Is the Feigenbaum constant transcendental?

Open problem. The Feigenbaum Constant δ ≈ 4.6692… is not known to be irrational, and a fortiori not known to be transcendental — despite being one of the most precisely computed constants in nonlinear dynamics.

What is known

δ is well-defined: it arises as an eigenvalue of the linearized renormalization operator for unimodal maps, and Lanford's computer-assisted proof (1982) established that the fixed point of that operator exists, making δ a bona fide mathematical constant rather than a numerical artifact of the Logistic Map. Thousands of digits have been computed, and empirically the digits pass the usual randomness tests. No closed form in familiar constants (\pi, e, algebraic numbers) has ever been found, and numeric searches for low-height polynomial relations have come up empty.

Why it is hard

Irrationality proofs typically lean on arithmetic structure — a series with controllable denominators, a continued fraction with a pattern, a functional equation. δ has none of these handles: it is defined spectrally, through a fixed point known only via rigorous numerics. Nothing currently connects that definition to the arithmetic of the number it produces.

What an answer would change

A proof either way would be the first arithmetic statement about a universality constant — a bridge between renormalization theory and transcendence theory, two fields that at present do not touch. Even a proof of irrationality would likely require a genuinely new technique for extracting arithmetic information from dynamically defined numbers.

See also

• Feigenbaum Constant — the constant itself, with a verified computation.
• Logistic Map — where the period-doubling cascade lives.