Vocabulary
- Affine Space
- "An affine space is a vector space that's forgotten its origin" (John Baez). In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point that serves as an origin.
- Circumcircle
- the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The centre of this circle is called the circumcenter.
- Face
- a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube. In convex geometry, a face of a polytope P is the intersection of any supporting hyperplane of P and P. From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. For example, a polyhedron R3 is entirely on one hyperplane of R4. If R4 were spacetime, the hyperplane at t=0 supports and contains the entire polyhedron. Thus, by the formal definition, the polyhedron is a face of itself.
- Facet
- A facet of an n-polytope is an (n-1)-dimensional face or (n-1)-face. For example:
- The facets of a polygon are edges. (1-faces)
- The facets of a polyhedron are faces. (2-faces)
- The facets of a polychoron (4-polytope) are cells. (3-faces)
- The facets of a polyteron (5-polytope) are hypercells. (4-faces)
- Hyperplane
- It is a higher-dimensional generalization of the concepts of a line in Euclidean plane geometry and a plane in 3-dimensional Euclidean geometry.
- Polytope
- Polytope means, first, the generalization to any dimension of polygon in two dimensions, polyhedron in three dimensions, and polychoron in four dimensions.
Dimension
of element
|
Element name (in a d-polytope)
|
| 0
|
Vertex
|
| 1
|
Edge
|
| 2
|
Face
|
| 3
|
Cell
|
| n
|
n-face - elements order n = 2, 3, ..., d − 1
|
| <math>\vdots</math>
|
<math>\vdots</math>
|
| d − 3
|
Peak - (d−3)-face
|
| d − 2
|
Ridge - (d−2)-face
|
| d − 1
|
Facet - (d−1)-face
|
- Simplex
- In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of (n + 1) affinely independent points in some Euclidean space of dimension n or higher. For example, a 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a pentachoron (in each case with interior).
- Supporting Hyperplane
- A hyperplane divides a space into two half-spaces. A hyperplane is said to support a set S in Euclidean space \mathbb R^n if it meets both of the following:
- S is entirely contained in one of the two closed half-spaces of the hyperplane
- S has at least one point on the hyperplane
Here, a closed half-space is the half-space that includes the hyperplane.
- Triangulation
- a subdivision of a geometric object into simplices. In particular, in the plane it is a subdivision into triangles,
References
Topology