Covering Maps of Triply Periodic Minimal Surfaces
The starting point in our enumeration is a
triply periodic minimal surface (TPMS).
These surfaces have the hyperbolic
plane (H²) as their universal
covering space.
This means that there is a continuous map from the
hyperbolic plane (H²) onto the surface
such that a small patch of the surface pulls back to a countable number of
isomorphic copies in the hyperbolic plane.
The following sequence of images shows how the hyperbolic plane wraps onto the
P minimal surface.

Step 7 

To proceed with our enumeration of
surface reticulations,
we need a
covering map,
that respects the symmetries of the TPMS (or at least some
chosen subgroup of symmetries). Within EPINET, we are interested in
translationally–periodic structures, so we use
covering maps,
that are based on the smallest translational unit that doesn't exchange
the two sides of the surface.
Consider once more Schwarz's
P
(primitive) surface, for example.
A translational unit cell for the P
surface can be as small as the portion on the left in the
illustration below,
(a translational unit for the nonoriented surface), or
large enough to incorporate many of these units (as the figure on the
right in the illustration shows).
The translational unit cell we use within EPINET is shown in the
middle of the illustration below.



P surface: nonoriented unit cell 
P surface: oriented unit cell 
P surface: four oriented unit cells 

The TPMS we use also have a high degree of intrinsic symmetry, so it is
convenient to work with a
covering map
that respects these higher
symmetries of the surface. For the
P (primitive),
surface and its close relatives, the
D (diamond) and
G (gyroid) surfaces,
the intrinsic surface symmetries have a natural representation in
the hyperbolic plane as a group of reflections in the sides of a
triangle with angles π/2, π/4,
and π/6. We use Conway's
orbifold notation,
and refer to this group as *246.


H² dodecagon 
P surface 



G surface 
D surface 

Now we have a well–defined map from the hyperbolic plane onto a
TPMS,
which also respects the chosen symmetries of the surface. This allows
us to work in the hyperbolic plane where the mathematics of tilings and
symmetry is much simpler. To get the corresponding three–dimensional
structures, we use the covering map to project the hyperbolic patterns
onto the
TPMS.
More mathematical background pages
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mathematical background
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