>[!abstract] >A fractal with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. ("Sierpiński triangle", 2025). What I find fascinating about the Sierpiński triangle (beyond its fractal nature — because fractals are never not cool) is that it is of dimension $d \approx 1.585$, and not $d \approx 2$ as you would expect from a regular triangle drawn on a screen or sheet of paper. How to find this value of $d$ is also interesting. Let's double the linear dimensions of shapes in various dimensions to understand this. - If you double the linear dimensions of the segment of a line (a one-dimensional shape), you end up with $2^1=2$ instances of that segment. - If you double the linear dimensions of a (non-fractal) equilateral triangle (as an example of a two-dimensional shape), you end up with $2^2=4$ instances of that triangle (three at each corner, plus one upside-down triangle in the middle). - If you double the linear dimensions of a a cube (a three-dimensional shape), you end up with $2^3=8$ instances of that cube (two cubes side-by-side in the $x$, two in the $y$, and two in the $z$ dimensions, all side-by-side, forming a $2\times2\times2$ larger cube.) - So, the general rule when doubling the linear dimensions of a shape is $2^d=n$ where $2$ is simply the doubling operation (i.e., we could use $3$ for tripling, etc.), $d$ is the number of dimensions of the shape, and $n$ is the number of shapes you end up with. Now, if we want to know $d$ given $n$, i.e., if we want to figure out the dimensions of a shape knowing how many instances of it we get from doubling the shape, we can use the *logarithm* to base $d$ (which is the inverse of exponentiation with base $d$) in the following formula: $d=\frac{\log{n}}{\log2}$. For example: - A line segment has $d=\frac{\log2}{\log2}=1$ dimension given that $n=2$ when doubling it. - A (non-fractal) equilateral triangle has $d=\frac{\log4}{\log2}=\frac{\log2^2}{\log2}=\frac{2\log2}{\log2}=2$ dimensions given that $n=4$ when doubling it. - A (non-fractal) equilateral triangle has $d=\frac{\log8}{\log2}=\frac{\log2^3}{\log2}=\frac{3\log2}{\log2}=3$ dimensions given that $n=8$ when doubling it. But the Sierpiński triangle is no regular equilateral triangle; it consists of three copies of itself in each corner, and an upside-down *empty* triangle of the same linear dimensions as the original in the middle. That the latter triangle is empty is *a defining characteristic* of the Sierpiński triangle. This means that if you double the linear dimensions of a Sierpiński triangle, you draw two more such triangles (one next to it, the other offset above), yielding three instances of the original, not four. The middle space remains empty. So now we can compute the dimensionality of the Sierpiński triangle: - A Sierpiński triangle has $d=\frac{\log3}{\log2}\approx1.585$ dimension given that $n=3$ when doubling it. In that sense, the Sierpiński triangle is a portal into a dimension other than our usual 1D, 2D, or 3D experiences. I also note that it is possible to perfectly draw the boundaries of that $\frac{\log3}{\log2}$-dimension shape on a $2$-dimension screen or sheet of paper; and these boundaries perfectly circumscribe the infinite self-similarity of the Sierpiński triangle. However, the fractal nature of it cannot be rendered to perfection, since it is physically impossible to draw infinitely many smaller Sierpiński triangles within each corner. To me, this situation is analogous to writing a rational number in closed form, for example $\frac{2}{3}$; it rigorously encapsulates the number "two thirds", and yet, it is physically impossible to write down the full decimal expansion of that same number ($0.666...$) because it goes on forever. This is an example of how "trivial" and its antonym "impossible" can be next-door neighbors in mathematics. Another example might be $y=\frac{1}{x^2}$ for $x\geq0$. It is trivial to calculate $y$ for any arbitrarily small value of $x$; for instance, $y(10^{-100}) = 10^{200}$. And, the difference between $10^{-100}$ and $0$ is infinitesimal. Yet, it is impossible to calculate $y(0)$, because dividing by zero is undefined. ## References - Sierpiński triangle. (2025, March 14). In *Wikipedia*. https://en.wikipedia.org/w/index.php?title=Sierpi%C5%84ski_triangle&oldid=1278360998