Steel towerhttp://img.alibaba....ion_Steel_Tower.jpg Has a Siepinski-like structure, but only 2 levels deep. [spidermother, Aug 25 2009]

Like this?http://en.wikipedia.../wiki/Menger_sponge The Menger Sponge - 3D (cubical) equivalent of the 2D Sierpinski Triangle (er, sort of...) [neutrinos_shadow, Aug 26 2009]

There is a three dimensional equivalent - a tetrahedron with an inverted 1/3 length tetrahedron removed from the middle, and so on. Not readily stackable though.

[edit] You must be referring to the Sierpinsky carpet. The 3D analog would be a cube with a square prism punched through each face, and the same done to each of the 20 remaining cubes, and so on. The faces would eventually be the same as a Sierpinsky carpet.

Some large steel towers have a structure that follows this principle to some extent (but using triangles, of course).

I thought that the point about a Sierpinski fractal was that,
as it was iterated more times, it filled a higher and higher
proportion of the available space? If so, then building a 3D
equivalent with the smallest elements of "nano" size (which
is what? nanometre?) would give you something denser than,
say, sandstone.

Not necessarily. Atomic spacing in nanostructures, and their corresponding densities, is malleable. Think aerogels. Getting them into this arrangement, however, is crystallographic magic and probably warrants an MFD. I wonder if some crystalline structures already do it to some extent.

//Not necessarily// Yes, necessarily. Your smallest
element is a single atom. The next largest will be a
cluster of 4 atoms in a tetrahedron, then a 32-atom
tetrahedron with twice the edge-lengths, etc. The
spacing of the atoms will be determined by the relevant
bond-lengths.

By building a 3D serpienski gasket (imagine starting with
the big tetrahedrons and working downward - it's easier to
imagine that way), you are starting with a very porous,
open structure (large tetrahedra corner-to-corner; 50%
empty). You then fill in 50% of each void with a half-sized
tetrahedron, leaving smaller voids, which you then half-fill
with smaller ones, etc etc. Eventually, you will have
created a fully-dense material.

You don't have to fill in any spacing and this is a ground-up paradigm, not the filling of a larger vessel that you describe. Besides, any material scientist will tell you that density is scale dependent. What's the density of an atom in the space of an atom? Very large. Carbon nanotubes are very dense if you only look at the volume of the tube, but when you have them loosely arranged in noncovalent networks they can take on a bulk density that is much much lower. I suspect you'll rebuke all this, though.

Consider that one wouldn't, and can't on physically terms, iterate the fractal pattern ad infinitum. This fact alone should convince you that lower densities are possible.

Yes yes, but you missed the point. The idea explicitly
states
that you "grow" bricks from nanosize (atom-size) up. So, if
the smallest element is atom-sized, my argument holds.
Just think about it a mo. The iteration is upwards from
atom size, not downward. So, it can be iterated til you
run out of stuff.

Also, any material scientist will not tell me that density is
scale dependent. Apart from probably hyphenating "scale-
dependent" (a moot point, granted), he/she would
probably
just say 'wha?'

Grow implies the structure has an intrinsic
knowledge of it's eventual form. An example
would be a snow flake. To make a block, say of snow
flake, the block
would have to have multiple starting centers. Each
center wouldn't have knowledge of how to
connect into another center. The total structure
would have to be planned so therefore 'built'
rather than grown.

Sounds like a hideously expensive way of manufacturing something with the approximate performance characteristics of aerated concrete besser blocks [-]

This isn't connected to that super simple reverse
osmosis idea is it? A sponge-like membrane acting like
solid salt so that only H20 can flow through. The big
spaces would be used for collection.

//It has a density of (20/27) ^N of the construction
material (where N is the number of iterations)// That
can't be right. That equation says that the material
density becomes lower with more iterations, whereas it
will in fact become higher.

[Ubie] The voids become smaller and more numerous as
you iterate (ie, if you start with a coarse pattern and add
smaller elements in the usual way), with the total area (or
volume) of void becoming less with each iteration. At
infinite iterations, there are an infinite number of
infintesimal voids, whose total area (or volume) is zero.
Of course, you can only go as far as atom-size elements
with a real material.

Of course [MaxwellBuchanan] is still working with
maths. In the real world, the atoms still have to get
to and be placed at, the right 3D positions to build
the form wanted.

//I'm still no clearer whether the addition of smaller blocks is
both outward and back inward into the central void/s. //
Ah, point taken. I'd assumed that it was both out and in.
But note also that the poster emphasizes starting with the
small elements.

//In the real world,// Yes, yes. That's why I said //you can
only go as far as atom-size elements with a real material.//

I don't think you would get them to the atom size
internally because to get the atoms to flow inside
would block up and deform the holes you are
trying to build. Placement, layer by layer, knowing
the positions before hand would build your 3D
wanted form but this is far from a crystal growth
and fractal iterations.

Then again, it might be possible to giggle a set of
atoms with a pattern that makes all the atoms sit
in the 3D space where you want them to bond. It
would be a pretty complex pattern of light,
though.

The voids created in each iteration remain, and further (smaller) voids are created in the remaining solid. The volume does indeed approach zero (while the surface area approaches infinity).

A real world example would obviously at best approximate the mathematical model for some finite number of iterations, and would have a small but non-zero density.

//The voids created in each iteration remain, and further
(smaller) voids are created in the remaining solid.// No,
the voids don't remain. Imagine we start with Very Big
tetrahedra, and iterate through successively smaller ones.

First we take three tetrahedra and lay them on the floor,
corner to corner. Suppose these first ones are of size 1.
Then we balance a fourth one on top of them, resting on
their points. You now have a tetrahedron of size 2, with a
hole of size 1 in the middle.

Now we take a tetrahedron of size 1/2* and we put it into
the hole. It leaves four new voids around it, but some of
the original void has been filled in, so the total amount of
void decreases.

We then take four tetrahedra of size 1/4*, and put them
into the new smaller voids. Each newly added tetrahedron
divides a void into four smaller voids, but also occupies
part of the original void.

Etsoforth.

As you fill in the voids with smaller and smaller tetrahedra,
the total number of voids increases, but the total volume
of voids decreases.

[*it's not actually 1/2, 1/4 etc, more like 2/3, 4/9 or
something; but the prinicipal is the same.]

[MaxwellBuchanan]; I think we're all talking at cross-purposes.
1: The idea - 'growing' a fractal solid with small particles. Volume increases faster than added mass (due to voids); density decreases.
2: Menger sponge (my fault...) - fixed volume, solid modified by removed mass; density decreases.
3: Your example - fixed volume, but with initial void, solid modified by adding mass into void; density increases.
All three are different; it took me a bit to get my head around why.
As to the idea itself, it would only be as strong as the material it was made of at a molecular/cellular level - ie: use carbon buckytubes to build a truss-like 'base unit', grow from there. Pretty much anything else will be too weak at the scale implied.

[neutinos shadow] Great thinking through, thanks.
1,2 and 3 should be the same given a set volume,
scale and subunit size, whether you approach the
construct from removing voids, adding varying sized
groups or adding the tiniest subunits.