The Hausdorf dimension of an object is a measure that allows us to describe things as not just 2D (flat) or 3D (taking up volume), but to assign fractional dimension to objects that are arguably both. If you crumble up a piece of paper, is it flat, or does it take up volume? The paper is flat, but the ball takes up space.
I don’t feel like looking it up, but if I had to guess, the fractional dimension of this paper ball is maybe 2.7 - the space it takes up is much more noticeable than the flatness of the paper it is made of.
Usually this fractional dimension is calculated by trying to approximate the shape with a lot of balls, and seeing how the number of balls needed to make the shape changes as the balls change in size. If you fill a swimming pool with balls (a ball pit!), then try replacing the balls with new ones with half the diameter, you would need 8 times as many balls. This tells us the pool is 3D, because log_2 (8) = 3; but if you try to cover the floor of the pool with a thin layer of balls, you will find that you need only 4 times as many balls when you use the smaller balls. Because log_2 (4) is 2, we know the floor of the pool is 2D.
If we tried approximating the shape of our crumpled ball of paper using really, really tiny (and shiny!) metal ball bearings, we would find that as we halve the diameter of the ball bearings, we need more than 4 times as many bearings, but less than 8 times as many of these shiny metal balls- if you actually counted them, and took the log_2 of the ratio, you would get a fractional dimension that’s more than 2 but less than 3.
Traditionally the fractional dimension is reported as a single number, the dimension we would measure as the balls become infinitely small. But really, we’re throwing away a lot of information by only returning that one number.
If we use ball-bearings that are as big as the crumpled up sheet of paper, we would find that on that scale, it’s actually pretty much just 3-dimensional. But if we used atomically small balls, we would probably find that the dimension isn’t much different from being perfectly flat. As our scale continuously, gradually shifts from the big picture to the small scale, we will find that at each scale, the dimension takes on a unique value, and that tells us something worth knowing about that shape at each degree of detail.
This means we should report the fractional dimension of an object not just as a single number, but use a continuous function that takes in a scalar describing the scale level, and telling us what the fractional dimension at that particular scale is.
This means we should report the fractional dimension of an object not just as a single number, but use a continuous function that takes in a scalar describing the scale level, and telling us what the fractional dimension at that particular scale is.
Also potentially relevant is the magnitude function, which is a function |tA| of a space A and a real-valued scale factor t, and asymptotically grows as O(tdimA) where dimA is A’s Minkowski dimension (which usually agrees with Hausdorff dimension).
I’m not sure this should be modeled as “fractional dimension”, though. It seems like standard “volume” in 3 dimensions is the thing you’re talking about, and your measurement granularity is determining how you include or exclude “pockets” in the crumple as part of the volume.
The Hausdorf dimension of an object is a measure that allows us to describe things as not just 2D (flat) or 3D (taking up volume), but to assign fractional dimension to objects that are arguably both. If you crumble up a piece of paper, is it flat, or does it take up volume? The paper is flat, but the ball takes up space.
I don’t feel like looking it up, but if I had to guess, the fractional dimension of this paper ball is maybe 2.7 - the space it takes up is much more noticeable than the flatness of the paper it is made of.
Usually this fractional dimension is calculated by trying to approximate the shape with a lot of balls, and seeing how the number of balls needed to make the shape changes as the balls change in size. If you fill a swimming pool with balls (a ball pit!), then try replacing the balls with new ones with half the diameter, you would need 8 times as many balls. This tells us the pool is 3D, because log_2 (8) = 3; but if you try to cover the floor of the pool with a thin layer of balls, you will find that you need only 4 times as many balls when you use the smaller balls. Because log_2 (4) is 2, we know the floor of the pool is 2D.
If we tried approximating the shape of our crumpled ball of paper using really, really tiny (and shiny!) metal ball bearings, we would find that as we halve the diameter of the ball bearings, we need more than 4 times as many bearings, but less than 8 times as many of these shiny metal balls- if you actually counted them, and took the log_2 of the ratio, you would get a fractional dimension that’s more than 2 but less than 3.
Traditionally the fractional dimension is reported as a single number, the dimension we would measure as the balls become infinitely small. But really, we’re throwing away a lot of information by only returning that one number.
If we use ball-bearings that are as big as the crumpled up sheet of paper, we would find that on that scale, it’s actually pretty much just 3-dimensional. But if we used atomically small balls, we would probably find that the dimension isn’t much different from being perfectly flat. As our scale continuously, gradually shifts from the big picture to the small scale, we will find that at each scale, the dimension takes on a unique value, and that tells us something worth knowing about that shape at each degree of detail.
This means we should report the fractional dimension of an object not just as a single number, but use a continuous function that takes in a scalar describing the scale level, and telling us what the fractional dimension at that particular scale is.
The relevant keyword is covering number.
Also potentially relevant is the magnitude function, which is a function |tA| of a space A and a real-valued scale factor t, and asymptotically grows as O(tdimA) where dimA is A’s Minkowski dimension (which usually agrees with Hausdorff dimension).
Thanks. The function I am describing can be derived from the covering number function, but is also distinct.
See also https://en.wikipedia.org/wiki/Coastline_paradox . At different scales, you’ll be measuring different things and different features or granularity of variance will matter.
I’m not sure this should be modeled as “fractional dimension”, though. It seems like standard “volume” in 3 dimensions is the thing you’re talking about, and your measurement granularity is determining how you include or exclude “pockets” in the crumple as part of the volume.
The metric I had in mind that the function would report is the log of the ratio of the measure at slightly different scales, not the measure itself.