Convex 3d surfaces and complex numbers

A complex number is basically a point in a 2d plane, but with the y direction
said to have the basic "one" unit being a multiple of i, the square root of -1.

These are also called Gaussian Integers because Mr Gauss did a lot of work
with them.

Gaussians are interesting because they can be factored, much like 'normal'
integers, and the factors are unique, again like normal integers.

The clever bit is that each factor is also a point on a 2d plane - thus,
a set of 2d points on a plan, can be associated with their "product", 
which is another 2d point on the plane.

The interesting bit is to consider these 2d points as a set, and what
shape they might make. Or... to point out that they have a unique
Delaunay Triangulation , after Boris Delaunay. 

And this triangulation forms a sequence of triangles - which can be seen
as the projection of a convex hull in three dimensional space, down
onto the two dimensional plane. 

In other words, a convex hull in three dimensional space, projected 
down, (well, it's bottom half anyways) imagined as points, and imagining 
these points "producted" together as Gaussian integers, we have their 
"product" integer.

In this way, we say that each point on the 2d plane is associated
with many a convex hull in three dimensional space, and vice versa,
except that each convex hull has one point on the 2d plane. 

Now, through Euclid's pythagorean triples, we could also associate
each Gaussian integer with a point on a unit circle - in this way many
3d convex hulls can be associated with a single point on a unit circle.



perimeter of 2d polygon of points vs product complex number

descartes circles vs product of complex number

recentering descartes circle triangle and recalculating product...