Please stand by while the maze is being constructed…

Move with arrow keys and insert/delete. (Also F1 to recenter.) Or click in the direction in which you want to move.

Projection:

Bread crumbs: ,

Beacons:

Show/hide explanations

✱ There is also a similar but different game (more of a classical maze) in the same world.

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 2`π`/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 `X`⁴+1, `X`⁴−(3/2)`X`²+(1/4)
and `X`⁴+(1/2)`X`²−(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.

David Madore