The classical geometry deals with objects of integer dimensions.
Fractals have fractional, noninteger values of dimension, and thus their properties are distinct from those of points, lines, surfaces, and solids that have integer dimensions.
We did the first experiments on physical systems that were truly fractal – that behaved as if they existed in universes with non-integer dimensions, such as 2.67.
For many years it was conjectured that the integer dimension subgroups of a group coincide with its lower central series.
The next highest dimension ((D+1)) is the minimum integer dimension is which the attractor can exist.
These spaces can be proven to have always an integer dimension (din mathbb Ncup {infty }), and can be considered as MMS whenever (d<infty ), by endowing them with their Hausdorff measure (mathcal {H}^d).
The same methodology will allow future studies to address fundamental questions about the effects of spin orbit interactions and magnetic fields on electrons in non-integer dimensions.
That means that the Laplacian in Eq. (1) is a symmetric matrix of dimension N with integer entries {0,1} for the off diagonal elements, and the diagonal elements equal to <img src="http://journals.plos.org/plosone/article/asset?id=info?doi/10.1371/journal.pone.0003479.e066.PNG" class= inline-graphic"/>, with i≠j.
Fractal shapes are interesting because they can have non-integer dimension (hence "fractal"!).
It could exhibit sensitive dependence on initial conditions (two very close trajectories tend to diverge from each other exponentially fast), its trajectories in phase space could be confined to an area of non-integer dimension (strange attractor), etc.
Etymologically, as its common root with "integer" makes clear, it suggests wholeness.
