P4Signon  Lattice Study of Polysigned Numbers

The Cartesian numbers form orthogonal lattices.
The polysigned numbers form nonorthogonal lattices at specific angles.
In general the angle between Pn sign poles is
   pi - arccos( 1 / (n - 1 ) ).
These are the angles from the center of a simplex to its vertices.
There is another important distinction:
The polysigned lattice is composed of unidirectional segments.
As we take unit steps in the Cartesian lattice we are free to choose a positive or negative step.
In the polysign system the signs are the orientations.
The only system with graphically opposed steps is at sign two (the real numbers).
In order to return to the same point in the polysign lattice a loop must be formed.
The simplest lattice loops travel one step in each sign resulting in a zero sum.
The set of unit loops taken as a whole form polyhedra which pack perfectly.
This is conjectured to hold at any sign level.

The P4 Signon

In P4 stepping -1, *1, #1, +1 forms a unit loop.
Any sequence of all sign components will form such a loop.
The shape has 12 faces and 14 vertices composing its exterior.
The exterior forms a rhombic dodecahedron.
These vertices include the points -1, -1*1, and -1*1#1.
These are merely instances which expose distance and angle characteristics of the signon.

View the P3 Signon
View the P5 Signon
View the P3 Lattice
View the P4 Lattice

The signon inherently contains a center point and paths leading to it and from it.
The unidirectional nature of the rays is not encompassed by traditional geometry.
Combinatorics are needed to analyze the lattice and the signon.
The signon is a natural consequence of the polysigned numbers.
The packing is exposed via special notation:
A coordinate representation of 
   [ a0, a1, a2, ... ] 
is used where the values are magnitudes and hold their sign order
with the identity sign as the a0 element and a1 being the minus element,
a2 the plus, etc. For instance
   [ 1.2, 3.4, 5.6 ] 
is a value in P3 representing
   * 1.2 - 3.4 + 5.6 .
   [ 1.1, 2.2, 0, 4.4 ]
is a value in P4 representing
   # 1.1 - 2.2 * 4.4 .
The bar notation [| x, y, z ] indicates a general solution that includes 
all 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 so that multiple barred instances 
used in an expression will retain synchronized permutations.
For instance in P3 unit signa may be found centered on
   [ | 2, 0, 1 ]
that will touch a unit signon centered on
   [ 0, 0, 0 ]
at  vertice
   [ | 1, 0, 0 ] 
and at vertice
   [ | 1, 0, 1 ] .
This notation creates six concrete instances one of which is a signon 
centered on
   [ 0, 2, 1 ] 
that touches an originated unit signon at vertices 
at [ 0, 1, 0 ] and at [  0, 1, 1 ] .
In general the unit signa which pack will be found at an offset of
   [| 2,0,1,1,1, ...]
The following equation details the adjacent positions:
   [|0,0,0,0,0] + [| 1,0,v,v,v, ...] = [| 2,0,1,1,1, ...] + [| 0,1,v,v,v, ...]  
where v's imply a variable coordinate which can be either 0 or 1.
This equation applies for P3 and higher sign spaces (P3+).

Return To Polysigned Numbers