Complementarity - Part I

Inspired by the diagram of George Duke's showing how the Falcon family was a multipath complement to the linear pieces, I began to examine spaces in between spaces described by the movement of pieces to see what patterns emerged.

At the same time, I made the trivial observation that the knight-guard can go to the same number of spaces as the alibaba-guard, 16. This led me to begin a catalogue of other pieces that can move sixteen spaces within a radius of two spaces. More trivial observations emerged. Here are the most symmetrical of the bunch along with the pieces described by the squares to which they don't go.

Pieces

Here is what I have so far, all the 16 space movers that start in the middle of a 5 x 5 space and move two spaces. Their complements describe either the knight's spaces or all possible combinations of any two of these Betzan atomic units, Ferz, Wazir, Alfil and Dababa:

************************************************************

The o spaces below describe the kNight's movement (N)

x o x o x
o x x x o 
x x . x x 
o x x x o 
x o x o x 

************************************************************

The alibaba (Alfil - Dababa) is represented by the o's (AD):

o x o x o
x x x x x
o x . x o
x x x x x
o x o x o

The x's (FWN) represent the kNight-Guard or Centaur. The same squares are accessed by the Doublemove Ferz-Wazir.

************************************************************

Here is the Guard, marked by o's (WF):

x x x x x 
x o o o x 
x o . o x 
x o o o x 
x x x x x 

..and (NDA) Squirrel, marked by x's. Also known as the Fortress.

************************************************************

o's mark the Alfil-Ferz (AF}

o x x x o 
x o x o x
x x . x x 
x o x o x 
o x x x o 

The x's (WDN) are spaces visited by the Minister.

************************************************************

o's mark the Woody Rook or Wazaba (DW):

x x o x x 
x x o x x 
o o . o o 
x x o x x 
x x o x x

x's (FAN) represent the High Priestess.

************************************************************

x x o x x 
x o x o x
o x . x o 
x o x o x 
x x o x x 

X's are WNA Alfil Knight Wazir.

************************************************************

o's represent the Waffle, also known as the Phoenix (WA)

o x x x o 
x x o x x 
x o . o x 
x x o x x 
o x x x o 

X's (DNF) are the Dababa Knight Ferz.

************************************************************

The latter two pieces described by x's are the only pieces I haven't yet encountered in a game, hence, I supply no name.

Notes

Here is one complement for a 7 x 7 section:

 

o x x o x x o
x o x o x o x
x x o o o x x
o o o . o o o
x x o o o x x
x o x o x o x
o x x o x x o 

The kNight-Camel-Zebra, also called Buffalo, goes to the squares not attacked by the 3 Queen or Sorceress.

Joe Joyce adds an additional diagram for us:

 S Z S H S Z S     
 Z S H H H S Z
 S H S H S H S
 H H H O H H H
 S H S H S H S
 Z S H H H S Z
 S Z S H S Z S

H = [D +/- W] S = [A +/- F]