Visualizing gnomon series for the figurate numbers modulo 10

In modular arithmetic, the integers modulo k form a closed figure -- a polygon with k vertices -- rather than a line. The decagon below represents the integers modulo 10. To count, start at +1 and follow the black lines clockwise. The numbers on the left side of the figure are negative because n ≡ n - 10 (mod 10); thus, 5 ≡ -5, 6 ≡ -4, 7 ≡ -3, and so on.

The number that may be added to the nth figurate number to yield the (n + 1)th is called a gnomon. To generate the series of triangular numbers, you start with 0, then add 1, then add 2, then add 3, and so on through the natural numbers. In other words, the gnomon series for triangular numbers is (1, 2, 3, 4, 5, 6, 7, ...) -- which is congruent (mod 10) to (+1, +2, +3, +4, ±5, -4, -3, -2, -1, 0) endlessly repeated. To get the gnomon series for the triangular numbers, start at +1 on the decagon and follow the black lines clockwise.

For the triangular numbers, the difference between the nth gnomon is the (n - 1)th gnomon is 1. I shall express this by saying that the gnomon interval for the triangular numbers is 1. For the squares, the gnomon interval is 2; for the pentagonal numbers, it is 3; and so on. The gnomon interval for the n-gonal numbers is always equal to n - 2.

The gnomon series for the n-gonal numbers modulo 10 may be read off our decagon by starting at +1 and going clockwise, reading every (n - 2)th vertex. It is readily apparent that there are only 10 possible gnomon series, since reading every (n + 10)th vertex is the same as reading every nth vertex. The gnomon series are:

• 3-gonal: black lines clockwise (+1, +2, +3, +4, ±5, -4, -3, -2, -1, 0)
• 4-gonal: red lines clockwise (+1, +3, ±5, -3, -1)
• 5-gonal: green lines clockwise (+1, +4, -3, 0, +3, -4, -1, +2, ±5, -2)
• 6-gonal: purple lines clockwise (+1, ±5, -1, +3, -3)
• 7-gonal: orange line (+1, -4)
• 8-gonal: purple lines counterclockwise (+1, -3, +3, -1, ±5)
• 9-gonal: green lines counterclockwise (+1, -2, ±5, +2, -1, -4, +3, 0, -3, +4, +1)
• 10-gonal: red lines couterclockwise (+1, -1, -3, ±5, +3)
• 11-gonal: black lines counterclockwise (+1, 0, -1, -2, -3, -4, ±5, +4, +3, +2)
• 12-gonal: only one vertex (+1)

After the 12-gonal numbers, the gnomon series repeat; the (n + 10)-gonal numbers are congruent to the n-gonal numbers (mod 10).

Which of these gnomon series will generate a repeating palindromic series? All those, and only those, whose representation on the decagon exhibits left-right symmetry -- that is, all figurate numbers except the 7-gonal (the orange line), the 12-gonal (a single non-centered point), and those congruent to them (the 17-gonal, 22-gonal, etc.).

When first looking for RPSs in figurate numbers mod 10, I only got as far as the 10-gonal numbers, so the 7-gonal numbers seemed to be the only exceptions to the RPS rule. This new postulate predicts that the 12-gonal numbers will also be an exception -- and indeed they are. The 12-gonal numbers mod 10 are (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) endlessly repeated, which is not a palindrome.

Two tasks remain: (1) proving what I have just asserted, and (2) devising a way to predict, for any modulus, which gnomon intervals will yield a left-right symmetrical pattern.