THE GOLDEN RATIO AND PYTHAGORAS

Old Guys and their formulas

Looking for Patterns

So what’s the pattern?

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.

Converging Quotient

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?

It’s The Golden Ratio!

, this is the exact value of the Golden Ratio, phi.

, this is the approximate value of the Golden Ratio.

Golden Ratio in Architecture

Accident or Knowledge?

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.

Old Greek Guys

Plato Archimedes Euclid Pythagoras

Plato (427 – 347 B.C.E)

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.

Archimedes (287 – 212 B.C.E)

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.

Archimedes Strategy:

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

Euclid

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.

Pythagoras

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).

Pythagorean Theorem

For any right triangle with side lengths A and B and hypotenuse C,

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.

Pythagorean example

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?

Pythagorean example

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 . Isolating the we get which leaves h = 120 feet.