Attack Fraction vs Distance: a Chart and Discussion

This chart compares the fraction of squares attacked vs. distance for selected pieces. All the FIDE pieces are here, along with the ancient "building block" pieces, and several newer pieces, most of them shortrange leapers, some of my own design.
The idea for this came from 2 sources. I was looking at the total number of squares attacked by several shortrange pieces and finding them grouped in multiples of 4 when I received an email from David Paulowich containing his figures on the average number of the immediately adjacent 8 squares attacked by the pieces in several games, ranging from 2.5 in shatranj and his Shatranj Kamil X to 5.0 for my Lemurian Shatranj [FIDE is 4.0], and his conclusions. I extended his idea and combined it with my numbers for this chart.
Joe Joyce

dist K-G F-W A-D W-E N FAD H-M J Sl Sb Kz dist #sq totsq
1 1 0.5 0 0.5 0 0.5 0.5 1 1 1 1 1 8 8
2 0 0 0.25 0.25 0.5 0.5 0.75 0.5 0.5 1 1 2 16 24
3 0 0 0 0 0 0 0 0 0 0 0 3 24 48
4 0 0 0 0 0 0 0 0 0 0 0 4 32 80
TotSq 8 4 4 8 8 12 16 16 16 24 24
dist H-S l Hb Sb L-O P T X Z B-R Q dist #sq totsq
1 0.5 0.5 0.5 0.5 1 0.5 1 1 0.5 1 1 8 8
2 0.25 0.75 0.25 0.25 0.5 0.5 1 1 0.25 0.5 2 16 24
3 0.17 0.17 0.5 0.17 0.33 0.5 0.17 1 0.17 0.33 3 24 48
4 0 0 0 0.13 0.25 0.25 0.13 0.5 0.13 0.25 4 32 80
5 0 0 0 0 0 0 0 0 0.1 0.2 5 40 120
6 0 0 0 0 0 0 0 0 0.09 0.17 6 48 168
7 0 0 0 0 0 0 0 0 0.07 0.14 7 56 264
8 0 0 0 0 0 0 0 0 0.07 0.13 8 64 328
Tot 12 20 20 16 32 32 32 64 32 64

Legend

In the first column, dist is distance and Tot is total squares attacked by the piece[s] in that column over the entire piece range.

K-G = king and guard
F-W = ferz and wazir
A-D = alfil and dabbabah
W-E = warmachine and elephant = DW & AF
N = knight
H-M = High priestess and Minister = NAF & NDW
J = DWAF; known as the squire, jumping general, and mammoth/mastodon in various games
Sl = linear sliding general or Q2 [2-square queen]
Sb = bent sliding general
Kz = Kozune = NDWAF
H-S l = [D+W]&[A+F] linear hero and shaman
H-S b = [D+/-W] bent hero
S b = bent shaman = [A+/-F]
L-O = lightningwarmachine and oliphant = [DW+DW]&[AF+AF]
P = parallel general = [DWAF+DWAF]
T = twisted knight = [AF +/- AF]
X = flexible knight = [DW +/- DW]
Z = zigzag general = [DWAF +/- DWAF]
B-R = bishop and rook
Q = queen

What does all this mean? Darned if I know yet, but there are some interesting patterns we can look at.

tot sq att'kd piece&range
4 = 4x1 W and F - 1; A and D - 2
8 = 4x2 K or G - 1; [DW] and [AF] - 2
12 = 4x3 FAD - 2; linear Hero and Shaman - 3
16 = 4x4 HiP, Min, JG, Sliding general - 2; L, O - 4
20 = 4x5 bent Hero and Shaman - 3
24 = 4x6 bent Sliding general, Kozune - 2
32 = 4x8 PG, TN, XN - 4; B, R - 8
64 = 4x16 ZZG - 4; Q - 8

Notes

1] If 2 pieces attack the same number of squares, and one has a shorter range than the other, that piece is more powerful. This is known in most cases, but indicates the FAD is stronger than the linear Hero and Shaman.
EDITS by author: The above statement was initially based on the values of the man [nonroyal king] and knight, which I took to be 4 and 3. Based on a value of about 3 for both of them, I no longer believe this statement of relative power is always true, merely sometimes true. The chart is a better indicator of piece power, and it's always true that the mean free path of game pieces favors infinite sliders the longer the game goes on and the longer that path gets.
2] If 2 pieces attack the same number of squares and have the same range, and one leaps, that piece is more powerful - trivial & obvious.
3] The orthogonal and diagonal pieces may be considered as pairs, with wazir and ferz being the 1st pair, then dabbabah, alfil, all the way up to rook and bishop. These pieces are all linear movers, and have exactly the same movement patterns, rotated 45 degrees. When you make a bent 2-step piece, this splits the pairs. The ortho partner becomes much stronger up close, and the diag partner becomes stronger at a distance - see the Hero-Shaman pairs, and the Lightningwarmachine-Oliphant vs fleXible-Twisted kNight pairs.
4] The "4xN" column just above these comments points up the difference between 2-stepped pieces with [the ability to choose] even steps compared to [forced] uneven steps. If N and range are odd, the steps are uneven. If N is odd and range is even (FAD), the piece is a single step piece with a choice of components. The king, N = 1, is a "collapsed" case, and drops out here.
5] The 4xN column implies there is a piece with N = 7 that would be a moderate-range uneven stepper - at a guess, a "super-bison" with step lengths of 2 and 3.
6] Pieces with N = 9 to 15 would fill in the zeros [or zeroes, if you prefer] in the lower left of the table - again, trivial and obvious.
7] Most pieces' fractions go down as the distance goes up. The exceptions are the "interesting" pieces.
8] Sure looks like you could "plug in" fractions to build a wide range of theoretical pieces, doesn't it?
9] That super-bison [N = 7] piece, if bent, should exhibit the same "fraction behavior" as the bent hero and shaman; ie: the fraction increases and then goes back down as the distance increases.
10] If you stop the calcs for the bishop-rook pair and the queen at a distance of 4, which is the maximum I've used for "short" range pieces, their total squares attacked numbers become 16 and 32, and they don't leap.
11] Note number 10, right above, implies that the "Mean Free Path" is the critical value factor for infinite sliders. For pieces, especially shortrange ones, that may leap at different points in their moves, the mean free path value has [greatly] reduced significance. Density approaching saturation and clustering of pieces would seem to be more determining factors here. In fact, the actual values of pieces changing as the game goes on, as the pieces in the game change, and as the board changes, are actually the only things I can be reasonably sure are true.