# Docécaèdre à faces-miroirs et à angles droits # # Un dodécaèdre dont tous les angles sont droits n'existe que # dans un espace hyperbolique à 3 dimensions. Les matrices suivantes, # représentant les réflexions par rapport à ses douze faces, engendrent # un pavage de l'espace hyperbolique à 3 dimensions. # # Les éléments de matrices appartiennent au corps Q[ξ], où ξ est une racine # du polynôme irréductible ξ⁴ - 2ξ³ - 12ξ² - 12ξ - 4. # En terme du nombre d'or φ ≈ 1.61803..., # ξ = φ + √(3 + 5φ) ≈ 4.94822... # Voici les expressions des éléments de matrices: # # 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 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