I suppose that answer is trivial and no any novelty can be discussed.

The 3D-Koch curve is obtained< for example, by parallel shifting along OZ axis the so-called Koch's island located in OXY plane.

But from another point of view this question might present an interest for specialists working in the fractional calculus. What kind of the spatial fractional integral one can expect in the result of the averaging a smooth function f(x,y,z) over 3D-Koch curve? This problem as far as I know is not solved.

I don't see a problem here. Start with a regular quadrahedron (pyramid) and add smaller quadrahedrons to each face sized and positioned to place a vertex midway on each edge of the original. The result now has 24 faces, all congruent equilateral triangles. Repeat step 2 on each face.

I completely agree with the Dr. James Langworthy point of view. But the suggestion of the correct averaging procedure of smooth functions over 3D fractal sets represents an important problem.

Hi Raoul!

If you're talking about simply extruding a Koch snowflake, then that's not what I had in mind. What I was interested in was ways of extending the Koch curve into higher dimensions in an analogous way to how the (2D) Sierpinski triangle can be extended to produce the (3D) Sierpinski pyramid.

I found one shape that I think meets that criterion (illustrated in the linked document and supplementary image), and was wondering it it had been published by any previous researchers, or whether there might be other solutions that could be treated as similarly powerful extensions of the curve.

Hi James!

That tetrahedron-based exercise is one that tends to be mentioned when people think about possible extensions of the Koch curve, but when the exercise completes, all you seem to end up with is a simple solid cube with some interesting fractal cracks running through it.

Those eight three-faced peaks of the 24-faced "star" become the eight corners of the final solid cube.

While its construction is obviously inspired by the construction of the Koch snowflake, the resulting tetrahedron-based solid doesn't obviously contain any Koch snowflakes or Koch curves within its structure (unless I missed them).

On the other hand the Sierpinski Pyramid and Menger Sponge are proper supersets of the Sierpinski Triangle and Sierpinski Carpet, and the 2D faces of those 3D solids are their 2D counterparts.

That's the sort of more ambitious connectivity that I was looking for.

The delta shape that I uploaded a description and image of seems to fit that criterion for the (generalised) Koch curve, and I'm wondering whether anyone else found it before me, and whether it has any sibling solutions.

Well if you follow what I specified it's a real fractal and doesn't reduce to a cube. Every step uses pyramids 1/3 the height of those in the previous step.

James:

(a) Yes, agreed, what you describe is a real fractal

(b) The volume does converge on the shape of a cube, with its surface cut by a fractal network of arbitrarily-narrow fissures. I agree that this is not an obvious result.

(c) You probably meant to say that the daughter-pyramids are half the height of their parents?

Right: 1/2 it is.

James:

FYI: There's a diagram showing how your tetrahedron-based fractal converges on the shape of a solid cube on page 20 of "Alt.Fractals: A visual guide to fractal geometry and design" (2011)

Figure 3-6: "Koch approach, applied to a tetrahedron".

That early 24-sided star shape has eight three-faceted points. If you draw lines linking each of those eight points with their immediate neighbours, the points become the eight corners of a cube. That cube is the bounding region of the fractal, and as the fractal gets more iterations, the daughter tetrahedra progressively fill in the empty spaces inside the cube.

Book Alt.Fractals: A visual guide to fractal geometry and design

Here's a picture of that (IMO off-topic) tetrahedral construction, demonstrating how it converges on the shape of a cube. The thing becomes obvious after the first few iterations.

Now that this side-issue is (hopefully!) disposed of, does anyone have any comments on the actual paper, or the "Delta" shape presented? The delta is the only shape I've ever seen that can claim to be a "proper" three-dimensional extension of a Koch curve, but since Koch curves can have different proportions, there's still the possibility of an alternative "sibling" solution based on a different construction method and on a differently-proportioned Koch. I'm assuming that we're not looking at "extremal" solutions that involve angles of 0/360 degrees, here.

Data Alt.Fractals, page20

Here's the delta shape:

Data "Delta" fractal, 3D extrapolation of Koch curve

I've just uploaded some 3D models in .stl format, for the benefit of anyone who wants to play with the shape: The basic six-faced building block (iteration zero) , and the fractal at three and eleven iterations.

The eleven-iteration version is quite big (about fifty Meg for the file) it might not load if your PC has less than four gigs of memory.

Data Delta fractal building block - 3D model

Data Delta fractal third iteration

Data Delta fractal eleventh iteration

Excellent, thank you for the .stl files, they will help me to understand things better. With your permission I may try 3D printing them, I'm just getting my new 3D printer going. I'm attaching the print ready .stl files, I fixed some problems to make them watertight. I hope they're helpful. I'm using Linux Mint 17.1 with netfabb and meshlab for testing/repairing, and Repetier-Host for testing, scaling and hopefully printing. I hope the zip comes through well, I wasn't allowed to attach the .stl files.

Hi John - free to do what you like with them, they're there to share! :)

The problem with 3D-printing the shape is that its parts are only joined by edges, so the thing will tend to fall apart when printed unless you "thicken" it somehow. A wireframe skeleton might work.

Thank you. I'm planning to add material at the intersections using solidworks if it doesn't "fall apart" importing the STL, I'll send the results. The wireframe skeleton is a good plan also, I might do both.