The Smooth and the Striated

Sierpensky’s sponge: more than a surface, less than a volume. The law according to which this cube was hollowed can be understood intuitively at a glance. Each square hole is surrounded by eight holes a third its size. And so on, endlessly. The illustrator could not represent the infinity of holes of decreasing size beyond the fourth degree, but it is plain to see that this cube is in the end of infinitely hollow. Its total volume approaches zero, while the total lateral surface of the hollowings infinitely grows. This space has dimension of 2.7268. It therefore lies between a surface (with a dimension of 2) and a volume (with a dimension of 3). ‘Sierpensky’s rug’ is one face of the cube; the hollowings are then squares and the dimension of the ‘surface’ is 1.2618.

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