# Lieb's square ice constant

 Binary 1.10001010001000110100010111001100… Decimal 1.53960071783900203869106341467188… Hexadecimal 1.8A2345CC04425BC2CBF57DB94EDCA6B2… Continued fraction $1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{5+{\cfrac {1}{1+{\cfrac {1}{4+\ddots }}}}}}}}}}$ Algebraic form ${\frac {8{\sqrt {3}}}{9}}$ Lieb's square ice constant is a mathematical constant used in the field of combinatorics to quantify the number of Eulerian orientations of grid graphs. It was introduced by Elliott H. Lieb in 1967.

## Definition

An n × n grid graph (with periodic boundary conditions and n ≥ 2) has n2 vertices and 2n2 edges; it is 4-regular, meaning that each vertex has exactly four neighbors. An orientation of this graph is an assignment of a direction to each edge; it is an Eulerian orientation if it gives each vertex exactly two incoming edges and exactly two outgoing edges.

Denote the number of Eulerian orientations of this graph by f(n). Then

$\lim _{n\to \infty }{\sqrt[{n^{2}}]{f(n)}}=\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8{\sqrt {3}}}{9}}=1.5396007\dots$ is Lieb's square ice constant. Lieb used a transfer-matrix method to compute this exactly.

The function f(n) also counts the number of 3-colorings of grid graphs, the number of nowhere-zero 3-flows in 4-regular graphs, and the number of local flat foldings of the Miura fold. Some historical and physical background can be found in the article Ice-type model.