# 鏡映直角正12面体 # # 全ての面の内角が直角であるような、正12面体は、3次元双曲型空間内に存在する。 # 次の12の行列は、それぞれ直角正12面体の、12の面を鏡映面とする鏡映変換であって、 # 3次元双曲型空間を、隙間無く直角正12面体で埋め尽くす、タイル張りを生成する。 # # 行列要素は、全て、体 Q[ξ] の元である。ここに、ξ とは、4次の既約方程式 # ξ⁴ - 2ξ³ - 12ξ² - 12ξ - 4 のひとつの根であって、黄金比 φ ≈ 1.61803... によって、 # ξ = φ + √(3 + 5φ) ≈ 4.94822... という関係で与えられている。 # 実際に、行列要素として現れる数は、次の4つである: # # 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 -1.61803398874989484820 0.00000000000000000000 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