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gica Milinković, Milenko Ćurčić, and Slađana Mitrović
Figure 1 The Pentagon ab
a+b
Figure 2 Golden Ratio in Geometry
to the second diagonal. The Pythagoreans used the pentagon (figure 1) as
a symbol of fraternity. They called it Health (Hygieia – the Greek goddess of
health) and saw mathematical perfection in it. They related Health, i.e. the
harmony of the body, to the harmony of the golden ratio (Herz-Fisher 1998;
Milojković 2009).
The first clear definition of what was later called the golden ratio was given
about 300 years BC by the founder of geometry as a formalized deductive
system, Euclid of Alexandria in his Elements. Euclid defined the proportion
derived from the simple division of a line into what he called its extreme and
mean ratio. He used to say that a line was divided in its extreme and mean
ratio when, as the whole line is to the greater segment, so is the greater to the
less. Knowing its extreme and mean ratio division, as Euclid called the golden
ratio, in his fourth book of Elements Euclid first constructed a golden trian-
gle as an isosceles triangle having each of the angles at the base double the
remaining one. Afterward, based on the previous construction, in the same
book of Elements, he inscribed a regular pentagon in a given circle (Herz-
Fisher 1998; Stakhov 2006; Lučić 2009; Milojković 2009).
The German mathematician, astronomer and astrologer, Johannes Kepler
(1571–1630) said: ‘Geometry has two great treasures; one is the Theorem of
Pythagoras; the other is a golden ratio. The first we may compare to a mea-
sure of gold; the second we may name a precious jewel.’ Kepler also calls the
golden ratio the divine proportion (Stakhov and Olsen 2009).
Depending on the context, the notion of a golden ratio may indicate a
point dividing a line such that as the whole line is to the greater segment,
280
Figure 1 The Pentagon ab
a+b
Figure 2 Golden Ratio in Geometry
to the second diagonal. The Pythagoreans used the pentagon (figure 1) as
a symbol of fraternity. They called it Health (Hygieia – the Greek goddess of
health) and saw mathematical perfection in it. They related Health, i.e. the
harmony of the body, to the harmony of the golden ratio (Herz-Fisher 1998;
Milojković 2009).
The first clear definition of what was later called the golden ratio was given
about 300 years BC by the founder of geometry as a formalized deductive
system, Euclid of Alexandria in his Elements. Euclid defined the proportion
derived from the simple division of a line into what he called its extreme and
mean ratio. He used to say that a line was divided in its extreme and mean
ratio when, as the whole line is to the greater segment, so is the greater to the
less. Knowing its extreme and mean ratio division, as Euclid called the golden
ratio, in his fourth book of Elements Euclid first constructed a golden trian-
gle as an isosceles triangle having each of the angles at the base double the
remaining one. Afterward, based on the previous construction, in the same
book of Elements, he inscribed a regular pentagon in a given circle (Herz-
Fisher 1998; Stakhov 2006; Lučić 2009; Milojković 2009).
The German mathematician, astronomer and astrologer, Johannes Kepler
(1571–1630) said: ‘Geometry has two great treasures; one is the Theorem of
Pythagoras; the other is a golden ratio. The first we may compare to a mea-
sure of gold; the second we may name a precious jewel.’ Kepler also calls the
golden ratio the divine proportion (Stakhov and Olsen 2009).
Depending on the context, the notion of a golden ratio may indicate a
point dividing a line such that as the whole line is to the greater segment,
280
