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 $\textstyle C$ be a closed simplicial cone in Euclidean space $\textstyle \mathbb^n$. The ''Klein polyhedron'' of $\textstyle C$ is the convex hull of the non-zero points of $\textstyle C \cap \mathbb^n$.

Relation to continued fractions

Suppose $\textstyle \alpha > 0$ is an irrational number. In $\textstyle \mathbb^2$, the cones generated by $\textstyle \$ and by $\textstyle \$ 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 $\textstyle \mathbb^n.$ Then the integer lengths of the edges of these two Klein polyhedra encode the continued-fraction expansion of $\textstyle \alpha$, one matching the even terms and the other matching the odd terms.

Graphs associated with the Klein polyhedron

Suppose $\textstyle C$ is generated by a basis $\textstyle (a_i)$ of $\textstyle \mathbb^n$ (so that $\textstyle C = \$), and let $\textstyle (w_i)$ be the dual basis (so that $\textstyle C = \$). Write $\textstyle D(x)$ for the line generated by the vector $\textstyle x$, and $\textstyle H(x)$ for the hyperplane orthogonal to $\textstyle x$.

Call the vector $\textstyle x \in \mathbb^n$ ''irrational'' if $\textstyle H(x) \cap \mathbb^n = \$; and call the cone $\textstyle C$ irrational if all the vectors $\textstyle a_i$ and $\textstyle w_i$ are irrational.

The boundary $\textstyle V$ of a Klein polyhedron is called a ''sail''. Associated with the sail $\textstyle V$ of an irrational cone are two graphs:
* the graph $\textstyle \Gamma_(V)$ whose vertices are vertices of $\textstyle V$, two vertices being joined if they are endpoints of a (one-dimensional) edge of $\textstyle V$;
* the graph $\textstyle \Gamma_(V)$ whose vertices are $\textstyle (n-1)$-dimensional faces (''chambers'') of $\textstyle V$, two chambers being joined if they share an $\textstyle (n-2)$-dimensional face.

Both of these graphs are structurally related to the directed graph $\textstyle \Upsilon_n$ whose set of vertices is $\textstyle \mathrm_n(\mathbb)$, where vertex $\textstyle A$ is joined to vertex $\textstyle B$ if and only if $\textstyle A^B$ is of the form $\textstyle UW$ where

: $U = \left( \begin 1 & \cdots & 0 & c_1 \\ \vdots & \ddots & \vdots & \vdots \\ 0 & \cdots & 1 & c_ \\ 0 & \cdots & 0 & c_n \end \right)$

(with $\textstyle c_i \in \mathbb$, $\textstyle c_n \neq 0$) and $\textstyle W$ is a permutation matrix. Assuming that $\textstyle V$ has been triangulated, the vertices of each of the graphs $\textstyle \Gamma_(V)$ and $\textstyle \Gamma_(V)$ can be described in terms of the graph $\textstyle \Upsilon_n$:
* Given any path $\textstyle (x_0, x_1, \ldots)$ in $\textstyle \Gamma_(V)$, one can find a path $\textstyle (A_0, A_1, \ldots)$ in $\textstyle \Upsilon_n$ such that $\textstyle x_k = A_k (e)$, where $\textstyle e$ is the vector $\textstyle (1, \ldots, 1) \in \mathbb^n$.
* Given any path $\textstyle (\sigma_0, \sigma_1, \ldots)$ in $\textstyle \Gamma_(V)$, one can find a path $\textstyle (A_0, A_1, \ldots)$ in $\textstyle \Upsilon_n$ such that $\textstyle \sigma_k = A_k (\Delta)$, where $\textstyle \Delta$ is the $\textstyle (n-1)$-dimensional standard simplex in $\textstyle \mathbb^n$.

Generalization of Lagrange's theorem

Lagrange

HOME


Content is Copyleft
Website design, code, and AI is Copyrighted (c) 2014-2017 by Stephen Payne


Consider donating to Wikimedia



As an Amazon Associate I earn from qualifying purchases