Transcript The Golden Ratio and Pythagoras

Old Guys and their formulas
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Let’s start with two numbers: 1 and 1.
Add these two values to get the next number in
the sequence (pattern).
Add the last two values to get the next number.
Continue until you want to stop.
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Let’s list the first several Fibonacci numbers.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …
Find the successive quotients with big divided
by little.
1/1=1, 2/1=2, 3/2=1.5, 5/3=1.666.., 8/5=1.6
13/8=1.625, 21/13=1.615, 34/21=1.619
55/34=1.6176, 89/55=1.618, 144/89=1.618
Do these quotients seem to be approaching a
single, fixed number? If so, what is it?
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1+ 5
,
2
𝜑=
this is the exact value of the Golden
Ratio, phi.
𝜑 ≈ 1.618, this is the approximate value of the
Golden Ratio.
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Did the ancient people know of the Golden
Ratio?
Did they really plan it out that way with grids,
blue prints, and measuring tapes?
We may never know.
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Plato
Archimedes
Euclid
Pythagoras
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Believed that the heavens must exhibit perfect
geometric form, and therefore argued that the
Sun, Moon, planets and stars must move in
perfect circles – proved incorrect in early 1600s.
Perfect Solids – aka Platonic Solids. These will
be discussed in more detail later in the chapter.
Invented a moralistic tale about a fictitious land
called Atlantis.
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Ancient people recognized that the
circumference of any circle is proportional to its
radius.
Archimedes’ method of discovery included
inscribing figures in a circle and circumscribing
that same figure around the circle. As the
figure took on a shape closer to a circle, he was
able to make a better estimate of the
circumference.
Using this method, Archimedes estimated π to
be 3.14.
Number of sides of
polygon
Perimeter of inscribed
polygon
Perimeter of
circumscribed polygon
4
2.8284
4.0000
8
3.0615
3.3137
16
3.1214
3.1826
32
3.1365
3.1517
64
3.1403
3.1441
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Wrote a book called “The Elements” which is
the basis of all standard Geometry texts used
today.
Books I – IV are all Geometry wherein he lays
down the fundamental definitions (can’t be
proved, must be accepted), postulates, and
notions. Discusses the basics of algebraic
geometry, and then goes on to discuss platonic
solids and prove the Pythagorean theorem.
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Was into math, music, and mysticism.
His society is responsible for many of the
sounds we hear in music today.
They believed fully in the power of the
pentagram.
Mathematically most known for the
Pythagorean theorem (even though it was
known in the Middle East and China several
centuries earlier).
For any right triangle with side lengths A and
B and hypotenuse C, 𝐴2 + 𝐵2 = 𝐶 2 .
 Notice that it must be a right triangle. That is,
there must be a 90 degree angle.
 The hypotenuse is always
across from the right angle.
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On a sunny warm day, Jack decides to fly a kite
just to relax. His kite takes off and soars. He
lets all 150 feet of the string out and attracts a
crowd of onlookers (this is how he met Diane).
There is a slight breeze, and a spectator 90 feet
away from Jack notices that the kite is directly
above her, Diane. Unlike real-life, the math
problem string of the kite is a straight line from
the Jack to the kite. How high is the kite from
the ground?
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On a sunny warm day, Jack decides to fly a kite
just to relax. His kite takes off and soars. He
lets all 150 feet of the string out. There is a
slight breeze, and a spectator 90 feet away from
Jack notices that the kite is directly above her.
How high is the kite from the ground?
Solution: The hypotenuse is 150 and one side
is 90 so we have 902 + ℎ2 = 1502 . Isolating
the ℎ2 we get ℎ2 = 14400 which leaves h =
120 feet.