MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Properties 67
These isometries form the dihedral group of order 6, D3, and its group
table also appears in Figure 2.56 where ρi stands for rotations and µi for
mirror images in angle bisectors [108].
(a) (b)
Figure 2.57: Color Symmetry: (a) Fundamental Domain. (b) Perfect Coloring.
[190]
Property 72 (Perfect Coloring). Starting from a fundamental domain (denoted
by I in Figure 2.57(a)), a symmetry tiling of the equilateral triangular tile
may be generated by operating on it with members of the group of isometries,
D3.
The five replicas so generated are labeled by the element of D3 that maps
it from the fundamental domain where, in our previous notation, R1 = µ1,
R2 = µ3, R3 = µ2, S = ρ2, S 2 = ρ1. A symmetry of a tiling that employs only
two colors, say black and white, is called a two-color symmetry whenever each
symmetry of the uncolored tiling either transforms all black tiles to black tiles
and all white tiles to white tiles or transforms all black tiles to white tiles and all
white tiles to black tiles. When every symmetry of the uncolored tiling is also
a two-color symmetry, the coloring is called perfect. I.e., in a perfect coloring,
each symmetry of the uncolored tiling simply induces a permutation of the
colors in the colored tiling. A perfect coloring of the equilateral triangular tile
is on display in Figure 2.57(b) [190].
Property 73 (Fibonacci Triangle). Shade in a regular equilateral triangular
lattice as shown in Figure 2.58 so that a rhombus (light) lies under each
trapezoid (dark) and vice versa. Then, the sides of successive rhombi form a
Fibonacci sequence (1,1,2,3,5,8,...) and the top, sides and base of each trapezoid
are three consecutive Fibonacci numbers [315].