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Good idea about starting a thread about 3D Chess.  I apologize to Stephane
for clutteringup his page with all the extraneous material and debate.

One of the mis-conceptions about planar pieces is that because of their
wide range of motion that this somehow diminishes their tactical value.
Basically turning a sniper position into a machine gun nest.

They definitely make defending the King on an open field(or at least
lightly-crowded field) almost impossible.  But during the opening portion
of the game, they are quite restricted in movement by the presence of
other pieces. They get in each others' way.

If they are only a small percentage of the ranked pieces on the field,
the player would be reluctant to risk them without cause. The linear 
pieces would then be relegated to the front lines, taking the threats of
theopposing planar pieces and blocking their potential moves.

There is also the potential that planar pieces could be short-range. For
example, their planes could be limited to 4x4.  This would still give
them quite an influence on the field.  On one of the central cells of the
8x8x8 field, they could still command the entire volume.

This might actually make for an interesting variant. Hmmmmmmmmm.

Another restriction might be that planar pieces make only capture moves. 
They could then threaten portions of the field and an opponent might
actually draw a planar piece into a negative position.

Needless to say, the potentials of the planar piece is far from totally
defined.