New 3D Leapers

By L. Lynn Smith


In 3D Chess, most are quite familiar with the three basic leapers.  These are
the Knight(1x2x3), the Hippogriff(2x2x3) and the Wyvern(2x3x3).  The first being
a planar leaper and the other two cubic.  And they are leapers within the
orthogonal pattern of the playing field.

What if leapers were restricted to the diagonal and triagonal patterns of the
3D playing field?

Let's use the planar leap of the Knight to demonstrate this.

Within a plane, the Knight is able to leap to eight cells and this is its
pattern ('O' represents the Knight and 'X' represents its potential destination):


[ ][X][ ][X][ ]
[X][ ][ ][ ][X]
[ ][ ][O][ ][ ]
[X][ ][ ][ ][X]
[ ][X][ ][X][ ]


This is the orthogonal pattern of this particular leap.

Now if the 1x2x3 pattern was expressed specifically within the diagonal 
pattern of that same plane;

[ ][ ][X][ ][x][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]
[X][ ][ ][ ][ ][ ][x]
[ ][ ][ ][O][ ][ ][ ]
[X][ ][ ][ ][ ][ ][x]
[ ][ ][ ][ ][ ][ ][ ]
[ ][ ][X][ ][x][ ][ ]


Looking at a single leap, comparing an orthogonal pattern to a diagonal:

      orthogonal                  diagonal

[ ][ ][ ][ ][ ][ ][ ]      [ ][ ][X][ ][ ][ ][ ]
[ ][ ][ ][-][X][ ][ ]      [ ][-][ ][-][ ][ ][ ]
[ ][ ][ ][-][-][ ][ ]      [ ][ ][-][ ][-][ ][ ]
[ ][ ][ ][O][-][ ][ ]      [ ][ ][ ][O][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]      [ ][ ][ ][ ][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]      [ ][ ][ ][ ][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]      [ ][ ][ ][ ][ ][ ][ ]



At first glance, this diagonal pattern leap appears to be the leap of the 
2D Camel.  But we are only looking at a single plane.  And in addition to 
two more orthogonal planes, the diagonal pattern has four diagonal plane 
patterns composed of axes from separate orthogonal planes in a 3D playing
field.  So that its potential leap pattern might appear thus:


[ ][ ][ ][ ][ ][ ][ ]
[ ][ ][X][ ][X][ ][ ]
[ ][X][ ][X][ ][X][ ]
[ ][ ][X][ ][X][ ][ ]  level 7
[ ][X][ ][X][ ][X][ ]
[ ][ ][X][ ][X][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]

[ ][ ][X][ ][X][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]
[X][ ][X][ ][X][ ][X]
[ ][ ][ ][ ][ ][ ][ ]  level 6
[X][ ][X][ ][X][ ][X]
[ ][ ][ ][ ][ ][ ][ ]
[ ][ ][X][ ][X][ ][ ]

[ ][X][ ][X][ ][X][ ]
[X][ ][X][ ][X][ ][X]
[ ][X][ ][ ][ ][X][ ]
[X][ ][ ][ ][ ][ ][X]  level 5
[ ][X][ ][ ][ ][X][ ]
[X][ ][X][ ][X][ ][X]
[ ][X][ ][X][ ][X][ ]

[ ][ ][X][ ][X][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]
[x][ ][ ][ ][ ][ ][X]
[ ][ ][ ][O][ ][ ][ ]  level 4
[X][ ][ ][ ][ ][ ][X]
[ ][ ][ ][ ][ ][ ][ ]
[ ][ ][X][ ][X][ ][ ]

[ ][X][ ][X][ ][X][ ]
[X][ ][X][ ][X][ ][X]
[ ][X][ ][ ][ ][X][ ]
[X][ ][ ][ ][ ][ ][X]  level 3
[ ][X][ ][ ][ ][X][ ]
[X][ ][X][ ][X][ ][X]
[ ][X][ ][X][ ][X][ ]

[ ][ ][X][ ][X][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]
[X][ ][X][ ][X][ ][X]
[ ][ ][ ][ ][ ][ ][ ]  level 2
[X][ ][X][ ][X][ ][X]
[ ][ ][ ][ ][ ][ ][ ]
[ ][ ][X][ ][X][ ][ ]

[ ][ ][ ][ ][ ][ ][ ]
[ ][ ][X][ ][X][ ][ ]
[ ][X][ ][X][ ][X][ ]
[ ][ ][X][ ][X][ ][ ]  level 1
[ ][X][ ][X][ ][X][ ]
[ ][ ][X][ ][X][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]

Obviously, this piece is much more powerful than the simple Camel.  And
it will always remain within its specific diagonal pattern.  Being a simple
planar leap, it might be the easiest to visualize.

The other two patterns, 2x2x3 and 2x3x3 are cubic and thus involve the
exploitation of three distinct axes.  These will be specific diagonal axes
within the diagonal patterns of the 3D playing field.

The following are single examples of each of these two forms of leaps
within a diagonal pattern.


        2X2X3                             2X3X3

[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][X][ ][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][-][ ][-][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][-][ ][-][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]   level 3   [ ][ ][ ][-][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]


[ ][ ][X][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]
[ ][-][ ][-][ ][ ][ ]             [ ][ ][-][ ][ ][ ][ ]
[ ][ ][-][ ][-][ ][ ]             [ ][-][ ][-][ ][ ][ ]
[ ][ ][ ][-][ ][ ][ ]   level 2   [ ][ ][-][ ][-][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][-][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]


[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]
[ ][ ][-][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]
[ ][-][ ][-][ ][ ][ ]             [ ][ ][-][ ][ ][ ][ ]
[ ][ ][-][ ][-][ ][ ]   level 1   [ ][-][ ][-][ ][ ][ ]
[ ][ ][ ][O][ ][ ][ ]             [ ][ ][-][ ][-][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][O][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]


Obviously, there are many more potential leaps for each of these
particular patterns.  But they will all remain with the defined
diagonal pattern of the playing field.

These same leaps may be applied to the triagonal patterns of the
3D playing field, creating three additional new 3D leapers.  I
will allow the reader to work out these particular patterns.

But I will give an example of a triagonal pattern Knight leap
to assist in initial visualization:


[ ][X][ ][ ]
[ ][ ][ ][ ]   level 4
[ ][ ][ ][ ]
[ ][ ][ ][ ]

[ ][ ][ ][ ]
[-][ ][-][ ]   level 3
[ ][ ][ ][ ]
[ ][ ][ ][ ]

[ ][ ][ ][ ]
[ ][ ][ ][ ]   level 2
[ ][-][ ][-]
[ ][ ][ ][ ]

[ ][ ][ ][ ]
[ ][ ][ ][ ]   level 1
[ ][ ][ ][ ]
[ ][ ][O][ ]


Now, we arrive at what to call these six new pieces.  A simple
form might be to prefix each piece with its particular pattern.
For example: 'o-' would denote orthogonal, 'd-' for diagonal and
't-' for triagonal.

So that the Knight operating in the orthogonal pattern would be the 
o-Knight, one within the diagonal pattern the d-Knight and the triagonal
as the t-Knight.

And the potential number of composite leapers has now 'jumped'
exponentially.  Although some combinations will be redundant, as 
they are expressed in others. For example, the o-Hippogriff is
present in the pattern of the d-Knight.

Now, I am not advocating the application of each and every one of
these leapers in any one particular game.  But a 3D developer might
consider one or two of them.  

A potential might be to create a 3D game in which the a variety of 
pieces are bound to particular patterns of the playing field.  For
example, if a majority of pieces were bound to the diagonal patterns, 
there exist the potential of two distinct games being played out.
The developer would then have to consider these dynamics when establishing
the goals of the game.

Comments

New 3D Leapers. Leapers restricted to the diagonal and triagonal patterns of the 3D field.

Charles Gilman wrote on Mon, Sep 6, 2004 03:38 AM EDT:Good ★★★★

Having looked into thesse pieces further, I am starting to warm to them. I notice that where the orthogonal-oblique piece is a two-coordinate leaper, m:n say, with a SOLL of m²+n², the diagonal-oblique piece has the three SOLLs {2(m²+n²[±mn])}, that is to say, the SOLL of the of the element of the di-ob piece that is itself another or-ob, and the numbers 2mn either side of it. For example, the standard Knight is a 2:1 leaper, SOLL 5, so the SOLLs of the d-Knight are {2(5[±2])} = {10, 6, 14}. 10 is the Camel SOLL, twice that of the standard Knight, and 6 and 14 are twice the 3 of the Wellisch hex Knight and the 7 of the Glinsky hex Knight.

📝Larry Smith wrote on Fri, Sep 3, 2004 05:49 AM EDT:

Here's an example of a d-Ferz, or a 1x2x2 planar piece in a diagonal
pattern:

[ ][ ][ ][ ][ ]
[ ][X][ ][X][ ]
[ ][ ][X][ ][ ]  level 3
[ ][X][ ][X][ ]
[ ][ ][ ][ ][ ]

[ ][X][ ][X][ ]
[X][ ][X][ ][X]
[ ][X][ ][X][ ]  level 2
[X][ ][X][ ][X]
[ ][X][ ][X][ ]

[ ][ ][X][ ][ ]
[ ][X][ ][X][ ]
[X][ ][O][ ][X]  level 1
[ ][X][ ][X][ ]
[ ][ ][X][ ][ ]

In relation to the orthogonal pattern, there are several apparent
'leaps'.  A developer might disallow those planar moves which result in
adjacent diagonal cells and create an interesting new leaper for the 3D
playing field.  A composite of an o-Dabbabah and an o-Hippogriff.

Or accept all its potential planar moves and thus have the o-Ferz moves
also present.  Most interesting.

📝Larry Smith wrote on Thu, Sep 2, 2004 08:24 AM EDT:

Charles,

Your application of angles is in relation to the orthogonal pattern.  If
you wish to be bound by such restrictions, feel free.

In this simple treatise, the diagonal and triagonal patterns are being
considered as seperate entities.  And all their planar and cubic
potentials are thus expressed in such regard.

Allow me to state that these suggestions are not to be taken as absolute,
either in their regulation or nomenclatures.  They are primarily attempts
to aid developers in thinking outside the box.

The logic of the 90 degree restriction collapses when applied to the cubic 
move within the diagonal pattern and with the simple planar move within 
the triagonal pattern.

Charles Gilman wrote on Thu, Sep 2, 2004 03:20 AM EDT:

Having studied these ideas offline, I can see your 'simple' diagonal-pattern pieces for the compounds that they are. In cubic-cell geometry, all the orthogionals are the same angle to each other; the diagonals are not. Turning between the move of two steps diagonally and that of 1 differs immensely depending on whether the turn is through 90°, 120°, or 60°. The first is indeed the dual of the usual Knight's move, and so a piece with such moves is indeed a Camel. The second is the dual of the move defining a hex Knight in Glinsky terminology, and the third that of the move defining a hex Knight in Wellisch terminology (which Glinsky would presumably consisder a Ferz). Would you consider the 'true' hex Knight (if such a thing exists!) the piece combining the moves of both hex pieces?

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