Two quantities are said to be in the gold-
en ratio if the “whole is to the larger as
the larger is to the smaller.” Euclid
expressed it as, “A straight line is said to
have been cut in extreme and mean ratio
when, as the whole line is to the greater
segment, so is the greater to the less.”
This is seen in the accompanying illustra-
tion, in which for two segments “a” and
“b,” the entire line is to the “a” segment
as “a” is to the “b” segment.
The symbol for the golden ratio is
(the Greek letter “phi,” or a circle with a
vertical slash through it); it is equal to about 1.6103398 and is considered an irrational
number. The calculation to reach the golden ratio is as follows:
/1 (1 )/
^1 12 1
This equals the quadratic equation:
^2 1 0
which results in:
1/2 5 /2 1.6103398What is the historical significanceof the golden ratio?
It is thought that over the centuries many architects and painters used the golden ratio
in their works. Some historians believe that the Great Pyramid of Cheops contains the
golden ratio. The ancient Greeks knew about the golden ratio from their works in geom-
etry, but they never truly believed it was as important as numbers such as pi (π). Many
works of art in the Renaissance are thought to have used the golden ratio within paint-
ings and sculptures, although it may have been subconsciously incorporated into their
compositions. In 1509 Luca Pacioli published the work Divina Proportione,which
explored the mathematics of the golden ratio, along with its use in architectural design.
Of course, humans aren’t the only ones who “practice” the golden ratio. It is also
seen in nature as the result of the dynamics of some systems. For example, the spac-
ing of sunflower seeds—and even the shape of the chambered nautilus shell—are
often claimed to be related to the golden ratio.In what way do some historians link mathematicsto the pyramids?
The pyramids in Egypt were built as royal tombs for the pharaohs—first along the
334 edges of cliffs as low rectangular structures called mastabas, then as tall, four-sided
The idea of the Golden Ratio is illustrated here by
the relationships between a, b, and a b.