Appear of top

In Geometry, a figure of top represents the arrangement of a whole of points connected of all the close tops, in a Polytope for a given top. This applies also well for the infinite Pavage S, or pavings filling space with polytopic cells .

A figure of top for a n-polytope is a (n-1) - polytope . For example, a figure of top for a Polyèdre is a polygonal figure , and the figure of top for a Polychore is a polyhedric figure .

By considering the connectivity of these close tops, a (n-1) - polytope completely imaginary can be built for each top of a polytope:

Each Sommet of the figure of top coincides with a top of the original polytope.
Each edge of the figure of top exists on or in a face of the original polytope connecting two tops alternated starting from an original face.
Each face of the figure of top exists on or in a cell of N - polytope original (for N >3).
… and so on for the higher elements of order in the polytopes of higher natures.

The figures of top are most useful for the uniform polytopes because a figure of top can imply the whole polytope.

For the polyhedrons, the figure of top can be represented by a notation of Configuration of top, by listing the faces in a continuation around a top. For example 3.4.4.4 is a top with a triangle and three squares, and it represents the Petit rhombicuboctaèdre.

If the polytope is of uniform Sommet, the figure of top will exist in a surface hyperplane N - space. In general, the figures of top do not need to be planar.

Like the not-convex polyhedrons, the figures of top can also be not-convex. The uniform polytopes can have faces in spangled polygons and figures of top for example.

Polytopes regular

If a polytope is regular, it can be represented by a Symbole of Schläfli and, the cell and the figure of top can both be extracted trivialement from this notation.

In general, a regular polytope with a symbol of Schläfli {has, B, C,…., there, Z} has cells {has, B, C,…, there}, and of the figures of top {B, C,…, there, Z}.

For a Polyhedral regular {p, Q}, the figure of top is {Q}, a q-gone.
* Example, the figure of top for a cube {4,3} is the triangle {3}.
For a regular Polychore or a paving filling space {p, Q, R}, the figure of top is {Q, R}.
* Example, the figure of top for a hypercube {4,3,3} is a regular tetrahedron {3,3}.
* the figure of top for a cubic Honeycomb {4,3,4} is octahedral regular the {3,4}.

Since the dual polytope of a regular polytope is also regular and is represented by the symbol of Schläfli whose indices are reversed, it is easy to see that the figure of top of dual is the cell of the dual polytope.

An example of figure of top of a honeycomb

The figure of top of a cubic Honeycomb truncated is a square Pyramide non-uniform. Octahedral and four cubic truncated meets at each top to form a paving filling space.

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