polytope love

# polytope

## Definitions

• n. A finite region of n-dimensional 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 n-dimensional geometry corresponding to a polygon or polyhedron.

## Etymologies

From German Polytop coined by Hoppe in 1882. Introduced to English by Alicia Boole Stott.. Composition: poly- ("many") + -tope ("surface")

## 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

• 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 2009-07-01

• 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

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

Guest Post: George Djorgovski, A New World Overture

• 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

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

Leonid Vitaliyevich Kantorovich - Autobiography

• The well-known 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.

Leonid Vitaliyevich Kantorovich - Autobiography

• 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 2-dimensional example of the more general polytope in any number of dimensions.

Muti

• 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.)

The Lopsided Universe