# Allegorical Hexagon

As an efficient construction basic unit in
nature, hexagons are widespread – the
structure of crystals and honey combs are
well-known examples. Consequently, the
hexagon has become a symbol of efficiency,
use of material, space and solidity, and
matured in almost all cultures to an
allegoric symbol.

The four colour theorem, first presented
by the English mathematician F. Guthrie in
1852, states that, put simply, on a surface
any map of autonomous regions can be
presented by using only four different
colours.

It took until 1976 for this theorem was
proven by K. Appel and W. Haken. It was,
by the way, the first theorem to be proven
by using a computer.

Given six equally sided and equally sized
triangles, on a plain surface there are 12
different arrangement possible. These 12
“pieces” therefore have the same surface
of six triangle-units, but only the hexagon,
being the most compact in shape, shows
a circumference of six side lengths. All
others, more or less odd shaped, show a
circumference of eight side lengths.

Despite its complexity, it is possible to fit
all 12 different pieces into a surface of a
rhomb with six-fold side length. Rotating
the rhomb structure around one of its large
angle corners generates a regular hexagon
again.