In the geometry of numbers, the Klein polyhedron, named after Felix Klein, is used to generalize the concept of simple continued fractions to higher dimensions.
Definition Let C {\displaystyle \textstyle C} be a closed simplicial cone in Euclidean space R n {\displaystyle \textstyle \mathbb {R} ^{n}} . The Klein polyhedron of C {\displaystyle \textstyle C} is the convex hull of the non-zero points of C ∩ Z n {\displaystyle \textstyle C\cap \mathbb {Z} ^{n}} .
Relation to continued fractions
Suppose α > 0 {\displaystyle \textstyle \alpha >0} is an irrational number. In R 2 {\displaystyle \textstyle \mathbb {R} ^{2}} , the cones generated by { ( 1 , α ) , ( 1 , 0 ) } {\displaystyle \textstyle \{(1,\alpha ),(1,0)\}} and by { ( 1 , α ) , ( 0 , 1 ) } {\displaystyle \textstyle \{(1,\alpha ),(0,1)\}} 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 Z 2 . {\displaystyle \textstyle \mathbb {Z} ^{2}.} Then the integer lengths of the edges of these two Klein polyhedra encode the continued-fraction expansion of α {\displaystyle \textstyle \alpha } , one matching the even terms and the other matching the odd terms.
Graphs associated with the Klein polyhedron Suppose C {\displaystyle \textstyle C} is generated by a basis ( a i ) {\displaystyle \textstyle (a_{i})} of R n {\displaystyle \textstyle \mathbb {R} ^{n}} (so that C = { ∑ i λ i a i : ( ∀ i ) λ i ≥ 0 } {\displaystyle \textstyle C=\{\sum _{i}\lambda _{i}a_{i}:(\forall i)\;\lambda _{i}\geq 0\}} ), and let ( w i ) {\displaystyle \textstyle (w_{i})} be the dual basis (so that C = { x : ( ∀ i ) ⟨ w i , x ⟩ ≥ 0 } {\displaystyle \textstyle C=\{x:(\forall i)\;\langle w_{i},x\rangle \geq 0\}} ). Write D ( x ) {\displaystyle \textstyle D(x)} for the line generated by the vector x {\displaystyle \textstyle x} , and H ( x ) {\displaystyle \textstyle H(x)} for the hyperplane orthogonal to x {\displaystyle \textstyle x} . Call the vector x ∈ R n {\displaystyle \textstyle x\in \mathbb {R} ^{n}} irrational if H ( x ) ∩ Q n = { 0 } {\displaystyle \textstyle H(x)\cap \mathbb {Q} ^{n}=\{0\}} ; and call the cone C {\displaystyle \textstyle C} irrational if all the vectors a i {\displaystyle \textstyle a_{i}} and w i {\displaystyle \textstyle w_{i}} are irrational. The boundary V {\displaystyle \textstyle V} of a Klein polyhedron is called a sail. Associated with the sail V {\displaystyle \textstyle V} of an irrational cone are two graphs:
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