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Punning by Numbers. A systematic set of names for cubic-cell oblique pieces.[All Comments] [Add Comment or Rating]

Mason Green wrote on Fri, Mar 18, 2005 11:38 PM UTC:

What's interesting about 3D elemental leapers as opposed to 2D ones is
there's a lot more diversity. In two dimensions, each (n:n) or (n:0)
leaper has 4 possible destination squares on an infinite board, arranged
like the vertices of a square. (n:p) leapers (where n and p are not zero
or the same) have 8, forming an octagon.

Finally, chiral (clockwise or counterclockwise) versions of the (n:p)
leapers have 4, arranged like a rotated square. A 'clockwise' knight can
only move to four out of the eight of a full knight, evenly spaced in a
clockwise pattern.

In three dimensions, we have many leapers whose potential destinations
resemble Platonic and Archimedean solids:

(0:0:n)         6 destinations, arranged like the vertices of an octahedron.
(0:n:n)         12 destinations, forming a cuboctahedron.
(n:n:n)         8 destinations, forming a cube.
(n:n:p), w/p>n  24 destinations, forming a rhombicuboctahedron.
(n:n:p), w/p<n  24 destinations, forming a truncated cube.
(0:n:p)         24 destinations, forming a truncated octahedron.
(n:p:q)         48 destinations, forming a truncated cuboctahedron.

And if that wasn't enough, there's MORE! It is possible to cut the
movement possibilities of an (n:p:q) leaper in half, by taking every other
destination away to form a chiral leaper (see above) who has 24
destinations, arranged like a snub cuboctahedron.

So there are 3 different fundamental 'types' of leapers in two
dimensions, and 9 in three dimensions.