Minimizing Self-Intersections of 3-dimensional Immersions of 5-dimensional Cubical Surfaces with Reinforcement Learning

TMLR Paper5341 Authors

Abstract: A closed cubical surface is a 2-dimensional cubical complex (analogous to a simplicial complex but with a cubical structure) such that each of its points has an open neighborhood homeomorphic to a disk. Aveni et.al. proved that up to isomorphism 2690 connected closed cubical surfaces can be built from the faces of a $5$-cube (sometimes called a penteract) and give a classification for closed orientable and non-orientable cubical surfaces. It is well known that non-orientable surfaces (of any kind) cannot be embedded in $\mathbb{R}^3$; their immersion will always have some self-intersection and in the context of cubical surfaces this also seems to be the case for some orientable surfaces. Therefore, given a cubical surface it is natural to ask: What is the smallest number of self-intersections it can have for any immersion in $\mathbb{R}^3$ using perspective projection and without deforming the $5$-cube? Given an initial immersion, can we calculate a sequence of $5$-dimensional rotations or perspective projections step-wise minimizing self-intersections efficiently? These questions are addressed using Reinforcement Learning and animation sequences are created to visualize the minimization strategies found by the agent.

Submission Length: Long submission (more than 12 pages of main content)

Changes Since Last Submission: I corrected a broken reference on the suplements.

Assigned Action Editor: ~Florian_Shkurti1

Submission Number: 5341