Euclidean geometry dictates that two-dimensional spaces, when properly connected, must be orientable for objects in two dimensions. This fundamental property of space makes it an attractive destination for scientists and engineers alike.
Surfaces cannot be shaped into objects by simply rotating or moving them around; they must have two sides which are known as positive and negative directions.
One way to test whether a surface is orientable is by looking at its ability to reflect itself back onto itself – in other words, how can it be mirrored without overlapping? In mathematics this property of the surface is known as its symmetry property.
This paper investigates a class of regular surfaces generated by rotating and twisting a rigid line segment. We demonstrate that its orientability can be accurately predicted using an Orientability Chart: that is, a chart with p such that all transition functions are orientation preserving with n > 0.
Second, the most famous and lasting enumerable chart is known as “Mirzakhani-Mott-Molemann,” whose integral kernels can be precisely described and mapped onto loop equations for a matrix with orthogonal symmetry.
This work has produced an astounding result regarding non-orientable 3-manifolds, which we prove by iteratively discovering infinitely many topologically distinct, irreducible closed 3-manifolds with none of their oriented covering spaces being prime.
