Net space (InstanceTopic, 3)
From Hi.gher. Space
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
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.