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Polar zonohedra are convex polyhedra bounded by n*(n-1) rhombs, where n is an
integer greater than two. Attached, a zipped archive containing two macros and
a scene file which shows their use.
The two most important parameters or whatever one would call them, are "n" as
defined above, and what I call pitch, a carpentry term applied to the slope of
a rafter. Pitch can be anything from 0 degrees to 90 degrees, although both 0
and 90 give "degenerate" polar zonohedra. It is interesting that when pitch
equals 35.26+ degrees, or arc tan sqrt[ 1/2 ], a polar zonohedron is an
orthogonal and isometric shadow of an n-dimensional cube, a solid shadow, cast
into three dimensions. When pitch is low, polar zonohedra are oblate and like
flying saucers; when pitch is high, polar zonohedra are prolate and like
spindles. If edge length is held fixed and one sends a polar zonohedron through
the whole range of pitch, it has maximal volume if and only if it is an
orthogonal shadow of an n-cube, i.e., when pitch equals arc tan sqrt[ 1/2 ].
Post a reply to this message
Attachments:
Download 'pov pz macro.zip' (4 KB)
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"Russell Towle" <rto### [at] inreach com> wrote:
> Polar zonohedra are convex polyhedra bounded by n*(n-1) rhombs, where n is an
> integer greater than two. Attached, a zipped archive containing two macros and
> a scene file which shows their use.
>
> The two most important parameters or whatever one would call them, are "n" as
> defined above, and what I call pitch, a carpentry term applied to the slope of
> a rafter. Pitch can be anything from 0 degrees to 90 degrees, although both 0
> and 90 give "degenerate" polar zonohedra. It is interesting that when pitch
> equals 35.26+ degrees, or arc tan sqrt[ 1/2 ], a polar zonohedron is an
> orthogonal and isometric shadow of an n-dimensional cube, a solid shadow, cast
> into three dimensions. When pitch is low, polar zonohedra are oblate and like
> flying saucers; when pitch is high, polar zonohedra are prolate and like
> spindles. If edge length is held fixed and one sends a polar zonohedron through
> the whole range of pitch, it has maximal volume if and only if it is an
> orthogonal shadow of an n-cube, i.e., when pitch equals arc tan sqrt[ 1/2 ].
Made a few changes.
Great spacecraft design @ n7 pitch5
(no maps and should render elsewhere, unlike my other mistakes)
Post a reply to this message
Attachments:
Download 'zonohedra7.zip' (2 KB)
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