N-Grams by Divisor 96ad

This arrangement of the N-Grams, where division by a given integer is represented in one row of the diagram, brings out how each pattern in the row repeats regularly. The length of the sequence of distinct patterns equals the divisor. The first pattern, where the divisor is the same as the number base, yields only the top-point pattern that is in every N-Gram [ N/N = 1.000…=0.999… ]. This is the “null” pattern, with minimal repetitive activity. This patter recurs whenever all the prime factors of the divisor are also prime factors in the number base. Examples are division by 4, 8, or 16 in any even-numbered base, or division by 9 or 27 in any base divisible by 3. When the number base is one more than the divisor, you get the same simple pattern for every divisor – a repetition of a single digit. For instance, in base ten, 1/9 = 0.111…, 2/9 = 0.222…, and so on. The set of N distinct patterns formed by division by N repeats starting at 2N, at 3N, and so on. For instance with division by 5, the patterns for[ 6, 11, 16, …] are basically the same, as are the patterns for [7, 12, 17, …] and [8, 13, 18, …] and [9, 14, 19, …].

To make it easier to see what changes with an increasing number base, I’ve constructed two ways to show division by 7 in bases 10, 17, 24, and 31. [The animated GIF is more nimble, the matched web pages is slow to change images.] Each diagram in the sequence shows division by 7 as a pattern in red. The complementary pattern (that touches all the other points around the circle) is represented by 3 black dots in base 10, and by blue lines in the other bases. The figure in base 31 and in base 10 (last and first figures in the sequence) have in common 3 anchor points which are in precisely the same points in the two diagrams.

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