:
I was just overlooking the obvious!
Jim
On Thu, Oct 22, 2020 at 5:15 PM Dan Asimov <dasimov@earthlink.net> wrote:
Yes (I presume this means in R^3).
To get on with all faces rectangles, drill a hole straight through a rectangular solid and look at its boundary.
Now apply a random linear mapping and all the rectangles will probably turn into parallelograms.
—Dan
Jim Propp wrote: ----- George Hart’s Celebration of Mind talk (just concluded) makes me wonder: is there a non-self-intersecting torus surface made of parallelograms? -----
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participants (3)
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Dan Asimov -
George Hart -
James Propp