MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Properties 59
(a) (b) (c)
Figure 2.42: Pascal’s Triangle (PT): (a) Arithmetical PT (1654). (b) Chinese
PT (1303) . (c) Fractal PT (1965).
It is in fact much older, appearing in the works of Mathematicians throughout
Persia, India and China. One such instance is shown in Figure 2.42(b),
which is taken from Chu shih-chieh’s Precious Mirror of the Four Elements of
1303. Can you find the mistake which is buried therein? (Hint: Look in Row
7, where rows are numbered beginning at 0.)
The mathematical treasures hidden within PAT are truly staggering, e.g.
the binomial coefficients and the triangular numbers, but we will focus our
attention on the following gem. Consider what happens when odd numbers
in PAT are darkened and even numbers are left blank. Extending PAT to
infinitely many rows and reducing the scale by one-half each time the number
of rows is doubled produces the previously encountered fractal, Sierpinski’s
Gasket (Figure 2.42(c)) [127]!
Property 62 (The Chaos Game). Equilateral triangular patterns can emerge
from chaotic processes.
Choose any point lying within an equilateral triangle, the vertices of which
are labeled 1 thru 3, and mark it with a small dot. Roll a cubic die to produce
a number n and set i = (n mod 3) + 1. Generate a new point located at
the midpoint of the segment connecting the previous dot with vertex i and
mark it with a small dot. Iterate this process k times always connecting the
most recent dot with the latest randomly generated vertex. The results for
(a) k = 100, (b) k = 500, (c) k = 1, 000 and (d) k = 10, 000 are plotted in
Figure 2.43. Voila, Sierpinski’s Gasket emerges from this chaotic process [239,
Chapter 6]!