Re: Fractal Sponges of Arbitrary Dimension 2 =< D =< 3

In message <1185908978.197602.189530@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, ranecurl@xxxxxxxxx writes
On Jul 31, 7:20 am, Stewart Robert Hinsley <stew...@xxxxxxxxxxxxxxxxx>
wrote:
In message <1185846755.877311.95...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
ranec...@xxxxxxxxx writes>I seek descriptions and generators for fully connected deterministic
>fractal sponges other than the Menger Sponge. They should exist with
>fractal dimensions from 2 to 3.

There's a tetrahedron. (Repeatedly remove the smaller tetrahedra defined
by drawing lines between the midpoints of edges). Similar figures are
composed of 10, 20 etc, rather than 4, self similar parts.

That's Mandelbrot's (1983) Plate 143, D =2, isn't it? I recalled
seeing it there
when you mentioned it. Do you know of one with the other limit, D=3?

I'll have to take your word for whether it's in Mandelbrot's book.

I hadn't noticed, but yes it is D=2 (4 copies, each half the size).

By analogy with D=2 plane figures I would think that sponges can approach arbitrarily close to D=3, but not reach it, but this may depend on what one accepts as a sponge. There are D=2 plane with voids - see for example

http://www.meden.demon.co.uk/Fractals/isotrianguloid.html

and I presume that similar figures can be created in R^3.

Presumably similar figures can be produced from square pyramids.

You can take the 27 part dissection of a cube, and remove the central
part, and so on. (Not that you can see the sponge-like nature of this
figure from the outside.)

You can take a cube, dissected into 125 (rather than 27) parts, and
remove the 13 (rather that 7 parts) joining the midpoints of the faces.
The same can be done with dissections into 7^3, 9^3 etc parts.

You can take the 125 part dissection of the cube, and replace the groups
of 8 parts at each corner with a single larger cube, giving a dissection
into 8 larger, and 61 smaller parts.

You can take a dissection of a cube into 64 parts, and then remove the
32 parts which touch the lines connecting the midpoints of the face.

Similarly with dissections into 6^3, 8^3 etc parts.

More figures arise when the pieces removed do not lie on the lines
connecting the midpoints of the faces, and yet more when any requirement
for symmetry is relaxed. (For an example of the former, take the 125
part dissection, and remove 4 symmetrically placed columns from each
face.)
--
Stewart Robert Hinsley

Thanks! I will work on visualizing these sponges. Do you know of
sources for nice
drawings of them?

Not offhand. But, IIRC, you can draw them using FractInt.

I gather that there are only series of discrete dissections. That is,
there is no
general way to construct a sponge given an arbitrary similarity
dimension?

It depends on what constraints you apply. The figures I described earlier are self-similar. If you relax the constraints to allow self-affine figures then you can dissect a cube into 8 corner pieces, 12 edge pieces, 6 face pieces, and 1 centre piece. Delete the transforms for the last 7 from the corresponding IFS and the attractor is a sponge (most of the holes of which are not square in cross-section). You can vary the size of the corner pieces from 0 < l < 0.5; the sizes of the remainder are fixed by l. (corner - l x l x l; edge 1-2l x l x l; face 1-2l x 1-2l x l; centre 1-2l x 1-2l x 1-2l; I think.)

I don't remember how to calculate the dimension when the figure is dissected into copies of different sizes, but obviously the dimension is a continuous function of l, with a value with approaches 3 as l approaches 0.5. It's not so obvious what the lower bound is.

Another example follows from the tetrahedron above. This is constructed by four self-similar transformations mapping the figure to its corners. You can add 4 self-affine transformations mapping the figure to low tetrahedrons placed on the interior faces of each of the original 4. As the height of the low pyramid increases the dimension increases, the dimension being a continuous function of the height, approaching 2 as the height approaches 0. Again I don't recall how calculate the dimension; however there is an upper limit to h (where the low pyramids intersect) and hence to the dimension. But when you've reached this point you can add 12 smaller low pyramids to the free faces of the first 4 low pyramids, and so on. Other contraints are introduced by this; the second and subsequent generations of low pyramids must not extend beyond the boundaries of the overall tetrahedron.

You don't have to add all 12, 36, 108, etc low pyramids at each stage, or to have a single height variable for all low pyramids of a single generation, though you may want to keep the figure symmetrical. It seems to me that you can bring the dimension arbitrarily close to 3, though it requires great numbers of transforms.

--Rane


--
Stewart Robert Hinsley
.