The binary representation of the square root of 2, perhaps depicted as black and white squares in a row, forms a one-dimensional half-infinite random pattern, with randomness definable in some proven or strongly conjectured way, for example, normality.
Is there some equally elegant way to generate a quarter-infinite two-dimensional pattern? Elegant is of course a highly subjective term. I don't consider laying out a one-dimensional pattern in two dimensions, for example along diagonals, to be elegant.
A fully infinite (in both directions) one dimensional pattern? Maybe p-adic numbers to the left of the decimal point. However, is there a way to do it without any position being the "origin" or decimal point?
Three dimensions?
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