Definitions
from The Century Dictionary.
 noun A form in ndimensional geometry corresponding to a polygon or polyhedron.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun geometry A
finite region of ndimensional spacebounded byhyperplanes ; thegeometrical entity represented by the general term of the infinitesequence "point ,line ,polygon ,polyhedron , ...".
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
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Examples

Computing power has been increased by incredible proportions, and the the environs of the “beanpole” has become a multidimensional polygon called a polytope—an abstraction that planners spend their adolescent years learning to handle.
Economic Principals David Warsh 1993

Batsheba used Wizard's mathematical data for the crystal visualization and explains: "This 3D projection is notable for showing all 240 vertices, arranged in the concentric circles that appear in the common 2D "spirograph" rendering of this polytope."
Own the universe  and a piece of Second Life  in Real Life Bettina Tizzy 2009

Batsheba used Wizard's mathematical data for the crystal visualization and explains: "This 3D projection is notable for showing all 240 vertices, arranged in the concentric circles that appear in the common 2D "spirograph" rendering of this polytope."
Archive 20090701 Bettina Tizzy 2009

Yet the homology sphere is a sphere with the rotational group SO (3) “modulo” a discrete set of rotations which describe a polytope or polyhedra.
The Lopsided Universe Sean 2008

Visualizations of the E8 Lie group polytope, an 8dimensional mathematical object projected into the 3D space, produced by Wizard Gynoid and Desdemona Enfield.
Guest Post: George Djorgovski, A New World Overture Sean 2008

The Karmarkar method starts in the inside of the polytope, then uses a technique called projective geometry to warp the whole structure, again and again, in effect changing the shape of the polytope, over and over, until the best solution is achieved.
Economic Principals David Warsh 1993

The wellknown general recommendation of calculus to compare the function values in the polytope vertices lost its force since the vertices number was enormous even in very simple problems.

Mathematically, it was a problem of maximizing a linear function on a convex polytope.

You should also look at beautiful older work on geometric aspects and elasticity of amorphous materials (the polytope model of glass) by David Nelson and my CMU colleague Mike Widom.
iMechanica  Comments Amit Acharya 2010

A polygon is a 2dimensional example of the more general polytope in any number of dimensions.
Muti 2008
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