Lightweight building material seen in Devs

Quantum physics slash Silicon Valley drama Devs features an interesting architectural idea, namely a floating laboratory.  In order make a laboratory float you need very lightweight construction materials and it seems that they've solved this problem by using Menger Sponge bricks, which are very lightweight indeed.

A Menger Sponge is a 3D version of this

To make a Menger Sponge you need to perform an iteration.  Start with any shape, then make 8 copies each scaled down by a factor of 3, and use them to construct the perimeter of a square.  Then take that shape and do the same operation, and keep repeating.  The picture above shows that you end up with the same thing whether you start with a solid square or a hollow square.  But actually it doesn't matter what shape you start off with (as long as it's bounded).

So, what's the area $A$ of the 2D version Menger Sponge shown above?  We can see from the first row in the picture that it's made up of 8 copies of itself, each of which is a 1/3 scaled down version, so

$$
A = 8\times\frac{A}{3^2}
$$

There's only one solution to this equation and that's $A=0$.  Perhaps it's a one dimensional object then.  So, let's work out what it's length is.   If we start off with two adjacent sides of a hollow square then at each stage the resulting object will have zero length overlapping so we can work out the length $L$ of the final result in a similar way to how we worked out the area

$$
L = 8\times\frac{L}{3^1}
$$

The only difference in this equation is that $A$ has been replaced by $L$ and the denominator is not squared.  So, maybe $L = 0$ too.  In fact there's another solution which maths purists would probably object to, but kind of makes sense in this context, and that's $L = \infty$.

So the 2D Menger Sponge has zero area and infinite length.  A reasonable interpretation of this is that it has a dimension between 1 and 2, a fractal dimension.  And it's pretty obvious from the calculations above that to find it we need to look for a $d$ solving

$$
1 = 8\times\frac{1}{3^d}
$$

Which is solved by $d = ln(8)/ln(3) \approx 1.89$.  So this object is $1.89$ dimensional.  The 3D version of Menger's Sponge is $ln(20)/ln(3) \approx 2.73$ dimensional.




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