Hyperbolic maze

Some comments about this maze

This is a maze in the hyperbolic plane, displayed using the Poincaré disk or Beltrami-Klein model (see this video for explanations and illustrations about projections of the hyperbolic plane). The fundamental pattern used to construct the maze is the uniform tiling of the hyperbolic plane by regular quadrilaterals (“squares”?) with angles of 2π/5=72° at each vertex (less than the 90° of a Euclidean square because the hyperbolic plane is negatively curved).

There are no obstacles of any kind in this maze, making obvious the fact that the hyperbolic plane is intrinsically mazelike, or at least, tree-like. (Specifically, going forward in two slightly different directions, or even in “parallel” directions from two slightly spaced points, will produce divergent trajectories, hence different regions that can be explored.)

Use the up and down arrows to move up and down, left and right to rotate left and right, and insert and delete to move left and right without turning. (Note that since the hyperbolic plane has curvature, even if one moves without turning, one can return to one's starting point with a different orientation. So it would be meaningless to have a “compass” in the hyperbolic plane.)

Since the hyperbolic plane is infinite, the maze has been made “periodic”, in much the same way that the Euclidean plane can be made “periodic”, that is, transformed into a flat torus by quotienting by a lattice of periods, typically a square lattice in computer games, so that one “loops around” if one goes to far. Here something analogous has been done, by quotienting out by a discrete subgroup Γ, acting without fixed points, of the group PSL(2,ℝ) of orientation-preserving isometries of the hyperbolic plane which is contained in the group Δ of symmetries of the tiling (the (2,4,5) triangle group); here, Γ is a normal subgroup and the quotient (PSL(2,ℝ)∩Δ)/Γ is the projective special linear group PSL(2,89) on the finite field with 89 elements: since #PSL(2,89) = 352440, the period domain consists of 88110 cells (each cell has 4 orientation-preserving symmetries because it is a square), 176220 edges (each edge has 2 orientation-preserving symmetries) and 70488 vertices (each vertex has 5 orientation-preserving symmetries). If one prefers, the maze lives on a compact Riemann surface of genus 8812 (having 352440 symmetries), whose universal cover is the hyperbolic plane (and Γ is its fundamental group). Or on the Cayley graph of PSL(2,89) for two generators R (rotation around a cell) and T (translation to the next cell).

The number 89 was chosen because it is the smallest mod which the three polynomials X⁴+1, X⁴−(3/2)X²+(1/4) and X⁴+(1/2)X²−(1/4) split completely (they are the minimal polynomials of the coefficients of R and T as elements of PSL(2,ℝ)), making it possible to define R and T in PSL(2,F) without requiring any extension.

To summarize the above, the maze world is large (consisting of 88110 square cells), although not infinite; but since there are a huge number of intrinsically different ways of “wrapping back” to one's starting point (the surface being of genus 8812), it is quite complex and difficult to describe.

Twenty-four orbs have been placed regularly on the surface (they have been placed at elements of a double coset x·H·y where H is a subgroup of PSL(2,89) isomorphic to the symmetric group on 4 objects), and colored domains have been constructed around them at random, of roughly the same size. The goal of the game is to collect all these orbs (by simply walking on the corresponding square). Or one can just explore around to get a feeling of what hyperbolic geometry looks like.