1

**Physics, Astronomy & Cosmology / Re: The Hopf gauge**

« **on:**10/04/2024 21:41:35 »

I'm one of the logicians who might say that a 1x1x1 cube is just the trivial case for an induction on nxnxn.

And that's only in three dimensions.

If you propose that a single, solid cube can be colored such that the colors can be 'printed' onto a plane surface, you have an algebra of face turns, and a color polynomial which tiles a section of the Euclidean plane. Moreover, there is a set of nets for a cube and the algebra above would partition this set, given some restrictions such as paths that must visit each face only once.

Mostly since a single cube is a geometric object, with symmetries. Coloring is then a way to coordinate the algebra, or be able to distinguish the symmetries. Complex vectors are just a handy way to linearise everything.

For instance, in electronics you analyse circuits in terms of a frequency response, a curve. Using complex frequencies gets you an additive domain, and simplification. You then recover a real response from the analysis. This is very common in engineering, even in transmission line design.

And that's only in three dimensions.

If you propose that a single, solid cube can be colored such that the colors can be 'printed' onto a plane surface, you have an algebra of face turns, and a color polynomial which tiles a section of the Euclidean plane. Moreover, there is a set of nets for a cube and the algebra above would partition this set, given some restrictions such as paths that must visit each face only once.

Mostly since a single cube is a geometric object, with symmetries. Coloring is then a way to coordinate the algebra, or be able to distinguish the symmetries. Complex vectors are just a handy way to linearise everything.

For instance, in electronics you analyse circuits in terms of a frequency response, a curve. Using complex frequencies gets you an additive domain, and simplification. You then recover a real response from the analysis. This is very common in engineering, even in transmission line design.