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 .
Can you have half dimensions?
You can have a set that’s really d-dimensional, but on a large scale it appears to be a different dimension. For example, a piece of paper is basically 2-D, but if you crumple it up into a ball it seems 3-D on a large enough scale.
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 dimension is Sierpinski’s Triangle?
The gasket is perfectly self similar, an attribute of many fractal images. Any triangular portion is an exact replica of the whole gasket. The dimension of the gasket is log 3 / log 2 = 1.5849, ie: it lies dimensionally between a line and a plane.
What is the highest dimension a fractal can be?
The dimension of the Mandelbrot’s boundary (picture above) is 2, which is the highest it can be, but there are more interesting (but less pretty) fractals out there with genuinely fractional dimensions, like the “Koch snowflake” which has a dimension of approximately 1.262.
What is the fractional dimension of a point?
Fractional Dimensions A point has dimension 0, a line has dimension 1, and a plane has dimension 2. But did you know that some objects can be regarded to have “fractional” dimension? You can think of dimension of an object X as the amount of information necessary to specify the position of a point in X.
What is the dimension of a fractional curve?
Most other “fractals” have fractional dimension; for instance a curve whose boundary is very, very intricate can be expected to have dimension between 1 and 2 but closer to 2.
Why does the Cantor set have a fractional dimension?
The standard Cantor set has fractional dimension! Why? 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.