Can there be 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 there be fractional dimensions?

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 .

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.

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 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 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.

What is fractional dimensional space?

We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. Models of Quantum Gravity, Field Theories in Lower Dimen- sions, Fractal Geometry.

Why do fractals have fractional dimensions?

Fractional dimensions are very useful for describing fractal shapes. In fact, all fractals have dimensions that are fractions, not whole numbers. If a line is 1-Dimensional, and a plane is 2-Dimensional, then a fractional dimension of 1.26 falls somewhere in between a line and a plane.

What does it mean if probability is 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. These distributions may apply to unobservable events or conditional probabilities.

Why can’t the probability of an event be negative?

No, because the number of trials in which the event can happen cannot be negative and the total number of trials is always positive.