## Wednesday, May 29, 2019

### [shqastbb] Towers of inaccessibility

Wander around the surface of a cube (a 2D manifold) rendered as a plane.  In the neighborhood of a vertex, the square face diagonally opposite the square you are on is black.  When you move to a different square, the black square also changes. In a 2.5-dimensional space like Minecraft, the squares become extruded into black towers.

Fly among cubes on the surface of a tesseract: there are some inaccessible black cubes.

We could also consider hyperbolic planes, for example 5 squares around a vertex.  The square face diagonally opposite the square you are on can be two possible squares, depending on which way around you go.  Visually when viewing it (in 2.5 dimensions) from a different square, it has black wall at an angle of 45 degrees extending from there vertex through the square.  Or, instead of 45 degrees, the line between you and the vertex, so you never see the face of the wall, just the discontinuity between either side.

For hyperbolic space, portions of cubes bounded by 45 degree planes.  Can't quite visualize, but doesn't seem too difficult.