The Step
The basic move is a single step to an adjacent cell. The step may be made across a side or edge or through a corner point or apex. The number of adjacent cells is a function of the angle subtended at the juncture of two cell sides.
| Cell Shape | Angle | Adjacent Cells |
|---|---|---|
| Trig | 60 | 12 |
| Square | 90 | 8 |
| Hex | 120 | 6 |
The step may be classified in two distinct ways:
- by using the colour of the step's from and to cells - chromatic typing
- by using the connection between the from and to cells - connective typing
There are two Chromatic types:
one where the cell colour changes, an allochromatic step (a-step); the other where the cell colour remains the same, an idiochromatic step (i-step).
There are also two Connective types:
one where the cells are connected at a point or apex, a pointwise or apical step (p-step); the other where the connection is along a line or edge, an edgewise or lineal step (e-step).
On a square based board p-steps are always i-steps and vice versa, and e-steps are always a-steps and vice versa. These identities are NOT true for the trig based board.
Diagram 1. illustrates this single step. From the red trig the blue trigs can be reached by an a-step, while the yellow trigs are reached by an i-step.
The Slide
This may be defined as
a series of steps of the same type all made in the same direction
and the direction of movement defined as
along a straight line drawn from the centre of the starting cell and passing through the adjacent cell that forms the first step of the slide.
Diagram 2. shows these directions: Red lines where the connection between immediately adjacent trigs alternates between edge and point (an alternating direction); Green lines where the connection is always an edge (a constant direction).
Examples of cells reachable by slides are shown in Diagram 3. Blue cells where the slide consists of a-steps in an alternating direction; Yellow where i-steps in a constant direction.
Knight's Move
This may be defined in 3 ways as
1. an a-step followed by an i-step continuing away from the starting cell; or
2. an e-step followed by a p-step continuing away from the starting cell; or
3. a p-step followed by an e-step continuing away from the starting cell
On a trig based board the order of chromatic steps is not significant, while with connective steps it is. On a square based board all 3 definitions lead to the same pattern.
The 3 variations of the knight's move are illustrated by Diagrams 4, 5 and 6 .
Diagram 6 shows the pattern referred to by Joe Joyce in
this comment for a piece termed a short knight.
Piece's and their pattterns off movement are discussed further here.
It should be noted that the number of cells reached by a knight's move decreases as the number of cell sides increases: 12 for trigs; 8 for squares; 0 for hexes. The hex board suffers from the fact that no i-steps are possible with the definition of adjacent given above. This is usually remedied by an extension to the definition of adjacent.
Adjacent Extension
The hexagonal board extension may be stated as follows:
Also considered as adjacent are those cells that are adjacent to two ordinarily adjacent cells that are themselves mutually adjacent, such that the same step-type exists between each pairing of the three cells
This extension adds no more cells when applied to squares, but adds 12 more cells when applied to trigs.
Diagram 7. shows the 6 extended a-step adjacent cells in a darker blue, while the 6 i-step ones are shown in orange. With the normally adjacent cells 3 are reached by e-steps and 9 by p-steps: with the extended adjacent cells 9 are reached by e-steps and 3 by p-steps.
Summary
With its ability to define steps in 2 unique ways and with a possible 24 cells capable of being considered adjacent the somewhat neglected trigonal board has much to offer the CV game designer.
Diagram 1.
Diagram 2.
Diagram 3.
Diagram 4.
Diagram 5.
Diagram 6.
Diagram 7.
Very nice so far; I like what you're doing, and where you're going looks productive. Great pix!
I noticed that the green lines in Diagram 2 go only through sides of triangles, and the red lines alternate: side, angle, side…
A bishop-type piece moving only along the green lines would cover half the board.
A rook-type piece moving along the same green lines covers the entire board, and should be worth double the "green bishop", as a first approximation.
Logically, you'd have analogous bishop and rook types moving only along the red lines. This gives you a nice split in pieces, turning 2, rook and bishop, into 4, green R&B, and red R&B.
The knight looks really good. It does what a knight should do, bend a little while moving, change color, and lay down a "circular" pattern of possible moves.
New pieces?
A "dual bishop" that moves on both the red and green lines of diagram 2. This would, on first pass, be worth as much as a rook.
A "dual rook" would probably be too powerful; this queen-type piece, if used, should probably be restricted to short range.
I'd develop some more short range pieces here, but I gotta go make dinner! More later.
Enjoy,
Joe
You're an inspiration, Joe!
I must admit I wasn't thinking of red/greenxbishop/rook combinations, but only of red=rook, green=bishop.
But you are so right! Both red lines (apical directions) and green lines (edgewise directions) could support i-sliders (bishops) and a-sliders (rooks).
Now need to come up with some catchy names (apical-bishop, edgewise-bishop, dual-bishop just don't cut the mustard). Perhaps apostle, evangelist and bishop? But as for the rook?
There's also the 2 queen-like possibilities of green-bishop+red-rook and red-bishop+green-rook - and then there's the knight combinations to consider! - Not forgetting your short range pieces…
Cheers
Graeme
PS - glad you like the pix - used the trig version of my Board Image Utility (which I really must post to the Wiki)
Hey, Graeme! Good pictures make it easy to see things.
How about a "short knight"? This piece hits 9 squares rather than the knight's 12; 6 squares overlap. In diagram 1, there are 13 trigs that make up the central figure, the red trig and the 12 surrounding blue and yellow ones. Of those, 9 share an edge with the outside, 6 yellow and 3 blue. The trigs they share edges with, 6 green and 3 pink trigs, are the destination spots of the short knight.
Gotta go - leaving in 6 hours for a 500 km drive to a family reunion. Then on to Niagara Falls, Can, then back to the US and probably home Wed. See you then. Maybe we'll have had some new ideas by then. ;-)
Enjoy,
Joe