Borromean Rings
Three rings linked so that removing any one frees the other two. No two are linked pairwise — only all three together. The linking is an emergent property of the triple, not reducible to any pair. This impossible-looking object exists in the topology of 3-space and shows up in molecular chemistry, particle physics, and the foundation of knot theory.
three is special
The Borromean rings are the simplest nontrivial example of a Brunnian link: a link of n ≥ 3 components in which every (n−1)-component sublink is trivial, but the whole is not. Cut any one ring and the other two fall apart unlinked.
Surprising fact: the Borromean rings cannot be drawn with actual round circles. Freedman & Skora (1987) proved no embedding of three round flat circles in ℝ³ has the Borromean property — you need elliptical or otherwise non-circular components. The famous "interlocked circle" image is always a topological approximation.
History: named for the Borromeo family of Renaissance Italy who used the symbol on their coat of arms (~1450). Found earlier in Norse Odin's three triangles ("valknut"), Buddhist three-jewel symbols, and Japanese mon. Mathematicians from Tait (1876) onward have used it as the canonical example of linkage that isn't pairwise.
In chemistry: J. Fraser Stoddart (Nobel 2016) synthesized Borromean molecules — three interlocked metal-ring complexes. The DNA Borromean ring was built by Seeman in 1997. The catenated 3-ring topology is now a recognized class of mechanical bonds.
In physics: Borromean nuclei are 3-body bound states (like ⁶He = ⁴He + n + n) where no 2-body subsystem is bound. The Efimov effect (Nobel-level prediction confirmed 2006) shows quantum Borromean states exist at unbounded sizes — an infinite tower of three-body bound states.
Knot theory: the Borromean rings have linking number 0 for every pair but a nonzero triple Massey product. This is the original example showing that links can have higher-order topology not captured by pairwise invariants.