Hey folks, I just joined and have two questions for y'all.

I have seen lovely things on this forum including:

((sqrt(y^2+z^2)-b)^2+(x-a)^2) * ((sqrt(((sqrt(3)x-y)/2)^2+z^2)-b)^2+((x+sqrt(3)y)/2+a)^2) * ((sqrt((-(sqrt(3)x+y)/2)^2+z^2)-b)^2+((x-sqrt(3)y)/2+a)^2) = c^6

And similarly the "tiger cage" in here viewtopic.php?f=24&t=1858

And the cube-like cut of the "triger" here viewtopic.php?f=24&t=801&start=390

These are of great interest to me because they are realizations of genus 3 and 4 surfaces in R3, and they are "nice"/"simple"/"smooth", maybe even "canonical" or "fundamental" . Does anyone have knowledge of how to create an implicit function for an arbitrary genus surface?

Contrast with

From https://www.youtube.com/watch?v=cermfDnqQ5M / http://www.math.uni-tuebingen.de/ab/GeometrieWerkstatt/

Object is made in S3 by starting with the clifford torus and having some kind of geometric flow. Then stereographically projected. Generalizes horribly, their other genus-n surfaces are revolting.

Other question

It is known that a genus-n surface can be acquired topologically by "gluing", see below or this video https://www.youtube.com/watch?v=G1yyfPShgqw. Does anyone know how to map points onto these surfaces?