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Number Systems in 3D

​
z = ( -1 , -0.76-0.64)
u = ( 0.921061 , 0.389418 , 0 )
r
plus
true
false
φ
ψ
n
levels
periodic cases for n=8
A
3
z
+ 0
2
z
+ 0z + 1 = 0
points
small
medium
large
color
1
2
3
axes
The points are the set
-l
∑
i=-l
i
z
a
i
:
a
i
∈{0,1,...,n-1}
, which is the first approximation of the fractional part in the base-
z
system with
n
digits.
This is a natural generalization of ordinary numeration systems with complex bases, but this time we treat the space as , where the first coordinate is along
u=(cosψ,sinψ,1)
and the second is in the plane orthogonal to
u
. Addition and multiplication are done component-wise using the coordinates, and
1=(1,1)
,
z=(r,r
φ
)
.
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