Hyperbolic Space Language

Imaginary Languqage

Hyperbolic Space Language
TANAKA Akio
September 5, 2011

[Preparation]

3 dimensional hyperbolic space H³

Spherical surface at infinity S_x

Ideal simplex that has vertexes on S_x Σ

Internal angles of triangle that is projected to plane z = 0

Internal angles of ideal simplex α, β,γ

Internal angles except α, β,γ α₁, β₁,γ₁

Volume of ideal simplex V

<Theorem>
V (Σ) = Λ(α) +Λ(β)+Λ(γ)

[Interpretation]
Language model is expressed by V (Σ).

Word is expressed by Σ.