Shape (EntityClass, 2)
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| - | A '''shape''' is any geometric entity. It is the most general term; there are many subsets of shapes, [[ | + | 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. |
| - | [[Category:Shapes| | + | == 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. | ||
| + | |||
| + | === 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). | ||
| + | |||
| + | == Powertope == | ||
| + | {{Mainarticle|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 == | ||
| + | {{Mainarticle|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 == | ||
| + | {{Mainarticle|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 [[digon]]s 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]]. | ||
| + | |||
| + | === Glaze === | ||
| + | {{Mainarticle|Glaze}} | ||
| + | |||
| + | == Stellar shape == | ||
| + | A ''stellar shape'' is a shape which is self-intersecting. | ||
| + | |||
| + | [[Category:Shapes| ]] | ||
Revision as of 00:06, 6 November 2009
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 elements. All elements of a polytope are lower-dimensional polytopes. Polytopes are grouped by their dimension into the polygons, polyhedra, polychora, polytera, polypeta, etc.
Uniform polytope
A uniform polytope is a polytope which is vertex-transitive and whose facets 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
Main article: 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).
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.
Glaze
Main article: Glaze
Stellar shape
A stellar shape is a shape which is self-intersecting.
