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.

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.

Copyright 2002 Wendy Krieger