Why the use of the term "snowflaking"?

Question

I've seen a few places in the literature (in particular in fractal geometry) where we consider a metric space $(X,d)$, and then for $0 < \epsilon < 1$ define a new metric space $(X,d^{\epsilon})$ where: $$ d^{\epsilon}(x,y) = d(x,y)^{\epsilon} $$ This new metric space is called an $\epsilon$-snowflaking of $X$.

My question: why this particular terminology? Is the new metric somehow reminiscent of snowflakes? Are there some nice illustrations that support this naming convention?

Answer 1

Take an interval, say $[0,1]$, with the usual distance function, i.e. $d(x,y) = |x-y|$. Now take $\alpha = \log(3)/\log(4)$, and consider the $\alpha$-snowflaked space $$ ([0,1], d^\alpha). $$ This new space is isometric to the classical von Koch curve, which resembles a snowflake. Heinonen has a short, but accessible discussion of this in his notes on Geometric embeddings of metric spaces. My recollection is that he has a more detailed discussion in his book Lectures on Analysis on Metric Spaces, but (thanks to the COVID shutdown), I don't have access to my copy of this book right now.

Citing another source, Tyson and Wu have the following to say:

Our terminology stems from the observation that the classical von Koch snowflake curve $C$, endowed with the planar Euclidean metric, is a $p$-snowflake with $p = \log 4 / \log 3$. Indeed, we may choose $d'(x,y) = \mathscr{H}^p(C_{xy})^{1/p}$, where $\mathscr{H}^p$ denotes the Hausdorff $p$-measure on $C$ and $C_{xy}$ denotes the minimal connected subset of $C$ containing $x$ and $y$. See Figure 1.