## Parallel transport in hyperbolic tilings

**Hyperbolic Geometry** is "the other kind of
hyperspace" - instead of higher-dimensional but
still flat Euclidean space - that I've been playing around a bit with
this year. Negatively curved space is weird, there is
"much more" space the further you go from your starting point. Hard
to explain without lots of diagrams and physical models that come to
pieces and bearded sandle-wearing lecturers all in tasteful shades of
orange and brown. Not much of this work/play has born productive fruit,
mostly because I don't think I understand it much at all yet.

**Hyperbolic Plane Tilings** (covering the hyperbolic
plane with identical regular polygons) turn out
to be interesting (for some values of interesting). Triangles in hyperbolic
space have angle sum less than 180 degrees (or
pi radians). The angle defect (how much less the sum is) depends on the
area of the triangle - the longer a triangle's sides the smaller its
angles are. I'll spare the equations... Regular polygons
can be made up of lots of congruent right-angled triangles (you split a
p-gon into 2p triangles with the right-angles at the midpoints of the
p-gon's edges) even in hyperbolic space. Combining the ability to pick
the angle sum (and thus the two other angles) of a right-angled triangle
by varying its size with polygon triangulation, it's possible to tile
the hyperbolic plane with regular polygons in many more ways than than
is possible in flat space or on a sphere.

**Colouring Hyperbolic Tilings** turns out to be quite
a challenge. I'll go into more detail in a later post, but for example
when you colour the {7,3} tiling with 8 colours each spread regularly
across the whole plane, you find that some of the 7-gons that are coloured
the same have their neighbouring colours in a different order around them.
If you take this into account you can colour {7,3} with 24 colours and be
satisfied that it's a perfect colouring, but similarly simple tilings
require a mindnumbingly huge number of colours to achieve the same goal.

**Animating Coloured Hyperbolic Tilings** is the eventual
goal of all of this, and while I have
some preliminary results
there's an annoying glitch where the rotations don't line up at the animation
loop point. This is caused by parallel transport. I think. At least I
don't have any other explanation at this stage!
**EDIT** (2008-12-27) I discovered the real cause: the "initial
conditions" where all arrows were aligned in Euclidean space. Having switched
the code to align each tile relative to its neighbouring tiles I have smooth
looping!

**Parallel Transport** is a concept that arises when
you think of the effect moving around inside a space has on your internal
notion of direction. In flat space, no matter which route you take they
all match up. In fact a definition of curved space is that your internal
notion of direction when you move through it depends on which route you
took. In the hyperbolic plane you get a rotation proportional to the area
of the loop you travel in.

Source code (currently a bit of a mess, probably missing stuff too) implemented in Haskell and mostly undocumented and horribly flawed is here:

svn co https://code.goto10.org/svn/maximus/2008/bolix bolix

Further reading: