An Alternate Way of Defining the Code

It is a curious fact that one can equally easily define the Fibonacci code by calling the wide element the "1", and the narrow element the "0". In fact, this is the way the code was first developed, because it seemed more natural to make the wider element (and hence, in a sense, the bigger element) have greater value. The mathematics works out in a similar fashion.

However, once it was realized that the code could be defined either way, it seemed better to define the code such that the "1" is represented by the narrow element. This removes all ambiguity about the weight attached to the "1", because the "1" occupies only a single cell. If the alternative code is used, in which the "1" is a wide element, then the weight attached to the "1" must be defined to be the sum of both cells occupied by the "1", assuming the cell weights are defined as we have done (with F0 the weight of the left-most cell). Alternatively, because of the sum-of-the-previous-two property of the Fibonacci numbers, one can shift the cell weights left one position (hence starting with F1), and just add in the weight of the rightmost cell containing each "1", or one can shift the weights left two positions (hence starting with F2), and just add in the weight of the leftmost cell containing each "1". With the "1" defined as the narrow element, there is only one obvious way to weight the cells, making that seem the preferable definition of the code, and that is the version described here.

On the other hand, the code in which the "1" element is wide does not have the property of having differing codes for the same value, depending upon whether the code is an even or an odd number of cells in overall length. This is, in essense, because any number of "0" elements can always be added to the most-significant end of the code when the "0" is narrow. In the code as we have described it, with wide "0" elements, adding a most-significant "0" when the code is enlarged by one cell also uses up the neighboring cell, and thus changes the entire code in the less-significant cells.

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