Polytopes

special kind of point that describes the corners or intersections of geometric shapes

line segment joining two adjacent vertices in a polygon or polytope

alt=From left to right: a point (marked 0), a line (marked 1), a triangle (marked 2), and a tetrahedron (marked 3).|thumb|The four simplexes that can be fully represented in 3D space.

{| class="wikitable" style="margin-left:1em" align="right" |- |50px |50px |50px |50px |50px |50px |- | colspan="6" | A polyhedron is a 3-dimensional polytope |} thumb|400px|right|A polygon is a 2-dimensional polytope. Polygons can be characterised according to various criteria. Some examples are: open (excluding its boundary), bounding circuit only (ignoring its interior), closed (including both its boundary and its interior), and self-intersecting with varying densities of different regions.

polyhedron whose vertices correspond to the faces of another one

Canadian mathematician (1907–2003)

Swiss geometer (1814-1895)

figure exposed when a corner of a polyhedron or polytope is sliced off

a projection of a polytope into a figure of dimension smaller by one

thumb|Construction of a stellated dodecagon: a regular polygon with [[Schläfli symbol {12/5}]] In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred",

tiling of 3-or-more dimensional Euclidean or hyperbolic space

convex hull of a finite set of points in a Euclidean space

algorithm for computing convex hulls by tracing the boundary of the hull

vertex-transitive polytope

Wikimedia template

method for constructing a uniform polyhedron or plane tiling

infinite-dimensional cube with a compact topology

the problem of partitioning a given shape into pieces that can be rearranged to form a second given shape

thumb|Projections of k-cells onto the plane (from k\in\{1,\dots{},6\}). Only the edges of the higher-dimensional cells are shown. In geometry, a hyperrectangle (also called a box, hyperbox, k-cell or orthotope), is the generalization of a rectangle (a plane figure) and the rectangular cuboid (a solid figure) to higher dimensions. A necessary and sufficient condition is that it is congruent to the Cartesian product of finite intervals. This means that a k-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. Every k-cell is compact.

operation that cuts polytope vertices, creating a new facet in place of each vertex

theorem in geometry

skew polygon derived from a polytope

feature of a polytope in the next-lower dimension

Image:CubeAndStel.svg Stella octangula as a faceting of the cube

process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points

algebraic partially ordered set or poset which captures the combinatorial properties of a traditional polytope, but not any purely geometric properties such as angles, edge lengths, etc

In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope is another polyhedron or polytope formed by replacing each facet of with a pyramid. In some cases, the pyramid is chosen to have regular sides, often producing a non-convex polytope; alternatively, by using sufficiently shallow pyramids, the results may remain convex. Kleetopes are named after Victor Klee, although the same concept was known under other names long before the work of Klee.

operation on a polytope where facets are separated and moved radially apart, and new facets are formed at separated elements

a polynomial whose values give the number of integer points in a scaled copy of a polytope

thumb|Associahedron (front) thumb|Associahedron (back) thumb| is the Hasse diagram of the [[Tamari lattice .]] thumb|The 9 faces of Each vertex in the above Hasse diagram has the ovals from the 3 adjacent faces. Faces whose ovals intersect do not touch.

number of windings of a polytope, in geometry

thumb|300px|The permutohedron of order 4

polytope whose facets are all simplices

set of points described by relative position in space

regular tilings of ≥3D spaces with hypercubes

thumb|A bitruncated cube is a truncated [[octahedron.]] thumb|A bitruncated cubic honeycomb - Cubic cells become orange truncated octahedra, and vertices are replaced by blue truncated octahedra.

𝑛‐dimensional polytope each of whose vertices are adjacent to exactly 𝑛 edges

Wikimedia template

thumb|A runcinated cubic honeycomb (partial) - The original cells (purple cubes) are reduced in size. Faces become new blue cubic cells. Edges become new red cubic cells. Vertices become new cubic cells (hidden).