Loaves & Fishes 27

Many parts of the human face
fit the proportions of the Golden
Ratio, on average. People seem
to find faces more attractive the
more closely they fit the Golden
Ratio. The Ratio also appears in
the proportions of the rest of
the body.
The nautilus shell is often offered
as an example of a Golden
Ratio spiral. While it doesn’t fit
the conventional Golden Spiral,
it does have the Golden Ratio in
the size of the outer spiral compared
to the inner spirals.
Because the dimensions of
the Golden Ratio are aesthetically
pleasing, artists often use
it in their works. Leonardo da
Vinci used the Golden Ratio in
multiple places in his famous
painting The Last Supper.
Mandelbrot Set
Compared to the Fibonacci
sequence, the concept of the
Mandelbrot Set is a bit more
difficult to understand and
explain. It was discovered in
1979 by Benoit Mandelbrot,
one of the first people to use
computer graphics to create
and display fractal geometric
images.
What is a set? A set is a group
of numbers that have a certain
property in common. The
Mandelbrot Set includes all the
numbers that have the characteristic
of Z staying small in this
formula: Z = Z 2 + C. We start
with Z=0, and the calculation
is iterated, meaning we use the
result of the calculation for the
value of Z, and then repeat the
calculation, over and over. In
this formula, the n shows that
Z is a series, and the calculation
is iterated: Zn+1 = Zn 2 + C.
Let’s try C=1. 0 squared + 1 = 1.
1 squared + 1 = 2. 2 squared plus
1 = 5. 5 squared plus 1 = 26. Because
Z continues to get larger,
1 is not in the Mandelbrot Set.
Let’s try C = 0. 0 squared + 0 =
0. 0 is in the set, because Z stays
at 0 when C = 0.
Let’s try C = -1. 0 squared + -1
= -1. -1 squared + -1 = 0. -1 is in
the set, because Z continually
cycles between -1 and 0, never
getting farther away from zero.
But the Mandelbrot set
doesn’t use only ordinary integers.
It uses complex numbers. A
complex number is a real number
multiplied by an imaginary
number. Even though mathematicians
call these numbers
imaginary, the numbers really
do exist. An imaginary number
is the square root of a negative
number. The letter i is used to
denote the square root of -1. You
may remember from math class
that a negative number squared
12 | Loaves & Fishes • Issue 27