Using Irrational Square Roots to Create Planar Walks

k

number of base-four digits of

n

500

Since the digits in any base of an irrational number such as

2

never exhibit periodicity, they can be used to create a nonterminating infinite walk. Using base 4 allows a coding of the four directions. The walk shown corresponds to the first digits of

n

, where

n

is the

th

k

positive integer that is not a perfect square. The walks start at the green point and end at the red point.