Net space (InstanceTopic, 3)

From Hi.gher. Space

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The '''net space''' of any object is the number of dimensions required to reference any [[point (object)|point]] in that object. In order to be physically made, the net space and [[bounding space]] of the object must be equal. In mathematical models however, we usually consider an object to have a lower net space if some dimensions are neglegible, for example, a [[Klein bottle]] has a 2D net space, even though it has a 4D bounding space (in fact, all [[manifolds]] have the same net space as the object they were created from).
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<[#ontology [kind topic] [cats Property]]>
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The '''net space''' of any [[shape]] is the number of dimensions required to reference any [[point (object)|point]] in that shape. In order to be physically made, the net space and [[bounding space]] of the shape must be equal. In mathematical models however, we usually consider a shape to have a lower net space if some dimensions are neglegible, for example, a [[Klein bottle]] has a 2D net space, even though it has a 4D bounding space (in fact, all [[manifolds]] have the same net space as the shape they were created from).
Net space is often specified as an adjective in the following way:
Net space is often specified as an adjective in the following way:
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*monoframe = the shape has a 1D net space (i.e. wireframe)
*monoframe = the shape has a 1D net space (i.e. wireframe)
*diframe = the shape has a 2D net space
*diframe = the shape has a 2D net space
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*triframe = the shape has a 3D net space
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*tetraframe = the shape has a 4D net space
etc.
etc.
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[[Category:Geometrical properties]]
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The net space of a shape is often represented as two numbers separated with an asterisk, the first representing the highest bounding space of the [[hypercell]]s in the shape, and the second representing the number of these dimension hypercells. For example, using this notation, a diframe cube would be 2*6, a nullframe cube would be 0*8, a tetraframe [[tesseract]] would be 4*1, a diframe [[octahedron]] would be 2*8, and a monoframe [[tetrahedron]] would be 1*6.
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== Fractional net spaces ==
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Unlike bounding spaces, net spaces may be fractional. Fractional net spaces arise when the net space of a [[fractal]] is calculated. The net space is fractional because, if the net space of a fractal is ''x'', then the fractal does not enclose any [[hypervolume]] of dimension ⌈''x''⌉, yet it is impossible to reference a point inside the fractal with only ⌊''x''⌋ numbers. Thus, the only way to represent a point inside a fractal is to reference it with a co-ordinate system in its bounding space.

Latest revision as of 22:39, 11 February 2014

The net space of any shape is the number of dimensions required to reference any point in that shape. In order to be physically made, the net space and bounding space of the shape must be equal. In mathematical models however, we usually consider a shape to have a lower net space if some dimensions are neglegible, for example, a Klein bottle has a 2D net space, even though it has a 4D bounding space (in fact, all manifolds have the same net space as the shape they were created from).

Net space is often specified as an adjective in the following way:

  • nullframe = the shape has a 0D net space (i.e. consists of vertices only)
  • monoframe = the shape has a 1D net space (i.e. wireframe)
  • diframe = the shape has a 2D net space
  • triframe = the shape has a 3D net space
  • tetraframe = the shape has a 4D net space

etc.

The net space of a shape is often represented as two numbers separated with an asterisk, the first representing the highest bounding space of the hypercells in the shape, and the second representing the number of these dimension hypercells. For example, using this notation, a diframe cube would be 2*6, a nullframe cube would be 0*8, a tetraframe tesseract would be 4*1, a diframe octahedron would be 2*8, and a monoframe tetrahedron would be 1*6.

Fractional net spaces

Unlike bounding spaces, net spaces may be fractional. Fractional net spaces arise when the net space of a fractal is calculated. The net space is fractional because, if the net space of a fractal is x, then the fractal does not enclose any hypervolume of dimension ⌈x⌉, yet it is impossible to reference a point inside the fractal with only ⌊x⌋ numbers. Thus, the only way to represent a point inside a fractal is to reference it with a co-ordinate system in its bounding space.