## Is it possible to have a dimension that is not an integer?

If you take a two-dimensional object in a three-dimensional ambient space (say, a piece of paper) and scale it up, then its area will increase by a factor of s2. This number in the exponent is basically the fractal dimension of that object, and it’s possible for it to be non-integral.

### What do you mean by fractal dimension?

In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured.

**What is the hausdorff method?**

The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric. Consider the number N(r) of balls of radius at most r required to cover X completely. When r is very small, N(r) grows polynomially with 1/r.

**How is Hausdorff dimension calculated?**

The Hausdorff Dimension We consider N=rD, take the log of both sides, and get log(N) = D log(r). If we solve for D. D = log(N)/log(r) The point: examined this way, D need not be an integer, as it is in Euclidean geometry. It could be a fraction, as it is in fractal geometry.

## Are there 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.

### What is meant by topological dimension?

The topological dimension \dim (X) is also called the Čech-Lebesgue covering dimension , or simply the Lebesgue dimension , of X. It is clear from its definition that topological dimension is a topological invariant, that is, one has \dim (X) = \dim (Y) whenever X and Y are homeomorphic topological spaces.

**What is the Hausdorff dimension of the set of natural numbers?**

Then the set ^Y*A of numbers which are normal to every base from R and to no base from S has Hausdorff dimension at least log (A – l)/log A. 2.

**What is Hausdorff dimension used for?**

Hausdorff measure takes the idea of looking at the volume of coverings by rectangles and generalizes it to arbitrary metric spaces and fractional α.

## What is the meaning of negative dimension?

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 a zero dimensional object?

Zero Dimensions: A point has zero dimensions. There’s no length, height, width, or volume. Its only property is its location. You could have a collection of points, such as the endpoints of a line or the corners of a square, but it would still be a zero-dimensional object.

**What is the topological dimension of Koch curve?**

Therefore for koch curve topological dimension is 1 but fractal dimension is 1.2618.