MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

70 Mathematical Properties
(a)
(b)
Figure 2.61: Padovan Spirals: (a) Padovan Whorl [138]. (b) Inner Spiral [293].
(c) Outer Spiral [139].
Property 76 (Padovan Spirals). Reminiscent of the Fibonacci sequence,
the Padovan sequence is defined as 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, . . ., where
each number is the sum of the second and third numbers preceding it [138].
The �ratio
of succesive � terms of this sequence approaches the plastic number,
p = 3
1
2
+ 1
6
� 23
3
+ 3
1
2
− 1
6
� 23
3
(c)
≈ 1.324718, which is the real solution of
p 3 = p + 1 [139]. The Padovan triangular whorl [139] is formed from the
Padovan sequence as shown in Figure 2.61(a). Note that each triangle shares
a side with two others thereby giving a visual proof that the Padovan sequence
also satisfies the recurrence relation pn = pn−1 + pn−5. If one-third of a circle
is inscribed in each triangle, the arcs form the elegant spiral of Figure 2.61(b)
which is a good approximation to a logarithmic spiral [293]. Beginning with a
gnomon composed of a “plastic pentagon” (ABCDE in Figure 2.61(c)) with
sides in the ratio 1 : p : p 2 : p 3 : p 4 , if we add equilateral triangles that grow in
size by a factor of p, then a truly logarithmic spiral is so obtained [139].
Property 77 (Perfect Triangulation). In 1948, W. T. Tutte proved that it
is impossible to dissect an equilateral triangle into smaller equilateral triangles
all of different sizes (orientation ignored) [309]. However, if we distinguish between
upwardly and downwardly oriented triangles then such a “perfect” tiling
is indeed possible [310].
Figure 2.62, where the numbers indicate the size (side length) of the components
in units of a primitive equilateral triangle, shows a dissection into 15
pieces, which is believed to be the lowest order possible. E. Buchman [35] has
extended Tutte’s method of proof to conclude that no planar convex region
can be tiled by unequal equilateral triangles. Moreover, he has shown that