I took a quick look at the arXiv paper behind this article, and it looks suspicious. I didn't take the time to be 100% sure it's vacuous, but it has many red fl

I took a quick look at the arXiv paper behind this article, and it looks suspicious. I didn't take the time to be 100% sure it's vacuous, but it has many red flags, such as being published in "Chaos, Solitons & Fractals," a poorly-regarded journal. Another red flag is the existence of theinfinitycomputer.com. Now for some real math related to this: The Cantor Dust and the Sierpinski Carpet really do have area (measure) 0 in the traditional sense. The normal area of the Sierpinski Carpet, during construction, starts at 1 (a full unit square), and is multiplied by 8/9 per iteration. The iterations never stop, so the area goes to zero. Its 2D measure is 0. There are infinitely many length-1 lines within the final product (any line with equation y=p for any constant p in [0, 1] with no 2's in it's base-3 representation), so it has infinite 1D measure. So we can look for a dimension between 1 and 2 that measures it, which is where the Hausdorff dimension comes in. If you double a unit (length-1) line, its measure doubles. If you double a unit square, its measure is multiplied by 4. If you double a unit cube, its measure is multiplied by 8. Here's a pattern we can generalize to non-integer dimensions! We work from the idea that any measurable set S, when multiplied (scaled) by scalar r, will have its measure (eg length, surface area, volume) multiplied by r^d, where d is the dimension of S. Using m(S) for the measure of S, we can write: m(S * r) = m(S) * r^d which means log [m(S * r) / m(S)] = d * log r and then d = log [m(S * r) / m(S)] / log r This isn't the formal definition of the Hausdorff dimension, but it is the idea behind it, and provides the correct dimension for iterative fractals like the Sierpinski Carpet. What is d for the Sierpinski Carpet? When we scale it up by 3, we get 8 copies of the original. So m(S * r) / m(S) = 8, and r = 3. So d = log 8 / log 3 = 1.892789... for this shape. If you wanted, you could use a factorial-based expression for the size of an n-dimensional sphere, along with the fractional version of !, the gamma function, and use those to get a reasonable number for the exact measure of such a shape. One more thing. There are numbers of infinite size that you can perform meaningful arithmetic on. My personal favorite are the surreals, championed by John Conway about 40 years ago. This number class basically starts with the ordinals (so including many sizes of infinity) and builds real-number-like operations on those. If omega is the first infinite number, then you can talk about sqrt(omega) in the surreals, and it's a meaningful quantity larger than any finite number and smaller than omega, for example. There are also hyperreal numbers which offer similar freedom of expression.