Perhaps the most fundamental place that the
Ö3 makes its appearance
is in a figure referred to in sacred geometry as the Vesica Pisces.
Draw a circle. Place a point on the circumference and
draw a second circle of the same radius as the first (see Figure 8a).
Figure 8. a) Vesica Pisces; b) two equilateral triangles in
a Vesica; c) the Vesica creates a triangular grid.
The region in common to the two circles has a length and
width in proportions Ö3:1
within which two equilateral triangles may be inscribed
(see Figure 8b). It is in this central region that images of
Christ were often placed (see Figure 8c). Four intersecting
circles create a triangular grid of ten vertices known in
ancient times as the tetractys (see Figure 8d).
Figure 9 shows a six pointed star with edges intersecting
in the ratio 1:2. Notice that numerous 1:Ö3
rectangles and star hexagons at smaller scales
are defined by this construction. Also notice that
30, 60, 90-triangles are defined by this star.
Figure 9. An hexagonal grid illustrating the root 3 system.
In Figure 10 a rectangle with proportions 1:
y is tiled by a triangular
grid with a small rectangular strip leftover.
Figure 10. Rectangle with proportions 1:
y tiled by a triangular grid with
a rectangular strip left over.
Figures 11a and b shows two examples in which the
Ö3 and Ö2
geometry are combined : 1) a right triangle whose hypotenuse
is the body diagonal and the two sides are the edge and face
diagonals of a cube; and 2) a rectangle discovered
by Doug Ailles, a high schook teacher at Etobicoke Collegiate Institute
in Etobicoke, Ontario,
including a 15, 75, 90 deg. right triangle with sides
and 2 [Vakil 1996]. Other designs with root 3 geometry are
shown in Figure 12 [Edwards 1968].
Figure 11. Root 2 and root 3
are combined in a) a cube; b) an Ailles rectangle.
Figure 12. Some root 3 patterns from Patterns and Designs with Dynamic
Symmetry by E.Edwards. Courtesy of Dover Press.
Coxeter  has shown that the fundamental domain
of the group of symmetries in the plane generated by
three mirrors can be either an equilateral triangle,
45 degree isosceles triangle, or 30,60,90 degree triangle.
Except for rectangular and square fundamental regions,
the fundamental domains of all symmetries of the plane
are subsets of these triangles as provided by the root 2
and root 3 proportional systems. The more recent development
of five-fold quasicrystal symmetry as exhibited by
Penrose tilings such as the one shown in Figure 13 makes
use of the f
-system of proportions.
Figure 13. A Penrose tiling illustrating 5-fold symmetry.