Last week saw a couple pleasant milestones. A paper I'd written describing a novel algorithm for analyzing river geometries was officially accepted to Computers and Geoscience, and a paper on which I am second-author paper, that looks at recurring slope lineae (RSL) on Mars following last year's global dust storm (or planet-encircling dust event), has been submitted to JGR-Planets.

In my current research, I've recently focused on clarifying my understanding of various terms related to fractal analysis. Surprisingly, although these terms have been used in the geologic literature, I was unable to readily find some simple, illustrative examples. Following a suggestion of my advisor, Dr. Catherine Neish, I decided that some such examples would be an asset to the paper we're working on, which looks at the fractal geometry of lava flow margins. Below is a figure showing all of these geometries, followed by plot illustrating their fractal dimension.

The first column is the classic Koch curve, which has a fractal dimension of ln(4) / ln(3) ~ 1.26. It is nominally constructed by starting with a line segment from (0, 0) to (1, 0) (not shown) and then replacing this segment with the "motif" (series of line segments) shown in the top left cell. At each generation after that, each line segment is again replaced by a scaled representation of the same motif. The second cell is the second generation, the third cell is third generation, and the fourth cell is the tenth generation. The fifth cell zoom-in on a small part of this tenth-generation construction; the pictured area comes from the center of the magenta circle in the fourth cell but is far small than that circle.

The Koch curve is very regular. At each scale, the same motif is repeated exactly, which also means that any region formed by a combination of motifs is likewise reproduced exactly at other scales. This regularity is called self-similarity.

The second column depicts a random Koch curve. This curve is constructed in exactly the same way as the classic Koch curve except that the motif is flipped (mirrored) across the its long axis randomly, with 50% frequency, The result by the 10th generation (again, the four cell down) is a much less symmetric geometry and, hence, once that appears more natural. Even though patterns are not exactly reproduced at different scales, on account of the random flipping, the geometry still has a repeatability with scale that can be verified statistically. Hence, it has statistical self-similarity.

The third column is very similar to the classic Koch curve except that triangular protrusion as the center of motif is less pronounced. As a result, the areal density of segments varies along the curve, and the curve is no longer even statistically self-similar. This inconsistency departs from the simple fractality that I've described before, which can be properly called monofractality. This geometry may instead be called multifractal, because a different fractal dimension may be observed in different regions. For our purposes, I will call this a spatial multifractal, because the fractal dimension varies from region to region, in contrast to the multifractals in the next two columns.

The fourth and fifth columns are a pair. The fourth column shows a construction that begins with the motif shown in the first cell, but in each successive generation, the motif gradually morphs to that shown in the first cell of the fifth column. Conversely, the fifth column shows a construction in the opposite sense: it begins with the subtle job shown in the first cell of that column and then, as each segment is replaced with a motif at each generation, that motif linearly evolves to the more jagged motif seen in the first cell of the fourth column. More casually, the construction in the fourth column starts rough at the coarsest scales but becomes smoother at finer scales, whereas the construction in the fifth column starts relatively smooth at the coarsest scales and becomes coarser at finer scales. These differences are clear in the 8th-generation views present in the bottom cell of each column.

When analyzed using the divider method, the scale-dependence of the measured fractal dimension for each geometry provides significant insights. The first two geometries are nearly indistinguishable, varying about the true fractal dimension value of ~1.26. (The periodicity is likely a consequence of geometry "short-circuiting" inherent to the geometry of the motif.) The spatial multifractal of the third column simply appears to have a muted, nearly constant fractal dimension with scale, likely because the smaller prominence of the triangular protrusion mutes the roughness of the geometry. Because these plots are the result of whole-geometry analysis, the spatial variation in fractal dimension is not detected. However, the scale-based variation in fractal dimension for the last two columns is detected, because this variation is expressed across the whole geometry. And the quantitative variation is approximate what would be theoretically expected. This close approximation is encouraging, because there are several simplifications that must be used in this analysis. The near correspondence to theory therefore indicates that these net effect of these simplifications is relatively small.