fractal gasket (idea)

The one-dimensional gasket is called the Cantor dust or the Cantor ternary set (it's noded under the second name).

The two-dimensional gasket is called the Sierpinski triangle if triangular, or the Sierpinski carpet if square, and either of them can be called the Sierpinski gasket, because they look like mechanical gaskets whether they're triangular or square, and that's why I've chosen this as the general name.

The three-dimensional analogue is called the Menger sponge, though it is equally possible to have a triangular one, which would then be called a Menger pyramid, I suppose. I don't know of a general term for higher-dimensional ones, but I've seen hypergasket, which seems as good as any.

The Cantor dust is formed by taking a line segment, removing the middle third, leaving two separate parts, then removing the middle third of each of those, leaving four separate parts. Clearly as this process iterates to infinity, the individual parts accumulate in number exponentially and their individual lengths vanish exponentially to zero. The final result is a set of measure zero having uncountably many points, none connected to any other.

The Sierpinski triangle is formed by taking a triangle, removing the middle triangle of four leaving three touching at their vertices, and so on down. Clearly, as we iterate, every remaining solid region gets its centre punched out, and the end result is a path connected gossamer web of zero area.

The Sierpinski carpet follows the same procedure, removing the middle one-third-size square from a square, leaving the solid shape formed from the remaining eight squares. Each of these eight then gets its middle third (or more accurately, middle ninth) removed. The end result at infinity is, like the Sierpinski triangle, path-connected and of zero area. But there's a difference. We'll get to that in a moment.

The Menger sponge is the three-dimensional equivalent: you take a cube, remove the cube at the middle with one third the side therefore occupying one twenty-seventh of the volume, and so on, to get a path-connected tracery with no volume.

Now for the difference: that's not how it's usually constructed. The pictures you see of fractals have it made by removing not just the 1/27 central cube, but those six cubes whose faces touch it, so that it's hollowed out from one end to the other whenever you look at any face. Why?

There's only one way of taking the central sections out of the minimal shapes, that's the Cantor dust, the Sierpinski triangle, and the Menger pyramid. But if you start with a square there are two ways you can apply the iterative gapping: take just the central one square of the nine, preserving the connectedness of the remainder, or remove a cross consisting of five of the nine smaller squares, leaving four disconnected corner squares. Either way, you can repeat the process ad infinitum, creating one of two objects, both of which have the original dust-like property of being of measure zero. But the more severe operation of taking out a cross will eventually create a true dust, that is something that is nowhere connected.

Now consider a cube, consisting of twenty-seven smaller cubes. At the next iteration you can remove (a) just the central one cube, or (b) the central cross of seven cubes, leaving a single connected three-dimensional hull, or (c) the three mutually orthogonal central slices, a total of nineteen smaller cubes, leaving the eight disjoint corner cubes. Clearly all three of these constructions iterate to produce a ghostly object of measure zero, but only the third reduces it to a disconnected dust, and the other two are different from each other.

A little secret. Strictly speaking, I just invented the term "fractal gasket". There doesn't seem to be a general term for these things already in existence, or perhaps I just haven't seen it. I was going to put this under gasket but thought that wouldn't get as much exposure, and fractal gasket is a better description.