Slice the Sierpinski tetrahedron at height c. The cross-section depends on what kind of number c is.
If c is a dyadic rational — a fraction whose denominator is a power of 2, like 3/8 or 7/16 — the slice has finitely many connected components, infinite first Cech homology, and trivial higher homology. The cross-section is a complex, richly connected object with infinite topological structure.
If c is anything else — irrational, or rational with an odd factor in the denominator — the slice is totally disconnected. Dust. Every positive-degree homology group vanishes. The cross-section has no topological structure at all.
Nakajima and Watanabe proved this dichotomy by analyzing the Cech cohomology groups at each height level of the Sierpinski tetrahedron. The transition is not gradual. There is no intermediate regime where the slice is “a little connected” or “mostly dust.” The topology switches cleanly between infinite complexity and total fragmentation, and the switch depends entirely on the number-theoretic properties of where you cut.
The mechanism is the self-similarity of the tetrahedron. The Sierpinski tetrahedron is built by iterative subdivision into halves — powers of 2 organize its structure at every scale. Heights that align with this dyadic structure intersect the fractal at points where the iterative construction produces genuine connections between components. Heights that do not align miss these connections entirely. The fractal is the same object at every height. What changes is whether the cutting plane's position resonates with the construction's arithmetic.
The structural principle: the topology of a cross-section encodes the relationship between the observer's position and the object's generative rule. Two observers looking at the same fractal from heights differing by an arbitrarily small amount — one at a dyadic rational, the other at an irrational — see categorically different objects. One sees infinite structure. The other sees nothing. The difference is not in the fractal. It is in the alignment between the cut and the construction. What you find depends on whether your vantage point is commensurate with the process that built the thing you are examining.