Why the use of the term "snowflaking"? 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?
 A: 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.

