The Process:
- List an original or initial integer sequence (space delimited as necessary and truncated at one’s discretion) along a single line.
- The difference, as an absolute value (intentional for this exercise), is then taken between the first and second integers of this initial sequence.
- That difference is recorded on the next successive line, positioned central to its minuend and subtrahend integer pair from the preceding line.
- The prior process is then carried out for the second and third integer pair of the initial sequence, and the same is repeated for all successive pairs until the initial sequence is exhausted.
- Next, the values of the aforementioned successive line are treated in the same manner as the original integer sequence which produces yet another new successive line of integer values.
- Step 5 is repeated until additional successive lines decay to a singular value. This, as the number of values in each successive line becomes one less than its predecessor.
- Finally, in regard to the completed table, the local region surrounding each individual value (perhaps a square shaped area) is color coded such that like values receive the same color. These now colored local regions effectively become contiguous “pixels” in the larger graphical representation.
The Result:
A graphical signature, one unique for the initial integer sequence, is thus mapped. The overall representation appears as an inverted triangle, one whose base is presented at the top of the page. Values within the depiction that degenerated to zero are viewed as voids or negative space and can be represented with a color matching that of the background. This, while values that retain natural numbers are viewed as positive spaces and are represented by a gradient of colors applied across the range of values. At close proximity, the representation is pixelated. Zoomed out, distinctive patterns emerge.
Unconnected but visually interesting nonetheless: While not fractal in nature, some integer sequences produce patterning reminiscent of the Sierpiński Sieve. As well, some produce patterning visually analogous to plots of Wolfram’s Rule 30.
In a digital form this signature approach could offer a visual tool, both human and machine readable, for the comparative analysis of different integer sequences. Perhaps patterning, as a result of the deconstructed elements found in particular sequences, could yield unknown points of commonality between those sequences thought unrelated.
Note:
The On-Line Encyclopedia of Integer Sequences® (OEIS), founded by Neil Sloane, offers a repository of nearly 400,000 unique sequences. While any initial integer sequence, including those generated randomly, can potentially be used in conjunction with this process, the OEIS represents a definitive source of mapping candidates.
Examples:
Below are truncated image examples of two famous integer sequences; the prime number sequence and the Fibonacci number sequence.
Many thanks to two colleagues and friends, both of extraordinary intellect, character, and perhaps most importantly, curiosity. John Sauvigné and Stephen Schermerhorn.
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