Why can fractal dimensions be non-integer values?

[Tracey3]

Hi there, so recently we had professor's assistant covering our class and he decided to talk about Fractal Dimensions. Maybe its just the concept or his explenation but we all left the class bewildered to say the least.

Could someone clarify for me, how do we refer to the number of dimensions? My understanding was always that they are uniform integers 1,2,3. Based on what my current comprehension is, we have 1-3 dimensions?

 

[TeethWhitener]

What you're looking for is the distinction between topological dimension (what you're familiar with) and Hausdorff dimension (what your prof was referring to). There's a pretty decent explanation here:
https://www.quora.com/In-laymans-terms-what-is-the-Hausdorff-dimension-a-measure-of

Caveat: I am not an expert in topology (at all).

 

[WWGD]

The Fractal dimension can be fractional and applies to spaces which may not be quite either 1- , 2- or n-dimensional, but somehow somewhere in-between.

 

[Mark44]

The term fractal was coined by Benoit Mandelbrot in 1975, to express the concept of dimensions that could be fractional. Common geometric objects such as lines (dimension 1), planes (dimension 2), and spheres (dimension 3) have integral dimensions, but there are objects that have dimension somewhere between these integer values. Some examples are the Koch snowflake and Sierpinski carpet. The Koch snowflake is made up of line segments, but the snowflake itself is so convoluted that it can be thought of as filling an area. See https://en.wikipedia.org/wiki/Fractal