First, I think it’s smooth at every point not just almost everywhere.
Second, I think it’s infinitely long because if iteration n is length L, iteration n + 1 is length m * L where m is greater than one. I haven’t worked out the details yet, but that’s the general idea.
An easier to way to construct what you want would be to take a function like x \sin(1/x) or \sqrt{x}\sin(1/x) that is not of bounded variation and look at its graph near x=0.
Actually, I’m not sure that @Lance_Turbo’s example quite works: If I’m not mistaken, the total length of straight segments remains constant in every iteration, and I think that the length of new curved segments in each iteration is a fraction less than 1 of the length of curved segments in the previous iteration, for a total length that converges.
But this can be remedied by increasing the straight fraction (i.e., making the curves smaller radius relative to the length), so that the straight length (and hence, the total length) increases with every iteration.
I had the same doubts, and the same fix in mind. It’s actually worse than you think. The total straight line length goes from x to 5x / 6 each iteration, but total length goes up. I do think the lengths of the curved portions will end up being a convergent geometric sum.
All of this is definitely fixable with smaller radius curves.
There’s also a small error in Arc 2 (I flipped a sign).
Should be…
Arc 2: The part of the circle with center (1/2, sqrt(3)/9) and radius sqrt(3)/36 that smoothly connects segments 2 and 3.
To be honest, I didn’t calculate the length of the diagonal straight segments-- I just assumed that since you made the ones on the axis 1/4 long, you did so for the diagonals, too.
But I must correct myself: If you had done that, then it would work. At each iteration, the total length of new curve is the same proportion of the total length of new segments. But the new segments are all of the segments, and thus have the same length at each iteration, and so the amount of new curved length is constant at each iteration, and so the curved length increases linearly with the iteration number.
The new catch that occurs to me, though, is that the final curve isn’t made up entirely of smaller copies of itself any more (it’s a combination of smaller copies of itself and circular arcs), which will make it trickier to calculate its dimensionality. I think that it might be the case that all of these “filleted snowflakes” (no matter how small the fillet) will turn out to be purely 1-dimensional.
I like this. New curve…
Segment 1: (0, 0) to (3/10, 0).
Segment 2: (7/20, sqrt(3)/60) to (9/20, 7 * sqrt(3)/60).
Segment 3: (11/20, 7 * sqrt(3)/60) to (13/20, sqrt(3)/60).
Segment 4: (7/10, 0) to (1, 0).
Arc 1: The part of the circle with center (3/10, sqrt(3)/30) and radius sqrt(3)/30 that smoothly connects segments 1 and 2.
Arc 2: The part of the circle with center (1/2, sqrt(3)/10) and radius sqrt(3)/30 that smoothly connects segments 2 and 3.
Arc 3: The part of the circle with center (7/10, sqrt(3)/30) and radius sqrt(3)/30 that smoothly connects segments 3 and 4.
Looks something like this.
Infinitely long with a well defined tangent line at every point.
If iteration 0 is a line segment of length 1, then iteration n has a total length of 1 + 4 * pi * n * sqrt(3) / 90.
Great illustration; thanks!
But the more I think about it, the more I think it’s not a fractal. Almost all of the curve is made up of circular arcs, because the total length is infinite, but the length of straight segments is finite. And any given circular arc is 1-dimensional: Once you find an arc (which you will eventually at almost every point), zooming in further won’t reveal any more detail.
We could make a new version of this curve (by decreasing the fillet), where the length of the straight segments grows without bound as iterations increase. But the curved segments will still grow faster, and so it will remain the case that the total figure is almost entirely circular arcs, and so the problem will remain.
It’s still quite an interesting figure, though, in that it has many of the properties of a fractal without (quite) being one.
It may or may not be a fractal, but my intention was not the creation of a fractal. It was this…
I’m pretty sure, however, that a version where the total length of the straight line segments grew geometrically would be a fractal by some definition of fractal. In particular, some define fractal as any set where the Hausdorff- Besicovitch dimension strictly exceeds the topological dimension.
If we had made one (which I probably will) where each straight line segment was replaced by four segments 3/10 as long as the original and connected them with arcs. We can compute the Hausdorff- Besicovitch dimension of just the straight line segment portion and that is…
log(4) / (log(10) - log(3)) \approx 1.15
I don’t think adding the arcs decreases the HB-dimension so we have a fractal by that definition.
Finally, I made a slightly cleaner version of the constant length of straight segments version. Here we have and object that is made of four identical 1/4 scale versions of itself connected by arcs that have the same radius. The HB-dimension of the straight line portion is exactly one, and I’m not sure what to do with the arcs.
Hm, true… A figure that consisted of just the disconnected line segments (in your 3/10 example), without the connecting curves, would be a fractal in every sense of the word (there’s no rule that a fractal has to be connected). And it’s hard to imagine any definition by which a one-dimensional figure could contain a part of greater than 1 dimension, so the whole, connected figure would have to be a fractal, too, I think? A collection of purely 1-dimensional arcs, connected by a Cantor-set-like fractal scattering of infinitesimal line segments.
To be clear here, “topological dimension” is that a connected set that’s disconnected by removal of a point is 1-d, a connected set that’s disconnected by removal of a 1-d figure is 2-d, and so on?
There may be some nuance to it that does not immediately leap to my mind, but I think what you wrote is correct or, at least, correct enough for this thread.
OK, in the hope of finding a definition of “what is a fractal?”, I cracked open a couple of books, and the second one I tried, Fractal Geometry by Falconer, straight-up refuses to give a formal definition and states in the introduction that
As for curves in the plane, there are a lot of propositions mentioned there, but as for Borel sets of Hausdorff dimension equal to 1 and of positive, finite Hausdorff measure, he breaks such a thing down as a union of its “regular”, “curve-like” (contained in a countable union of rectifiable curves) part, plus the Cantor-set-like “curve-free” part. It is proved that a “regular” set has a well-defined tangent at almost all of its points, while at almost all points of an “irregular” set no tangent exists.