# Mirrored Right-Angled Dodecahedron # # A dodecahedron with all right angles lives in 3-dimensional # hyperbolic space. The following matrices, representing # reflections across each of its twelve faces, generate # a tiling of hyperbolic 3-space. # # The matrix entries live in the field Q[ξ], where # ξ is a root of the irreducible quartic polynomial # ξ⁴ - 2ξ³ - 12ξ² - 12ξ - 4. # In terms of the golden ratio φ ≈ 1.61803..., # ξ = φ + √(3 + 5φ) ≈ 4.94822... # The specific matrix entries are # # φ = -(1/4)ξ³ + ξ² + (3/2)ξ ≈ 1.61803... # φ + 1 = -(1/4)ξ³ + ξ² + (3/2)ξ + 1 ≈ 2.61803... # √( φ² - 1) = -(1/4)ξ³ + (1/2)ξ² + (7/2)ξ + 2 ≈ 1.27201... # √(2φ² - 1) = (1/2)ξ³ - (3/2)ξ² - 4 ξ - 2 ≈ 2.05817... -1.61803398874989484820 1.61803398874989484820 0.00000000000000000000 -2.05817102727149225032 1.61803398874989484820 0.00000000000000000000 0.00000000000000000000 1.27201964951406896425 0.00000000000000000000 0.00000000000000000000 1.00000000000000000000 0.00000000000000000000 2.05817102727149225032 -1.27201964951406896425 0.00000000000000000000 2.61803398874989484820 -1.61803398874989484820 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