# Dodecaedro de caras reflectantes y ángulo rectos # # Un dodecaedro en el que todos sus ángulos son rectos, # se encuentra en un espacio hiperbólico 3‑dimensional. # Las siguientes matrices, que presentan # reflexiones en cada una de sus doce caras, # generan un espacio hiperbólico 3‑dimensional. # # Los elementos de las matrices están en el campo Q[ξ], # donde ξ es una raíz del polinomio de cuarto grado # ξ⁴ - 2ξ³ - 12ξ² - 12ξ - 4. # En términos de la razón áurea φ ≈ 1.61803..., # ξ = φ + √(3 + 5φ) ≈ 4.94822... # Los elementos específicos de las matrices son: # # 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 -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 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 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 0.00000000000000000000 0.00000000000000000000 1.61803398874989484820 1.27201964951406896425 0.00000000000000000000 1.00000000000000000000 0.00000000000000000000 0.00000000000000000000 1.61803398874989484820 0.00000000000000000000 -1.61803398874989484820 -2.05817102727149225032 -1.27201964951406896425 0.00000000000000000000 2.05817102727149225032 2.61803398874989484820 0.00000000000000000000 0.00000000000000000000 -1.61803398874989484820 -1.27201964951406896425 0.00000000000000000000 1.00000000000000000000 0.00000000000000000000 0.00000000000000000000 -1.61803398874989484820 0.00000000000000000000 -1.61803398874989484820 -2.05817102727149225032 1.27201964951406896425 0.00000000000000000000 2.05817102727149225032 2.61803398874989484820 0.00000000000000000000 0.00000000000000000000 -1.61803398874989484820 1.27201964951406896425 0.00000000000000000000 1.00000000000000000000 0.00000000000000000000 0.00000000000000000000 -1.61803398874989484820 0.00000000000000000000 -1.61803398874989484820 2.05817102727149225032 -1.27201964951406896425 0.00000000000000000000 -2.05817102727149225032 2.61803398874989484820 0.00000000000000000000 0.00000000000000000000 1.61803398874989484820 -1.27201964951406896425 0.00000000000000000000 1.00000000000000000000 0.00000000000000000000 0.00000000000000000000 1.61803398874989484820 0.00000000000000000000 -1.61803398874989484820 2.05817102727149225032 1.27201964951406896425 0.00000000000000000000 -2.05817102727149225032 2.61803398874989484820 1.00000000000000000000 0.00000000000000000000 0.00000000000000000000 0.00000000000000000000 0.00000000000000000000 -1.61803398874989484820 1.61803398874989484820 -2.05817102727149225032 0.00000000000000000000 1.61803398874989484820 0.00000000000000000000 1.27201964951406896425 0.00000000000000000000 2.05817102727149225032 -1.27201964951406896425 2.61803398874989484820 1.00000000000000000000 0.00000000000000000000 0.00000000000000000000 0.00000000000000000000 0.00000000000000000000 -1.61803398874989484820 -1.61803398874989484820 -2.05817102727149225032 0.00000000000000000000 -1.61803398874989484820 0.00000000000000000000 -1.27201964951406896425 0.00000000000000000000 2.05817102727149225032 1.27201964951406896425 2.61803398874989484820 1.00000000000000000000 0.00000000000000000000 0.00000000000000000000 0.00000000000000000000 0.00000000000000000000 -1.61803398874989484820 -1.61803398874989484820 2.05817102727149225032 0.00000000000000000000 -1.61803398874989484820 0.00000000000000000000 1.27201964951406896425 0.00000000000000000000 -2.05817102727149225032 -1.27201964951406896425 2.61803398874989484820 1.00000000000000000000 0.00000000000000000000 0.00000000000000000000 0.00000000000000000000 0.00000000000000000000 -1.61803398874989484820 1.61803398874989484820 2.05817102727149225032 0.00000000000000000000 1.61803398874989484820 0.00000000000000000000 -1.27201964951406896425 0.00000000000000000000 -2.05817102727149225032 1.27201964951406896425 2.61803398874989484820