Shape (EntityClass, 2)

From Hi.gher. Space

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Latest revision as of 17:09, 9 March 2011

A shape is any geometric entity. It is the most general term; there are many subsets of shapes. Common subsets are summarized below, with the point being the simplest element of each set.

Polytope

Main article: Polytope

Powertope

Main article: Powertope

A powertope is a shape which can be written as a shape raised to the power of another shape. Technically, all shapes would be powertopes since they can all be written as themselves raised to the power of digon, so in order to be labeled a powertope, both the base and exponent shapes must be at least two-dimensional. This of course means that there are no powertopes in prime dimensions.

Rotope

Main article: Rotope

The rotopes are the classic set of shapes enumerated by Alkaline, later extended by others. Unfortunately these extensions have produced a quantity of undesirable "shapes", which pollute the set of rotopes and make it difficult to use. However, it is still to this day the most intuitive construction-based set known of. The set of rotopes includes the hypercube, hypersphere and simplex of each dimension.

Bracketope

Main article: Bracketope

The bracketopes are a set enumerated by Keiji, which was initially an attempt to replace rotopes, but did not fare much better as once again it produced a quantity of undesirable shapes, which in this case were duplicates of legitimate ones. The set of bracketopes includes the hypercube, hypersphere and cross polytope of each dimension.

Convex shape

A convex shape is a shape which is its own convex hull; it has a genus of zero and all digons connecting two points in the shape also lie inside the shape.

Concave shape

A concave shape is a shape which is not convex but also is not self-intersecting.

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+<[#ontology [kind class] [cats Entity]]>
 A '''shape''' is any geometric entity. It is the most general term; there are many subsets of shapes. Common subsets are summarized below, with the [[point]] being the simplest element of each set.
 == Polytope ==
-A ''polytope'' is a shape with no [[curved]] [[element]]s. All elements of a polytope are lower-dimensional polytopes. Polytopes are grouped by their dimension into the ''polygons'', ''polyhedra'', ''polychora'', ''polytera'', ''polypeta'', etc.
+{{Mainarticle|Polytope}}
-=== Uniform polytope ===
-A ''uniform polytope'' is a polytope which is [[vertex-transitive]] and whose [[facet]]s are lower-dimensional uniform polytopes.
-The uniform polytopes in three and four dimensions can be indexed by their [[Bowers acronym]] or [[Kana mnemonic]].
-==== Regular polytope ====
-{{Mainarticle|Regular polytope}}
-A ''regular polytope'' is a uniform polytope whose elements within each dimension are all [[congruent]] (that is, all edges are congruent to all other edges, all faces are congruent to all other faces, and so forth).
-=== Convex regular-faced polytope ===
-A ''convex regular-faced polytope'' (abbreviated as ''CRF polytope'') is an ''n''-polytope which is strictly convex (i.e. for any two points in the shape, the line segment between them is also entirely inside the shape, and there are no two ''n-1''-cells which are in the same ''n-1''-plane) and all its faces are regular.
-Excluding the infinite set of prisms and antiprisms, there are [[Table of convex regular-faced polyhedra|precisely 110 CRF polyhedra]]. These (exhaustively) include the [[Platonic solids]], the [[Archimedean solids]] and the [[Johnson solids]].
-It is currently unknown how many CRF polychora there are, though there are at least 238, [http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468 including]:
-*all 64 convex uniform polychora
-*92 prisms of the Johnson solids
-*82 pyramidal forms, (4 x 21 polyhedra, minus 2 uniform results)
-*various cupolae
-=== Zonotope ===
-{{Mainarticle|Zonotope}}
-A ''zonohedron'' is a convex polyhedron whose faces all have [[point symmetry]]. A ''zonotope'' is either a zonohedron or a convex ''n''-polytope (with ''n'' ≥ 4) whose facets are all zonotopes themselves. Zonotopes can be constructed as the [[Minkowski sum]] of a set of vectors.
 == Powertope ==
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 == Stellar shape ==
 A ''stellar shape'' is a shape which is self-intersecting.
-[[Category:Shapes| ]]