Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A finite region of ndimensional space bounded by hyperplanes; the geometrical entity represented by the general term of the infinite sequence "point, line, polygon, polyhedron, ...".
from The Century Dictionary and Cyclopedia
 n. A form in ndimensional geometry corresponding to a polygon or polyhedron.
Etymologies
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.

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."

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.

Visualizations of the E8 Lie group polytope, an 8dimensional mathematical object projected into the 3D space, produced by Wizard Gynoid and Desdemona Enfield.

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.

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

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.

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.

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

These configurations to me were about “surfaces,” and if we were to allot a progression to the “projective geometries” here in relation to higher dimensional thinking, “as the polytope [E8]” (where Coxeter [I meant to apologize for misspelling earlier] drew us to abstraction to the see “higher dimensional relations” toward Plato’s light.)
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bertilak commented on the word polytope
May 16, 2007