Penrose stairs have been popularized by M. C. Escher in "Ascending and Descending". This is based on the 3D model of Penrose stairs by Lucian B. Such a staircase is impossible in Euclidean geometry; the 3D model uses an optical illusion to hide this fact. Here, we put this model in Nil geometry, which is a non-Euclidean geometry which makes architecture of this kind possible.

There is a natural system of directions in Nil, i.e., we can assign North, East, West, South, Up and Down directions at every point in a natural way. We always move upwards here (however, Up/Down is not along the fibers of Nil here -- the system is rotated by a fixed angle).

In the first half of the video, we make two loops around the staircase, which is viewed in the native geodesic projection (i.e., we assume that light rays travel along geodesics, i.e., locally shortest paths, in Nil). In the second half of the video, we also make two loops, but the staircase is viewed in a projection which assumes that light rays travel in a fixed direction. Since the geodesics in Nil are weird, and also our method of moving architecture to Nil is based on fixed-direction paths rather than geodesics, the second projection looks cleaner.

We will probably also make a video explaining the math behind this. See this for now:    • Playing with Impossibility (D. Celińs...  

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00:00 native geodesic projection
00:34 native geodesic projection again
01:08 fixed-direction projection
01:42 fixed-direction projection again

3D model: https://3dwarehouse.sketchup.com/mode...

Music: "R'Lyeh" by Shawn Parrotte from the HyperRogue soundtrack. This music is based on Shepard tones, which are often seen as the musical counterpart of Penrose stairs.

Made using the HyperRogue engine: http://roguetemple.com/z/hyper/