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The Sevenly Flat

The sevenly flat is the heptagonal version of the Fibonacci series. Unlike the pentagonal version, the sevenly flat is a 2-dimensional series. This brings with it things that a regular series does not have.

In a regular series, one can go this way or that, and it may converge in this way, and not in that. Convergence is generally very good: one often gets forms of the type a+(1/a) as a goes to infinity.

In the hevenly flat, the numbers cover a grid, and there is no single direction. Convergence is now in a sector, with a peak convergence in a particular direction. Since the hevenly flat is of the form x+x'+x'', where x.x'.x'' is an integer, it follows the best convergence is to be had when that integer is one, and x is about the square of 1/x'.

Here's a portion of the hevenly flat. The axies are the coordinates of the elements.

                          a^n
             -2  1  0  1   2   3   4   5    6
            ---------------------------------  How to make the series grow
        -1 | -3 -2  0 -1   1  -1   2  -1    4
         0 |  3  1  1  0   1   0   2   1    5          z-y    y-x
         1 | -2 -1  0  0   1   1   3   4    9           x      y   x+z
         2 |  2  1  1  1   2   3   6  10   19      z-x  z     z+y
   b^m   3 | -1  0  1  2   4   7  13  23   42      x+y x+z+y
         4 |  2  2  3  5   9  16  29  52   94
         5 |  1  3  6 11  20  36  65 117  211
         6 |  5  8 14 25  45  81 146 263  474
         7 |  9 17 31 56 101 182 328 591 1065

This series converges towards the bottom left. The thing actually converges on two different values. Going across, the values converge on a number that I designate as a, and going down the table, it converges on a value of b.

The numbers a and b are the chords of a heptagon, of edge 1. The values are (approximately).

a = 1.80193773580483825247220463901489010233183832426371430010712
b = 2.24697960371746706105000976800847962126454946179280421073110

The area of convergence is where there are large powers of b. For example,

a ~ 117/65 = 1.80000
b ~ 146/65 = 2.24615

The hevenly flat works in much the same way as the fibbinacci series, except instead of evaluating powers of {$i "tau"}, expressions for (a^n)(b^m) are found.

For example, a⁴b⁵ is thus found. The line at the intersection of the column 4 and the row 5 is "65". The number to the immediate right is 117, immediately below it is 146. The number we seek is 65+117a+146b or 603.8857372: cf a^4*b^5 = 603.8857372.

Just as there are generalised Fibbinacci series, there are generalised sevenly flats. The generating kernel of the flat is expressed as the three numbers that make up an "L-shape", in the form (x,y,z). y is directly to the right of x, and z is directly below it. Here are some of the important series.

(1,0,0): Sevenly flat: As with the Fibbonacci series, this is the most important series.
(-1,1,1): Symmetric: This reflects the symmetric nature of the Lucas series. Here the symmetry is a distorted trigonal symmetry.
(3,1,2): Lucas-Like: This is the series to which the convergent region most closely approximates. Note that this series, like the Lucas series, represents the sum of squares of the chords.
(1,a,b): Logrithmetic: When these values are used, the numbers given are direct powers of the axies, in much the same way as the series based on 1 and 1.61803398875 are strictly the powers of tau.

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