Table of Contents
Can something have negative dimensions?
Dimension of a (finite dimensional) vector space is defined as the cardinality of a basis for the vector space. Since the cardinality cannot be negative, negative dimension for vector spaces is meaningless.
Can fractal dimensions be negative?
A new notion of fractal dimension is defined. When it is positive, it effectively falls back on known definitions. But its motivating virtue is that it can take negative values, which measure usefully the degree of emptiness of empty sets.
What is the negative first dimension?
On negative first dimension, there is no point on this space. In abstract polytopes, the -1D shape is the null polytope; this is a consequence of abstract polytopes also being sets since the null polytope corresponds to the empty set.
Do fractional dimensions exist?
The standard Cantor set has fractional dimension! Well it is at most 1-dimensional, because one coordinate would certainly specify where a point is. However, you can get away with “less”, because the object is self-similar. At each stage, you only need to specify which 2 out of 3 segments a point is in.
What does negative dimension mean?
For a dimension to be negative it would need a value of quality to the opposite, which do not appear to exist. The distance between two points exists or not, the distance is a positive numerical expression, a distance between two points does appear to be able to be negative.
What is fractal dimension used for?
Fractal dimension is a measure of how “complicated” a self-similar figure is. In a rough sense, it measures “how many points” lie in a given set. A plane is “larger” than a line, while S sits somewhere in between these two sets.
Can dimensions be fractions?
Starting with fractional dimensions, yes. An object can have a fractional dimension, it is called its fractal dimension . There are many similar definitions for fractal dimension, but a very general one is the Hausdorff dimension .
Can you have 2.5 Dimensions?
There’s no such thing as “2.5 dimensional universe”. When we talk about the “dimension of space” we’re talking about the number of completely different directions that are available, not the whole “sphere thing”.
Can the probability be negative?
The probability of the outcome of an experiment is never negative, although a quasiprobability distribution allows a negative probability, or quasiprobability for some events.
How do you make a dragon curve?
Dragon Curve
- Cut eight strips of paper, two strips of each of the four colors.
- Fold a strip in half by bringing the right edge on top of the left edge.
- Fold the strip in half again right edge on top of left edge.
- Fold the strip in half again two more times for a total of four folds always folding in the same direction.
Are fractals 2d?
The fractal dimension of a curve can be explained intuitively thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional.
Is it possible to have a negative dimension?
However, if you consider dimension as the value of some sort of integration which, in vector space case, coincides with the above definition, then a negative dimension is possible (for example, you can use all types of measures for integration, negative, complex, etc).
What is the negative dimension of a vector space?
Dimension of a (finite dimensional) vector space is defined as the cardinality of a basis for the vector space. Since the cardinality cannot be negative, negative dimension for vector spaces is meaningless.
Can a homology complex have a negative dimension?
If the homology groups have a concept of dimension, we define the strict reverse homology groups to have negative dimension. You can have complexes with mixed forward and reverse homology.
Can Grassmann-odd directions have negative dimension?
For a supergroup, one should use the supertrace, so Grassmann-odd directions can in some sense be viewed as having negative dimension. See also e.g. Ref. 1. K-theory, which is relevant for e.g. string theory and integer quantum Hall effect.