On the plane,
every hexagon (which has six sides) is surrounded by six others
and they all fit neatly together to exactly fill the space. To make
a soccer ball, we replace some of the hexagons with pentagons (which
only have five sides), thereby causing the now fewer number of hexagons
to close up towards one another and wrap into a sphere.
In the hyperbolic version of this model we make the opposite move.
Rather than replacing some of the hexagons with pentagons, we replace
them with heptagons (which have seven sides). Now, instead of closing
up, the surface opens out – for the heptagons add to, rather
than subtract from the space, and what we get is an excess of surface.
The effect is similar to what we see in lettuce leaves and certain
types of kelp where the vegetable surface expands outward from a
modest start generating a ruffled effect. |