Klein polyhedron
In the geometry of numbers, the Klein polyhedron, named after Felix Klein, is used to generalize the concept of continued fractions to higher dimensions. Definition. Let formula_1 be a closed simplicial cone in Euclidean space formula_2. The "Klein polyhedron" of formula_1 is the convex hull of the non-zero points of formula_4. Relation to continued fractions. Suppose formula_5 is an irrational number. In formula_6, the cones generated by formula_7 and by formula_8 give rise to two Klein polyhedra, each of which is bounded by a sequence of adjoining line segments. Define the "integer length" of a line segment  to be one less than the size of its intersection with formula_9 Then the integer lengths of the edges of these two Klein polyhedra encode the continued-fraction expansion of formula_10, one matching the even terms and the other matching the odd terms. Graphs associated with the Klein polyhedron. Suppose formula_1 is generated by a basis formula_12 of formula_2 (so that formula_14), and let formula_15 be the dual basis (so that formula_16). Write formula_17 for the line generated by the vector formula_18, and formula_19 for the hyperplane orthogonal to formula_18. Call the vector formula_21 "irrational" if formula_22; and call the  cone formula_1 irrational if all the vectors formula_24 and formula_25 are irrational. The boundary formula_26 of a Klein polyhedron is called a "sail". Associated with the sail formula_26 of an irrational cone are two graphs: Both of these graphs are structurally related to the directed graph formula_35 whose set of vertices is formula_36, where vertex formula_37 is joined to vertex formula_38 if and only if formula_39 is of the form formula_40 where (with formula_42, formula_43) and formula_44 is a permutation matrix. Assuming that formula_26 has been triangulated, the vertices of each of the graphs formula_28 and formula_31 can be described  in terms of the graph formula_35: Generalization of Lagrange's theorem. Lagrange proved that for an irrational real number formula_10, the continued-fraction expansion of formula_10 is periodic if and only if formula_10 is a quadratic irrational. Klein polyhedra allow us to generalize this result. Let formula_67 be a totally real algebraic number field of degree formula_68, and let formula_69 be the formula_68 real embeddings of formula_71. The simplicial cone formula_1 is said to be "split" over formula_71 if formula_74 where formula_75 is a basis for formula_71 over formula_77. Given a path formula_51 in formula_35, let formula_80. The path is  called "periodic", with period formula_81, if formula_82 for all formula_83. The "period matrix" of such a path is defined to be formula_84. A path in formula_28 or formula_31 associated with such a path is also said to be periodic, with the same period matrix. The generalized Lagrange theorem states that for an irrational simplicial cone formula_87, with generators formula_12 and formula_15 as above and with sail formula_26, the following three conditions are equivalent: Example. Take formula_103 and formula_104. Then the simplicial cone formula_105 is split over formula_71. The vertices of the sail are the points formula_107 corresponding to the even  convergents formula_108 of the continued fraction for formula_109. The path of vertices formula_110 in the positive quadrant starting at formula_111 and proceeding in a positive direction is formula_112. Let formula_101 be the line segment joining formula_96 to formula_115. Write formula_116 and formula_117 for the reflections of formula_96 and formula_101 in the formula_18-axis. Let formula_121, so that formula_122, and let formula_123. Let formula_124, formula_125, formula_126, and formula_127. Generalization of approximability. A real number formula_5 is called "badly approximable" if formula_143 is bounded away from zero. An irrational number is badly approximable if and only if the partial quotients of its  continued fraction are bounded. This fact admits of a generalization in terms of Klein polyhedra. Given a simplicial cone formula_16 in formula_2, where formula_146, define the "norm minimum" of formula_1 as formula_148. Given vectors formula_149, let formula_150. This is the Euclidean volume of formula_151. Let formula_26 be the sail of an irrational simplicial cone formula_1. Then formula_165 if and only if formula_166 and formula_167 are both bounded. The quantities formula_168 and formula_169 are called "determinants". In two dimensions, with the cone generated by formula_170, they are just the partial quotients of the continued fraction of formula_10. Klein polyhedron In the