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With hexagonal cells, however, the exact solution of the so-called hard hexagon lattice gas model in 1980 showed that h ≃ 0.481 is the logarithm of the largest root of a degree 12 polynomial.
They look in many ways similar to ordinary random walks, but their limiting distribution is no longer strictly Gaussian, and their root mean square displacement after t steps varies like t 3/4 .
(In the case of rule 225, the width of the overall pattern does not grow at a fixed rate, but instead is on average proportional to the square root of the number of steps.)
Linkages consisting of rods of integer lengths always trace out algebraic curves (or algebraic surfaces in 3D) and in general allow any algebraic number (as represented by Root ) to be constructed.
In particular, it is usually assumed that performing some standard mathematical operation, such as taking a square root, cannot have a significant effect on the system one is studying.
(In 8D the lattice also corresponds to the root vectors of the Lie group E 8 ; in 24D it is the Leech lattice derived from a Golay code, and related to the Monster Group).
The most common are ones based on repetition or iteration, classic examples being Euclid's algorithm for GCD (page 915 ), Newton's method for FindRoot and the Gaussian elimination method for LinearSolve .
In Mathematica, however, functions like Root provide more convenient ways to access such results.
Most of the 2 n possible states have unique predecessors; for large n , about 2 0.76 n or Root[# 3 - # 2 - 2 &, 1] n instead have 0 or 2 predecessors.
Any iterative procedure (such as FindRoot ) that yields a constant multiple more digits at each step will take about Log[n] steps to get n digits.
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