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



)

.