I could use some advice on a packing checking algorithm.
-Tim

I've developed a methodology for finding adjacent signa (plural of
signon).
It is showing that P5 signa do not pack exclusively.
For an introduction to the signon( requiring polysigned numbers) please
see:
http://bandtechnology.com/PolySigned/Lattice/Lattice.html
It is easy to place signa within a lattice, but to find the ones that
pack is less easy.
For example in any lattice it is clear that another signon exists two
units away in any sign orientation that will touch at a vertice:
A signon centered at [| 2, 0, ... ]
will touch one centered at [ 0, 0, ... ] at [| 1, 0, 0, ... ] .
The notation [| x, y, z ] indicates a general solution that includes
ordered permutations.
This means that [| x, y, z ] could mean:
[ x, y, z ],
[ y, x, z ],
[ z, y, x ],
[ x, z, y ],
[ y, z, x ],
[ z, x, y ] .
The ordering is maintained in a context so in the concrete example
above a signon centered at [ 2, 0, 0 ] would touch the [ 0, 0, 0 ]
signon at [ 1, 0, 0 ] .
Like wise [ 0, 2, 0 ] touches [ 0, 0, 0 ] at [ 0, 1, 0 ] .
This I think is a complete and noncryptic definition of the notation [|
x, y, z ] .
Now it is immediately apparent that this set of signa do not pack:
http://bandtechnology.com/PolySigned/Lattice/P3Lattice.png
There will be triangles in between them that are not covered.
Instead we need to choose signa at [| 2, 1, 0 ] .
These will pack perfectly.
Stepping up to P4 we see that we will need to find a matching signon
for each external face. A face can be expressed using the following
notation:
[ v, v, 1, 0 ] .
Here the v's imply a variable coordinate which can be either 0 or 1.
Hence the above notation expands out to:
[ 0, 0, 1, 0 ], [ 1, 0, 1, 0 ], [ 0, 1, 1, 0 ], [ 1, 1, 1, 0 ]
which is an external face of the signon centered at [ 0, 0, 0, 0 ].
In fact the expression
[| v, v, 1, 0 ]
represents all of the external faces since
[| v, v, 0, 0 ] and [| v, v, 1, 1 ]
are internal faces.
We can express the external faces of the signon centered at [ a, b, c,
d ] as:
[ a, b, c, d ] + [| v, v, 1, 0 ] .
We would like to show that these faces each correspond to one other
signon.
Thus the expression:
[| 0, 0, 0, 0 ] + [| v, v, 1, 0 ] = [| 1, 1, 2, 0 ] + [| v, v, 0, 1
]
is reached. The permutable nature is enforced accross each instance so
that they are all in synch. So if we use
[ 0, 0, 0, 0 ] + [ 1, 0, v, v ]
on the left we must use
[ 2, 0, 1, 1 ] + [ 0, 1, v, v ]
on the right.
This is the P4 signon we are studying and it is known to match the
rhombic dodecahedron which is already known to pack.
But we can test this by stepping arbitrarily many times to facially
adjacent signa:
[ 0, 0, 0, 0 ] + [ 1, 1, 2, 0 ] + [ 1, 2, 1, 0 ] + [ 1, 0, 1, 2 ] +
[ 1, 1, 2, 0 ] .
No matter what combinations one should never land a center at a face.
That would prove that these signa did not pack. This is the method that
I can use to demonstrate that P5 signa do not exclusively pack.
The P5 signon exists in four dimensions.
Here is an animated image projected down to two dimensions:
http://bandtechnology.com/PolySigned/Lattice/P5Signon.gif
In P5 the external faces take on two forms:
[| 0, 0, 0, 0, 0 ] + [| v, v, 1, 0, 0 ] = [| 1, 1, 2, 0, 0 ] + [| v,
v, 0, 1, 1 ]
and
[| 0, 0, 0, 0, 0 ] + [| v, v, 1, 1, 0 ] = [| 1, 1, 2, 2, 0 ] + [| v,
v, 0, 0, 1 ] .
Starting from the originating signon S0 I can move to [ 1, 1, 2, 0, 0 ]
(S1) .
from S1 I can move by [ 1, 0, 1, 2, 2 ] to [ 1, 0, 2, 1, 1 ] (S3).

From S3 I can move by [ 0, 2, 0, 1, 1 ] to [ 0, 1, 1, 1, 1 ] (S4).

But S4 is centered at a face vertice of S0.
So the argument that P5 signa do not pack perfectly is made.
It is quite a maze of signa.
Is the concept of internal and external faces flawed?

I've been sitting on this for awhile and don't think I can get much
further with it.
If it is true then P5 breaks a natural progression in terms of perfect
packing.

Every level has these embedded nonexclusive signa beyond P2 but P5 is
the first to allow such a transition to them via the face rules layed
out here.