How the smallest hyperbolic universe hides an arithmetic structure inside its shape equations
There is a polynomial that governs the shape of the smallest possible hyperbolic universe — the Meyerhoff manifold. It has degree 8, integer coefficients, and has been attached to this manifold in the SnapPy database for years.
This week I proved what it actually is. It is not a mysterious degree-8 object. It is a norm — the product of four Galois conjugates of a simple quadratic polynomial defined over the trace field of the manifold.
Let K = ℚ(w), w⁴ = w+1, and define
Then
Verified by exact symbolic computation in SageMath. Zero floating-point.
The norm construction takes a quadratic over a degree-4 field and multiplies all four Galois conjugates together. The result has degree 4×2 = 8 and lands in ℤ[y].
The four conjugates correspond to the four embeddings σᵢ : K ↪ L̄, where L is the degree-24 splitting field of x⁴−x−1. Over L, the polynomial p₈ splits completely into these four quadratic factors.
Inside K, three elements — w³, −w, and w⁻⁴ — form a period-3 orbit under the Möbius transformation T(z) = 1/(1−z). All three are pure powers of the fundamental unit u₁ = 1−w³.
All three orbit elements are pure powers of the fundamental unit u₁ = 1−w³. Exponent sum = 0.
At the geometric complex embedding of K, the Bloch–Wigner function evaluated at w³ equals minus the volume of the Meyerhoff manifold — to every digit we can compute.
Conjecture: this is exact — [w³] generates B(K)⊗ℚ and the Borel regulator gives r([w³]) = −vol(M) precisely.
The full picture connects six levels, from the trace field arithmetic down to the hyperbolic volume.
Click each step to reveal the chain of reasoning.